Proof. The under-\(A\) homotopy supplies a map of cofiber sequences \[\begin{tikzcd}[column sep=large] A \arrow[r,"a"] \arrow[d,equal] & B \arrow[r] \arrow[d,"\tau"] & \mathop{\mathrm{cofib}}(a) \arrow[r,"\partial_a"] \arrow[d,"\bar\tau"] & \Sigma A \arrow[d,equal]\\ A \arrow[r] & LA \arrow[r] & \Sigma N \arrow[r,"-\Sigma i"] & \Sigma A. \end{tikzcd}\] The right square is (5); its sign is the rotation sign fixed above. If \(N\) is \(M\)-local, so is \(\mathop{\mathrm{cofib}}(A\to LA)\simeq\Sigma N\). Its \(M\)-localization map is then an equivalence, and exactness identifies the localized cofiber with \(\mathop{\mathrm{cofib}}(MA\to MLA)\). ◻

Proof. Constructing the middle map. The cofiber \(Y\) is \(E(2)\)-local and \(K(2)\)-acyclic by (i), hence \(E(1)\)-local. Exactness identifies it naturally with \(\mathop{\mathrm{cofib}}(L_1w)\). Its height-one fracture square is \[Y\simeq L_{K(1)}Y\mathbin{\times}_{L_0L_{K(1)}Y}L_0Y.\] The two proposed maps from \(U_{\mathrm{mid}}\) to its corners are \[U_{\mathrm{mid}}\longrightarrow L_{K(1)}U_{\mathrm{mid}}\xrightarrow{\kappa}L_{K(1)}Y, \qquad U_{\mathrm{mid}}\longrightarrow L_0U_{\mathrm{mid}}\xrightarrow rL_0Y.\] By (2), the specified rational homotopy in (6) identifies their composites to the overlap as \(S\)-module maps. The pullback therefore gives \[ h:U_{\mathrm{mid}}\longrightarrow Y,\qquad L_{K(1)}h=\kappa,\quad L_0h=r. \tag{8}\] Here each equality includes the homotopy obtained from the corresponding pullback projection: localizing that homotopy and using idempotence gives the displayed identification. In particular, \(h\) is a \(K(1)\)-equivalence.

The first four stages. Let \(\partial_w:Y\to\Sigma W\) be the boundary of \(w\), and put \[d=\partial_wh:U_{\mathrm{mid}}\longrightarrow\Sigma W,\qquad d_i=d|_{U_i}\quad(i=3,4).\] Define \[ F_0=0,\quad F_1=W_1,\quad F_2=W,\quad F_3=X\mathbin{\times}_YU_3,\quad F_4=X\mathbin{\times}_YU_{\mathrm{mid}}. \tag{9}\] The first nonzero transition is the wedge inclusion \(W_1\to W\). The identification \(X\times_Y0\simeq W\) makes the next transitions the pullbacks of \(0\to U_3\to U_3\vee U_4\). In a stable category a pullback square is also a pushout square. Their cofibers are consequently \(U_3\) and \(U_4\), and \[F_3\simeq\mathop{\mathrm{fib}}(d_3:U_3\to\Sigma W),\qquad F_4\simeq\mathop{\mathrm{fib}}(d:U_{\mathrm{mid}}\to\Sigma W).\] Naturality of the cofiber sequence for each pullback gives \[ \partial_1=0,\qquad \partial_2=0,\qquad \partial_3=d_3,\qquad \partial_4=\Sigma(F_2\to F_3)d_4. \tag{10}\] For the last formula, the quotient of the pullback square for \(U_3\to U_3\vee U_4\) identifies its quotient with \(U_4\). There is a map of cofiber sequences from \(W\to F_4\to U_{\mathrm{mid}}\) to \(F_3\to F_4\to U_4\), whose components are \(W\to F_3\), the identity of \(F_4\), and the projection \(\operatorname{pr}_4:U_{\mathrm{mid}}\to U_4\). Its boundary square gives \(\partial_4\operatorname{pr}_4=\Sigma(W\to F_3)d\). Restricting to the summand \(U_4\) gives the displayed formula.

We next express \(d\) through the height-two fracture map. Choose coherent inverse data for \(L_{K(2)}w\), and form \[\tau_2:L_1X\longrightarrow L_1L_{K(2)}X \xrightarrow{(L_1L_{K(2)}w)^{-1}}L_1L_{K(2)}W.\] It is a map under \(L_1W\), so it induces \[\bar\tau_2:Y\simeq\mathop{\mathrm{cofib}}(L_1w) \longrightarrow\mathop{\mathrm{cofib}}(L_1W\to L_1L_{K(2)}W).\] Let \(j_W:\mathop{\mathrm{fib}}(W\to L_{K(2)}W)\to W\) be the fiber inclusion. Then \[ \begin{split} d:\ U_{\mathrm{mid}}\xrightarrow hY\xrightarrow{\bar\tau_2} &\mathop{\mathrm{cofib}}(L_1W\longrightarrow L_1L_{K(2)}W)\\ &\simeq\Sigma\mathop{\mathrm{fib}}(W\longrightarrow L_{K(2)}W) \xrightarrow{-\Sigma j_W}\Sigma W. \end{split} \tag{11}\] Indeed, (3) identifies the fiber of \(W\to L_{K(2)}W\) with the fiber of \(L_1W\to L_1L_{K(2)}W\); it is \(E(1)\)-local. 2, with \(L=L_{K(2)}\) and \(M=L_1\), then says that the composite from \(Y\) in (11) is exactly \(\partial_w\). This proves the formula and fixes its minus sign.

The rational quotient. Let \(a:F_4\to X\) be the pullback projection and set \(Z=\mathop{\mathrm{cofib}}(a)\). Taking the cofiber of the pullback square defining \(F_4\) gives a natural equivalence \[ Z\simeq\mathop{\mathrm{cofib}}(h:U_{\mathrm{mid}}\longrightarrow Y). \tag{12}\] This cofiber is \(E(1)\)-local and \(K(1)\)-acyclic by (8), hence rational. Thus \(a\) is an equivalence after localization at \(K(1)\) or \(K(2)\), and therefore after \(L_J\). Rationalizing (12) identifies \(Z\simeq\mathop{\mathrm{cofib}}(r)\).

Write \(z:X\to Z\) for the quotient. Since \(Z\) is rational, (2) gives its factorization \(\widetilde z:L_0X\to Z\). Under the preceding cofiber identification, the map in (iii) is \[ t=\widetilde z\,b:V\xrightarrow{\simeq}Z. \tag{13}\] This records the factorization used to compare the last attachments with fracture.

The last four stages. Let \(V_{\leq j}\) be the first \(j\) summands of \(V\), for \(0\leq j\leq4\), and use the restriction of \(t\) to define \[ F_{4+j}=X\mathbin{\times}_ZV_{\leq j}. \tag{14}\] At \(j=0\), the cofiber sequence of \(a\) identifies this definition with the previous \(F_4\), over \(X\). The cofiber of \(V_{\leq j-1}\to V_{\leq j}\) is \(V_{4+j}\). Exact pullback gives the same cofiber for \(F_{4+j-1}\to F_{4+j}\).

Let \(\partial_a:Z\to\Sigma F_4\) be the boundary of \(a\), put \(g=\partial_at\), and write \(g_i=g|_{V_i}\). The quotient-of-a-pullback argument used above gives every remaining connecting map: \[ \partial_i=\Sigma(F_4\to F_{i-1})g_i: V_i\longrightarrow\Sigma F_{i-1},\qquad 5\leq i\leq8. \tag{15}\]

The map \(g\) is described by the positive-height fracture square. Choose coherent inverse data for \(L_Ja\), and let \(j_a:\mathop{\mathrm{fib}}(F_4\to L_JF_4)\to F_4\) be the fiber inclusion. The formula is \[ \begin{aligned}[b] g:\ V\xrightarrow bL_0X\longrightarrow L_0L_JX &\xrightarrow{(L_0L_Ja)^{-1}}L_0L_JF_4\\ &\longrightarrow\mathop{\mathrm{cofib}}(L_0F_4\longrightarrow L_0L_JF_4)\\ &\simeq\Sigma\mathop{\mathrm{fib}}(F_4\longrightarrow L_JF_4) \xrightarrow{-\Sigma j_a}\Sigma F_4. \end{aligned} \tag{16}\] We verify in particular why this roof begins with \(b\), rather than only with the quotient equivalence \(t\). Put \[C_a=\mathop{\mathrm{cofib}}(F_4\to L_JF_4),\qquad \tau_J=(L_Ja)^{-1}\circ(X\to L_JX),\] and let \(q_a:L_JF_4\to C_a\) be the quotient. The chosen inverse homotopy makes \(\tau_Ja\) the localization unit of \(F_4\). 2 gives \(\bar\tau_J:Z\to C_a\), with a specified homotopy \[\bar\tau_Jz\simeq q_a\tau_J.\] Both \(Z\) and \(C_a\) are rational: for \(C_a\) this follows from (4). If \(\rho:C_a\xrightarrow{\simeq}L_0C_a\) is its localization map, the rational adjunction transports the last homotopy to \[\rho\bar\tau_J\widetilde z\simeq (L_0q_a)(L_0\tau_J).\] Using \(t=\widetilde z b\), this becomes \[\rho\bar\tau_Jt\simeq (L_0q_a)(L_0L_Ja)^{-1}\circ (L_0X\to L_0L_JX)\circ b.\] Exactness identifies \(L_0C_a\) with the cofiber in (16). The boundary formula \(\partial_a=(-\Sigma j_a)\bar\tau_J\) now identifies that entire roof with \(\partial_at=g\), including the second minus sign.

Finally, (13) makes the projection \[e:F_8=X\times_ZV\longrightarrow X\] an equivalence. All stages use finite limits or colimits of \(E(2)\)-local \(S\)-modules, so they remain in that category. Their maps to \(X\) extend \(w\); hence \(F_2\to F_8\xrightarrow eX\) is \(w\). ◻

Proof. The proof of 3 established (10) and (15) under these ordered cofiber identifications. It identified their components by (11) and (16) using precisely the stated inverse data. The collected formulas therefore apply to the same \(h,F_i,a,Z,t,e\). ◻

Proof. An element of order \(p\) in \(D_n^\times\) would generate a copy of \(\mathbb Q_p(\zeta_p)\). Its degree \(p-1\) must divide \(n^2\), since \(D_n\) would be a vector space over that subfield. For \(n=1,2,3\) and \(p\geq5\), this is impossible except for the apparent numerical possibility \(n=2,p=5\); in that case a commutative subfield of a degree-two central division algebra has degree at most two, whereas \(p-1=4\). Equivalently, the usual subfield-degree theorem rules out all these cases at once. An element of order \(p\) in \(\mathop{\mathrm{GL}}_2(\mathbb Q_p)\) would give a faithful two-dimensional representation of the nontrivial irreducible factor of \(X^p-1\), whose degree is \(p-1>2\), and is likewise impossible.

The Lie dimensions are the stated ones. Reduced norm has surjective differential, so its kernel has dimension eight. Conjugation has determinant one on a central simple algebra; its restriction to the trace-zero summand also has determinant one. The same calculation applies to the matrix groups. Thus their adjoint orientation characters are trivial.

For a compact \(p\)-adic analytic group without \(p\)-torsion, \(p\)-cohomological dimension equals its Lie dimension (Serre 1965, sec. 1, Corollary (1)). Writing \(\Lambda=\mathbb Z_p[[G]]\), Venjakob (Venjakob 2002, 275–76) records both Noetherianity of \(\Lambda\) and \(\operatorname{pd}_{\Lambda}\mathbb Z_p=\operatorname{cd}_p(G)\), with agreement between finitely generated compact and abstract modules. Successive finite free presentations therefore have finitely generated kernels and a projective final syzygy, giving the asserted finite projective resolution. The associated duality is used with the continuous coefficient models specified below. ◻

Proof. The residue-field trace is surjective. Lifting a trace-one element to \(W(k')\) and dividing by its unit trace produces \(a\in W(k')\) with \(\sum_{\gamma\in\Gamma}\gamma(a)=1\). For a semilinear module \(M\), the finite sum \[P(m)=\sum_{\gamma\in\Gamma}\gamma(a)\,\gamma(m)\] is an additive projection onto \(M^\Gamma\). If an invariant element in a quotient has a lift, applying \(P\) to that lift gives an invariant lift. This proves exactness, and the same formula works modulo \(p^\ell\). For an equivariant homogeneous cochain \(f\) of positive degree, the operator \[(hf)(g_0,\ldots,g_{q-1}) =\sum_{\gamma\in\Gamma}\gamma(a) f(\gamma,g_0,\ldots,g_{q-1})\] is still equivariant, by semilinearity and reindexing the sum. The alternating homogeneous differential satisfies \(dh+hd=\mathrm{id}\) in positive degrees, since the sum of the coefficients is one. This proves the higher vanishing. All sums and scalar operations are finite and continuous. No division by \(|\Gamma|\) has been made. ◻

Proof. One precise definition of the closure is the image of the finite \(A_m\)-algebra of \(r\)-tuples in \(\mathcal G[p^l]\) in the reduced coordinate algebra of the indicated generic locus. It is therefore finite and reduced, the map \(A_m\to B_l\) is injective, and every irreducible component dominates \(\mathop{\mathrm{Spec}}A_m\). At the closed point all torsion coordinates are nilpotent. Consequently \(B_l\) is local with residue field \(k\), and its maximal ideal is generated by the height parameters and the \(t_{i,l}\).

Write \(d=p^m\) and \(q=p^3\). Put \(a_i=[p^{l-1}]_{\mathcal G}(t_{i,l})\), and for \(v=(v_1,\ldots,v_r)\in\mathbb F_p^r\) put \(a_v=\sum_{\mathcal G}[v_i]a_i\). If \(W_p(T)\) is the distinguished polynomial for \([p]_{\mathcal G}(T)\), then \[ W_p(T)=\prod_{v\in\mathbb F_p^r}(T-a_v)^d =\prod_{v\in\mathbb F_p^r}(T^d-a_v^d). \tag{21}\] Indeed, over each geometric generic field the connected part has length \(d\) and the \(a_v\) enumerate the distinct étale points. Both sides are monic of degree \(q\). Equality of their coefficients there implies equality in the reduced generic closure.

On \(A_m\), the series \([p](T)\) factors through \(T^d\). It follows that every \(a_v^d\) is a series over \(A_m\) in \(w_i=t_{i,l}^{d^l}\), with zero constant term as a series in the \(w_i\). The algebra \(C_l=A_m[w_1,\ldots,w_r]\) is finite over the complete ring \(A_m\), hence complete. Equation (21) is an identity over \(C_l\). Modulo \((w_1,\ldots,w_r)\) it becomes \(W_p(T)=T^q\). The preparation unit for \([p](T)\) lies over \(A_m\); thus every coefficient below degree \(q\) in \([p](T)\) vanishes in this quotient. The successive height-coordinate congruences \[[p](T)\equiv u_iT^{p^i} \pmod{(u_m,\ldots,u_{i-1},T^{p^i+1})},\qquad m\leq i\leq2,\] now imply, successively, that all \(u_i\) vanish. Hence the maximal ideal of \(C_l\) is generated by the \(r\) elements \(w_i\). The ring has dimension \(r\), because it is integral over \(A_m\). It is a regular local ring. The surjection \(k[[W_1,\ldots,W_r]]\to C_l\), \(W_i\mapsto w_i\), is an isomorphism: a nonzero kernel would lower dimension in the regular domain on the left. Applying the identical maximal-ideal argument with \(t_{i,l}\) in place of \(w_i\) proves \(B_l=k[[t_{1,l},\ldots,t_{r,l}]]\). This also proves the asserted identification of the inclusion \(C_l\subset B_l\), rather than just abstract regularity of both rings.

Over \(u_m\ne0\), relative Frobenius of order \(ml\) identifies the quotient of \(\mathcal G[p^l]\) by its connected part with its image under \(T\mapsto T^{d^l}\). Here is a check that retains the prepared finite algebra. Write \([p^l](T)=g_l(T^{d^l})\) and prepare \(g_l(S)=U_l(S)V_l(S)\), with \(U_l\) a unit and \(V_l\) monic of degree \(p^{rl}\). Its linear coefficient \(a_l\) is a unit multiple of \(u_m^{1+d+\cdots+d^{l-1}}\). Invariant differentials for this Frobenius-divided homomorphism give \[g_l'(S)=a_l\frac{h_{\mathrm{source}}(S)} {h_{\mathrm{target}}(g_l(S))},\] where both \(h\) are unit series. In the finite algebra defined by \(V_l\), this says that \(U_lV_l'\) becomes a unit after inverting \(u_m\). Thus that algebra is étale of rank \(p^{rl}\). The kernel of relative Frobenius has rank \(d^l\) and is the connected part. Taking bases gives exactly \(S_l=C_l[u_m^{-1}]\).

Over \(S_l\), the scheme of lifts of its universal basis is finite locally free of rank \(d^{lr}\). Its coordinate algebra surjects onto \(B_l[u_m^{-1}]\): the latter is generated by the same lifted coordinates. But the identified power-series rings show that \(B_l[u_m^{-1}]\) is free over \(S_l\) of this same rank, with the monomials \(\prod_i t_{i,l}^{e_i}\), \(0\leq e_i<d^l\), as a basis. A surjection between finite locally free modules of the same rank is an isomorphism. This proves the assertion about the full lift scheme; in particular the earlier generic reduction has removed no infinitesimal part on the exact-height locus.

Every étale basis at level \(j\) lifts to a basis at level \(l\) after a geometric field extension. The transition map on generic coordinate algebras is therefore injective. Its restriction \(B_j\to B_l\) is injective by generic density and finite since \(B_l\) is finite over \(A_m\). Finally \(t_{i,j}=[p^{l-j}]_{\mathcal G}(t_{i,l})\) factors through \(t_{i,l}^{d^{l-j}}\). Substituting the expressions for the base parameters in \(C_l\) gives the last assertion. ◻

Proof. Choose a power-multiplicative spectral norm and \(\lambda<1\) bounding the specialized parameters. All coefficients of \(\mathcal G_a-\Gamma_3\) and of \(P_a(T)-T^q\), \(q=p^3\), have norm at most \(\lambda\). Integral formal series are \(1\)-Lipschitz on the open unit ball. Since \(P_a\) factors through \(T^d\), \(d=p^m\), we have \[|P_a^{\circ n}(v)-P_a^{\circ n}(w)|\leq |v-w|^{d^n}.\] Consecutive approximations in the first formula differ by at most \(\lambda^{d^n}\). Their limits exist, are topologically nilpotent, and satisfy \[|\Phi_a(z)_j-z^{1/q^j}|\leq\lambda, \qquad P_a(\Phi_a(z)_{j+1})=\Phi_a(z)_j.\] For a compatible sequence \((t_j)\), consecutive terms in the second formula differ by at most \(\lambda^{q^j}\). Its limit \(z\) satisfies \(|z-t_j^{q^j}|\leq\lambda^{q^j}\). Substitution in the first formula and the displayed contraction show in both directions that the formulas are inverse. This reasoning never requires a common strict upper bound on all the \(|t_j|\).

The coefficients of \(\Gamma_3\) are fixed by the \(q\)th-power map. The difference between the two addition laws is at most \(\lambda\); after \(n\) applications of \(P_a\) it is at most \(\lambda^{d^n}\). Taking the limit proves additivity. If \(g\) changes the deformation framing, its evaluated coordinate change differs from its Honda reduction by coefficients of norm at most \(\lambda\), after increasing \(\lambda\) if necessary. The same contraction proves \(\Phi_{ga}(gz)=g\Phi_a(z)\). Uniform convergence on smaller balls proves continuity and compatibility with maps of perfectoid algebras.

Over an algebraically closed field the points with \(t_0=0\) are exactly the compatible \(p^j\)-torsion points. A finite connected group over a perfect field has just its identity as a geometric point, so these points form the Tate module of the étale quotient. The zeroth projection is surjective: preparation applied to \([p](T)-w\) gives a monic distinguished polynomial, each of whose roots is small when \(w\) is small. Successively choosing roots gives a compatible sequence with prescribed \(t_0\). ◻

Proof. First retain the reduced finite-level closures, including their higher-height boundary. Perfection identifies a point of the étale quotient with its unique connected generator lift, so their perfected inverse tower is the tower from 8.

We check the complete algebras after extension to an algebraically closed perfectoid field \(C^\flat\); these field extensions form v-covers, and all constructions are defined over \(k\). Put \(q=p^3\), \(d=p^m\) and \(y_{i,l}=t_{i,l}^{q^l}\). For \(0<\rho<1\) in the value group, let \(X_l(\rho)\) be the closed polydisc \(\max_i|y_{i,l}|\leq\rho\). If \(\lambda\) is the maximum norm of the height parameters at a point of this polydisc, then 8 at level one gives \[\lambda\leq\|t_{\cdot,1}\|^d,\qquad \|t_{\cdot,1}\|\leq\max(\rho^{1/q},\lambda).\] The second inequality follows by finite telescoping of the normalized division coordinates. If \(\lambda>\rho^{1/q}\), the two inequalities would give \(0<\lambda\leq\lambda^d<\lambda\). Therefore \[ \lambda\leq\rho^{d/q}. \tag{23}\] For consecutive levels it follows that \[\|y_{\cdot,l+1}-y_{\cdot,l}\| \leq\lambda^{q^l} \leq\rho^{d q^{l-1}}<\rho\qquad(l\geq1).\] The finite transition maps take \(X_{l+1}(\rho)\) to \(X_l(\rho)\). Conversely every point of \(X_l(\rho)\) has a lift, by finiteness, injectivity and lying-over, and the last strict inequality forces every lift to remain in \(X_{l+1}(\rho)\). They are thus surjective maps of these closed polydiscs. Pullback is isometric for their spectral norms, before and after completed perfection.

In the completed direct limit of the perfected Banach algebras, the \(y_{i,l}\) converge to elements \(z_i\). The homomorphism from the perfected Gauss algebra of a polydisc of radius \(\rho\) sending its coordinates to \(z_i\) is isometric. Indeed, for any polynomial in finitely many fractional powers, evaluation on \(y_{\cdot,l}\) has exactly its Gauss norm: \(t_{\cdot,l}\) are free polydisc coordinates and \(q^l\) is a power of \(p\). These evaluations converge to evaluation on \(z\). A nonzero limiting polynomial therefore has precisely that norm. Completing preserves the isometry.

Its image is all the completed direct limit. For \(l\geq j\), write \[t_{i,j}=F_{i,j,l} (t_{1,l}^{d^{l-j}},\ldots,t_{r,l}^{d^{l-j}}), \qquad F_{i,j,l}\in k[[T_1,\ldots,T_r]].\] The formulas in 9 show that replacing \(t_{i,l}\) by \(z_i^{1/q^l}\) changes this expression by at most \[\lambda^{d^{l-j}} \leq \rho^{(d/q)d^{l-j}},\] which tends to zero. Its substituted series converges on the \(z\)-polydisc. Thus the closed image contains every \(t_{i,j}\) and, by applying roots to the same estimates, all its \(p\)-power roots. These elements generate the completed direct limit. Varying \(\rho\) proves the comparison for the open balls.

It remains to identify the open locus. For \(a=(a_1,\ldots,a_r)\in\mathbb Z_p^r\), form the compatible points \(t_{a,j}=\sum_{\mathcal G}[a_i]t_{i,j}\). The normalized limit is \(\sum_{\Gamma_3}[a_i]z_i\) by 9. If it vanishes, then \(|t_{a,l}|\leq\lambda\) at every level, and hence \[|t_{a,j}|=|[p^{l-j}](t_{a,l})| \leq\lambda^{d^{l-j}}\longrightarrow0.\] Thus every \(t_{a,j}\) vanishes. Conversely a relation in the Tate module gives the same relation among the \(z_i\). At height exactly \(m\), (21) gives a basis of the étale quotient modulo \(p\), and the compatible points give an injective map \(\mathbb Z_p^r\) into its Tate module. At greater height that Tate module has rank less than \(r\), so there is a relation. Multiplying a rational relation by a power of \(p\) reduces to the integral assertion. Exact height \(m\) is therefore exactly the independent locus. This identifies the corresponding open subsheaves; it is not only a bijection on their field-valued points.

Equivariance is part of 9. All the estimates are spectral estimates on affinoid perfectoid algebras. They apply to continuous functions with a profinite parameter by taking their uniform norm, so the same isomorphisms retain those parameters. ◻

Proof. For the unramified degree-\(s\) extension of \(\mathbb Q_p\), the Lubin–Tate universal cover is the section sheaf of \(\mathcal O(1)\) on the curve with that coefficient field (Fargues and Scholze 2024, Proposition II.2.2). Unramified pushforward gives \(V_s\). Reduction of the universal cover and tilting identify it with the Honda universal cover; one can check reduction directly by the contracting formulas of 9, using \(|[p](v)-[p](w)|\leq\max(|p|\,|v-w|,|v-w|^2)\) before reduction. This proves the first assertion with its addition and endomorphisms. For the local-form assertion, the vector-bundle equivalence and pure-module theorem identify a pure family of slope \(c/d\) in lowest terms, after adjoining \(\mathbb F_{p^d}\), with a \((c,d)\)-\(\mathbb Q_p\) local system (Kedlaya and Liu 2015, Theorems 8.7.8 and 8.5.12). Its semilinear Frobenius supplies the action of the constant cyclic division algebra (Kedlaya and Liu 2015, Definition 8.5.7). These local systems trivialize in the pro-étale topology (Kedlaya and Liu 2015, Lemma 9.1.11); a module basis over that division algebra gives the specified basic local form. Its automorphism sheaf is therefore the locally profinite \(D_s^\times\). The slope-zero case is also (Kedlaya and Liu 2015, Corollary 8.7.10). To retain the ground field \(k\), observe that every geometric Honda endomorphism \(f(T)=\sum a_iT^i\) commutes with \([p](T)=T^{p^s}\). Hence \(a_i^{p^s}=a_i\), so all integral endomorphisms, and then all of \(D_s\), are defined over \(\mathbb F_{p^s}\subset k\). The actual Honda bundle over \(k\) therefore defines a map from \(B\underline{D_s^\times}\) to the rank-\(s\), degree-one basic locus. After extending \(k\) to its algebraic closure, the preceding local-form theorem and its identification of automorphisms make this an equivalence. Equivalences of v-stacks descend along that cover, so the frame torsors already have descent over \(k\).

For the required cohomology, on the curve with unramified coefficient field of degree \(s\), the bundle \(\mathcal O(1)\) is the positive geometric twist of the globally étale bundle \(\mathcal O\). Thus its \(H^1\) vanishes on every affinoid perfectoid test object (Kedlaya and Liu 2015, Corollary 8.8.7 and Theorem 8.7.13). Unramified finite pushforward gives the same assertion for the standard \(V_s\). A family of that basic type first standardizes pro-étale locally, so its \(H^1\) sheaf vanishes. For \(\mathcal O^a\), its \(H^0\) sheaf is \(\underline{\mathbb Q_p}^{\,a}\). An extension of \(\mathcal O\) by \(\mathcal O^a\) is pointwise of slope zero; the local-system equivalence makes it split pro-étale locally. This proves vanishing of its \(H^1\) sheaf, without asserting vanishing over an unrefined affinoid. Higher cohomology on affinoid tests is zero by (Kedlaya and Liu 2015, Theorem 8.7.13).

The needed full faithfulness is relative: for any small v-stack \(S\), the functor \[R\tau_*:\operatorname{Perf}(X_S)\longrightarrow D(S_v,\mathbb Q_p)\] is fully faithful (Anschütz and Le Bras 2025, Corollary 3.11). For the basic positive and slope-zero bundles just considered, the preceding vanishing identifies \(R\tau_*E\) with its section sheaf in degree zero. Taking degree-zero morphisms therefore identifies bundle maps with \(\mathbb Q_p\)-linear maps of section sheaves. This construction is compatible with every base change on \(S\), so it also applies to relative forms of the basic bundles. ◻

Proof. First work on an absolute curve. Let \(M\) be the saturation of the generic span of \(r\) independent sections and put \(b=\mathop{\mathrm{rank}}M\). A wedge of \(b\) generically independent sections gives a nonzero section of \(\det M\), so \(\deg M\geq0\). Stability of \(V_3\) gives \(\deg M\leq b/3<1\). Since degree is integral, it is zero. The wedge has no zeros, because a nonzero effective divisor has positive degree. The selected sections trivialize \(M\). Since \(H^0(X,\mathcal O)=\mathbb Q_p\), independence of the original sections forces \(b=r\); hence their map is a subbundle map. All slopes of its quotient are positive: a quotient of nonpositive slope would give a nonzero map from \(V_3\) to a bundle of slope at most zero. A positive bundle of rank \(m\leq2\) and degree one is \(V_m\), by the classification of bundles and integrality of degrees (Fargues and Scholze 2024, Theorem II.2.14).

An extension of \(V_m\) by \(\mathcal O^r\) has no negative-slope quotient, since neither end has a nonzero map to a negative bundle. A nonnegative bundle of rank three and degree one is either \(V_3\) or has a quotient \(\mathcal O\). Such a quotient restricts nontrivially to \(\mathcal O^r\), because \(\mathop{\mathrm{Hom}}(V_m,\mathcal O)=0\). The resulting nonzero linear functional on \(\mathbb Q_p^r\) kills the extension class. Conversely, if a nonzero functional kills that class, its pushout extension splits and gives a quotient \(\mathcal O\). This proves the equivalence. Fiberwise surjectivity of a morphism of vector bundles is a relative subbundle condition, and 11 supplies local standard forms and the relative extension sheaf, so the argument gives the asserted relative loci. ◻

Proof. Put \(R_m=A_m[u_m^{-1}]\). Apply (Lurie 2010, Theorem 1 and its proof, pp. 1–4) to the formal completion \(\widehat{\mathcal G}_{R_m}\) of \(\mathcal G\) over \(R_m\) and to \(\Gamma_m\), both of exact height \(m\). That theorem defines the isomorphism ring using every coefficient of the full identity \[F_{\Gamma_m}(f(X),f(Y))=f(F_{\mathcal G}(X,Y))\] and expresses it as a filtered union of injective finite étale \(R_m\)-algebras. Pullback to \(\mathcal X_m\) therefore gives the pro-étale tower of genuine connected markings; any finite set of Taylor coefficients is defined at one finite stage. Composition makes it a torsor under the automorphism sheaf of \(\Gamma_m\). The Honda coefficient calculation in 11 makes the latter the constant profinite group \(P_m=\mathop{\mathrm{Aut}}_k(\Gamma_m)\): its finite étale coefficient stages have all geometric coefficients in \(\mathbb F_{p^m}\subset k\).

For the needed bound, pull a genuine marking \(f(T)=\sum_{n\geq1}a_nT^n:\widehat{\mathcal G}_R \xrightarrow{\sim}\Gamma_{m,R}\) to a characteristic-\(p\) affinoid perfectoid test algebra \(R\) on this tower, with its power-multiplicative spectral norm. Put \(q=p^m\). The factorization of the \(p\)-series used in 8 reads \[[p]_{\mathcal G}(T)=P(T^q),\qquad P(S)=u_mS+\sum_{j\geq2}c_jS^j,\] where \(u_m\in R^\times\cap R^{\circ\circ}\) and \(c_j\in R^\circ\); these bounds come from the integral deformation coefficients. The necessary identity \(f([p]_{\mathcal G}(T))=f(T)^q\) gives \[a_1^{q-1}=u_m,\qquad a_n^q-u_m^na_n=\sum_{1\leq i<n}a_i[S^n]P(S)^i \quad(n\geq2).\] Here \([S^n]\) denotes coefficient extraction. First \(\|a_1\|=\|u_m\|^{1/(q-1)}<1\). If the earlier coefficients have norm at most one and \(A=\|a_n\|>1\), the second equation gives \(A^q\leq\max\{\|u_m\|^nA,1\}\leq A\), a contradiction. Thus every \(a_n\) belongs to \(R^\circ\). The series \(f(z)\) converges in \(R^{\circ\circ}\), uniformly on each closed ball of radius less than one. The full formal-law identity then evaluates on small points and proves additivity; evaluation commutes with maps of perfectoid test algebras. Compose it with the zeroth projection in 9. The resulting additive sheaf map \(H_3\to H_m\) is \(\mathbb Q_p\)-linear: it commutes with integral formal-group multiplications and with inverse multiplication by \(p\). It kills the \(\mathbb Q_p\)-span of the Tate basis: it kills all \(p\)-power torsion because \([p]\) is injective on \(H_m\). By [h3:geom:independent-extensions,h3:geom:curve-inputs], it therefore factors through the section sheaf of the quotient bundle. By the relative full faithfulness in 11, this is the section map of a unique relative bundle morphism from the quotient to \(V_m\). On every geometric fiber the section map is nonzero: restrict near the origin, where the formal isomorphism has nonzero linear coefficient. After identifying that fiber of the quotient with \(V_m\), the bundle map is a nonzero element of the division algebra \(D_m\), hence invertible. A bundle morphism invertible on every geometric fiber is an isomorphism. Uniqueness under relative full faithfulness makes these frames compatible with base change and with the group actions.

Connected markings form a \(P_m\)-torsor, and the constructed frames change by that same maximal compact subgroup. The relative frame theorem shows that their only possible discrepancy with the determinant-lattice reduction of the quotient is its integer valuation. This integer is locally constant on the Tate-basis space and invariant under \(P_3\), \(P_m\) and \(\mathop{\mathrm{GL}}_r(\mathbb Z_p)\): all their determinant changes are units. For \(m=2\), that space is \(H_3\setminus\{0\}\) by 10. It is geometrically connected: after extending the base field it is exhausted by intersecting connected closed annuli, and perfection preserves their underlying topology. The integer is therefore constant, and we choose the fixed determinant identification to absorb it. For \(m=1\), let the locally constant valuation discrepancy be \(\nu_{\mathrm{val}}\). Multiplication of the quotient frame by \(p^{-\nu_{\mathrm{val}}}\) corrects it. Since \(D_1^\times=\mathbb Q_p^\times\) is central and \(\nu_{\mathrm{val}}\) is invariant under all three actions, this correction is equivariant and glues without any connectedness assumption. These choices give the common determinant-lattice normalization.

Now mark both the connected group and the integral Tate basis. By 10 this tower is the sheaf of exact sequences \[0\to\mathcal O^r\to V_3\to V_m\to0\] whose induced determinant identification is a unit multiple of the chosen one. The left frame changes are exactly \(\mathop{\mathrm{GL}}_r(\mathbb Z_p)\times P_m\). Forgetting the middle frame instead gives \(\mathcal E_m\), and its middle frames preserving that determinant lattice form exactly a \(P_3\)-torsor. This pair of torsor presentations proves (24). Descent of the displayed sequence proves (25); taking its determinants proves (26). Pushout at the kernel and pullback at the quotient give the stated action on extension classes. This construction uses actual frames throughout, so the maps to the classifying stacks are part of the equivalence.

For \(r=1\), the fixed determinant identification and quotient frame identify the kernel line with \(\mathcal O\). On the integral Tate-basis torsor its determinant multiplier is a unit. Dividing the generator by that multiplier gives the unique generator with multiplier one. Uniqueness glues this normalization and shows that \(g\in P_3\) changes it by \(\mathop{\mathrm{Nrd}}(g)\), with the inverse if all frame conventions are reversed. It is therefore fixed by \(H\). Forgetting this now determined generator identifies the connected marking tower with all surjections \(V_3\to V_2\). Indeed, a surjection \(q\) determines its unique kernel generator \(i_q\) of determinant multiplier one. Coordinate comparison on the entire nonzero Honda ball reconstructs a deformation and a primitive integral Tate generator from \(i_q\), at every valuation. Scaling \(i_q\) by a power of \(p\) can change that deformation; the comparison always returns a basis for the reconstructed one. Choose a connected marking locally and let \(q_h\) be its quotient frame. The constant discrepancy already normalized above ensures that the determinant multiplier of \((i_q,q_h)\) is a unit. Write \(q=dq_h\) with \(d\in D_2^\times\). Since \((i_q,q)\) has multiplier one, \(v_p(\mathop{\mathrm{Nrd}}d)=0\), so \(d\in P_2\). Thus the required change of marking is an integral Honda automorphism and yields a unique connected formal isomorphism. Allowing the middle bundle to vary while keeping its determinant frame gives precisely the nonsplit extensions of \(V_2\) by \(\mathcal O\), namely \(N_2^*\). Changing the middle determinant scales the kernel generator by its full reduced norm. This norm maps \(P_3\) onto \(\mathbb Z_p^\times\), as is seen on the units of the unramified cubic subfield of \(D_3\). This proves (27), with the full norm quotient action. ◻

Proof. Every \(B_i\) and every \(B_i^{1/p^n}\) is a finite product of complete valued fields. First let \(e\in C\) be an idempotent. Choose \(a\in B_i^{1/p^n}\) with \(\|a-e\|<1\). Then \(\|a\|\leq1\) and \(\|a^2-a\|<1\), and \(2a-1\) is a unit whose inverse has norm at most one. Hensel’s lemma in this one complete finite product gives an idempotent \(e_i\) with \(\|e_i-a\|<1\). Two idempotents at distance less than one are equal: each of \(e(1-e_i)\) and \(e_i(1-e)\) is an idempotent of norm less than one and hence vanishes. Thus \(e=e_i\). Purely inseparable extensions create no idempotents, so \(e_i\in B_i\). Any finite list of idempotents descends to a common stage.

The algebra \(E\) descends to a finite étale algebra at some \(B_i\). After a finite idempotent partition it admits a monogenic étale presentation: each component field of \(B_i\) is infinite, so a primitive element may be chosen for its finite product of separable extensions. The preceding paragraph descends the idempotents chosen by a section to \(C\). It remains to descend its finitely many roots.

Suppose \(f(z)=0\) and \(f'(z)\) is invertible in \(C\), with \(f\in B_i[T]\). Approximate both \(z\) and \(f'(z)^{-1}\) simultaneously by \(a,c\in B_j^{1/p^n}\), for one \(j\geq i\) and one \(n\). We may require \(\|1-cf'(a)\|<1\). The geometric series in this complete finite product makes \(f'(a)\) invertible with inverse norm at most \(\|c\|\). Put \(C_0=\max(1,\|c\|)\), and let \(M\geq1\) bound the quadratic Taylor remainder of \(f\) on the unit ball about \(a\). By making the approximations closer, require \(\|f(a)\|C_0^2M<1\) and \(\|f(a)\|C_0<\min(1,(C_0M)^{-1})\). Newton iteration stays in \(B_j^{1/p^n}\), its inverse derivatives stay bounded by \(C_0\), and its residual norms are bounded successively by \(M C_0^2\|f(a)\|^2\) and their iterates. It converges there to a root \(b\) with \(\|b-a\|\leq C_0\|f(a)\|\). Taking the original approximation inside the same simple-root ball makes \(b=z\): the divided difference \((f(b)-f(z))/(b-z)\) is a unit throughout that ball. The element \(b\) is separable over each component of \(B_j\) and lies in its purely inseparable extension, so \(b\in B_j\). The finitely many roots use a common \(j\). Injectivity follows from \(L\hookrightarrow C\), proving the assertion. ◻

Proof. For each \(l\) the normalized generator is a point of \[\operatorname{Isom}_L ((\mathbb Z/p^l)_L,\mathcal G^{\mathrm{et}}[p^l]_L).\] By 14, it descends to \(L\) at one finite separable stage. Compatibility and equivariance descend through the injective coefficient extension, so the descended points give the asserted integral marking. This argument descends a finite étale marking, not a splitting of the connected–étale extension.

The fiber over this marked generator in \(\mathcal G[p^l]\) is a torsor for its connected kernel. Over the étale generator cover, 8 identifies its two fields as \[k((t_l^{p^{2l}}))\subset k((t_l)).\] They have purely inseparable degree \(p^{2l}\). The marked generator supplies a map from the separable field on the left to \(L\). Root extensions commute with separable scalar extension, so the torsor algebra after this base change is exactly (28); no degree is lost. The connected marking identifies its group with \(\Gamma_2[p^l]\). The equivariance already proved fixes both markings under \(H\) and changes them by constant automorphisms under \(P_2\), giving the stated actions on translations.

To prove the torsor assertion, choose a cofinal system of normal open subgroups \(\Pi'\subset\Pi\). The algebra \(B_{\Pi'}=L^{\Pi'}\) is a finite étale \(\Pi/\Pi'\)-torsor over \(B=L^\Pi\). Perfection commutes with finite étale base change, and completion commutes with finite modules. Consequently \[A_{\Pi'}:=B_{\Pi'}\otimes_B B^{\mathrm b} \simeq\widehat{B_{\Pi'}^{\mathrm{perf}}}\] is again a finite étale \(\Pi/\Pi'\)-torsor. Their transition maps are surjective. The affinoid perfectoid inverse limit is \(\operatorname{Spa}(L^{\mathrm c})\): its algebra is the uniform completed union of the \(A_{\Pi'}\), equal to \(\widehat{L^{\mathrm{perf}}}\). Passing the finite torsor identities to inverse limits gives \[\operatorname{Spa}(L^{\mathrm c}) \mathbin{\times}_{\operatorname{Spa}(B^{\mathrm b})} \operatorname{Spa}(L^{\mathrm c}) \simeq\underline\Pi\times\operatorname{Spa}(L^{\mathrm c}).\] Each geometric fiber is a nonempty inverse limit of finite nonempty sets with surjective transition maps. Thus the map is surjective and is the pro-étale \(\Pi\)-torsor in (29). These arguments permit arbitrarily many factors at later finite stages. Finally \(H\) and \(\Pi\) commute, so quotienting this torsor and applying (27) gives (30). No cohomological Galois-descent assertion is used in this deduction. ◻

Proof. The integral canonical-line theorem gives the first isomorphism (Strickland 2000, Theorem 20). In that theorem \(\omega=E_2\), and the determinant is the reduced norm, so its line and action conventions are the ones just fixed. If \(g(U)=q_gU\), comparison of the first nonzero graded \(p\)-series coefficient gives \(g(x)=q_g^{-(p-1)}x\) modulo \(p\). Equivalently the graded coefficient \(v_1=xU^{p-1}\) is invariant there. Hence \((x)/(x^2)\simeq\omega^{-(p-1)}\) on the height-two slice. Taking determinants in its conormal sequence yields \[\Omega^1_{k[[y]]/k,\mathrm{cts}} \simeq\omega^{3+(p-1)}[\det] =\omega^{p+2}[\det].\] Localizing at \(y\) proves the second isomorphism.

There is no change of differential theory at the generic point. Every power series in one variable over a perfect field can be grouped uniquely by its exponent modulo \(p\); thus \(y\) is a \(p\)-basis of \(K\) and \(\Omega^1_{K/k}=K\,dy\). Separable extension gives \(\Omega^1_{L/k}=L\otimes_K\Omega^1_{K/k}\). Transport a nonzero invariant differential on \(\Gamma_2\) along the tautological connected marking. This gives \(U_0\). The \(P_3\) action transports the marking, so it fixes this frame. The image of \(U_0^{p+2}\otimes1_{\det}\) under (31) is \(\eta\). Its remaining character is the reduced norm, trivial on \(H\). Its coefficient and that of \(U_0\) occur at a finite separable stage, since each belongs to the algebraic union \(L\). The action of \(P_2\) is the constant change of Honda marking, and differentiation gives the asserted constant characters. ◻

Proof. On the ordinary locus the relative Frobenius kernels define the connected height-one subgroup; dividing by them gives the finite étale quotients. These compatible finite groups give \(\mathcal C\) and \(\mathcal E\). A height-one one-dimensional connected \(p\)-divisible group becomes \(\mu_{p^\infty}\) on its connected-marking cover; at each finite level the required cover is étale. Thus \(\mathcal C[p^l]\) is of multiplicative type.

The \(p^l\)-torsion sequence is fpqc exact. Locally choose a basis of \(\mathcal E[p^l]\) and lift its two generators to \(\mathcal G[p^l]\). Every lift is killed by \(p^l\), so the two lifts define a homomorphism and a splitting. Two splittings differ by a unique homomorphism to \(\mathcal C[p^l]\). Consequently the splitting scheme is a finite flat torsor for the displayed group, of rank \(p^{2l}\).

The étale basis cover is \(B'=C_l[x^{-1}]\), with \(m=1\) in 8. On this cover, the splitting scheme is the scheme of generator lifts, whose algebra is \[B_l[x^{-1}] =k[[t_{1,l},t_{2,l}]][x^{-1}] =(C_l[x^{-1}])^{1/p^l}=(B')^{1/p^l}.\] Relative Frobenius on an étale algebra is an isomorphism. Therefore \[(B')^{1/p^l}\simeq B'\otimes_{B_0}B_0^{1/p^l}.\] These local identifications descend canonically. More explicitly, in the splitting algebra the \(p^l\)th power of every element lies in the base algebra, as can be checked after the faithfully flat cover \(B'\). Send that element to the unique \(p^l\)th root of its power in \(B_0^{1/p^l}\). The resulting algebra map is an isomorphism after the cover, hence an isomorphism. This description shows both its uniqueness and its independence of all basis and connected-marking choices.

The splitting construction is functorial for changes of deformation framing. Its canonical root identification is therefore \(P_3\)-equivariant. Restricting a splitting at level \(l+1\) to level \(l\) is the quotient by the kernel of the corresponding homomorphism of the translation groups. Under the unique root identifications it is the inclusion \(B_0^{1/p^l}\subset B_0^{1/p^{l+1}}\). Frobenius compatibility follows from the same uniqueness and the natural relative Frobenius maps of the finite groups. Thus all the stated compatibilities hold over \(B_0\) itself, rather than only on an auxiliary étale extension. ◻

Proof. On the Fargues–Fontaine curve with coefficient field \(F\), the divisor of the chosen untilt gives \[0\longrightarrow\mathcal O_F(-1)\longrightarrow\mathcal O_F \longrightarrow i_*\mathcal O_{S^\sharp}\longrightarrow0.\] The relative section sheaf of \(\mathcal O_F\) is \(\underline F\), its relative first cohomology vanishes, and \(\mathcal O_F(-1)\) has no relative sections. These are the slope-zero and negative-slope cases of the relative bundle calculation in 11 and (Fargues and Fontaine 2018; Kedlaya and Liu 2015). Pushforward along the unramified coefficient extension identifies \(\mathcal O_F(-1)\) with \(V_m^\vee\): its isocrystal has rank \(m\), degree \(-1\), and the cyclic Frobenius of slope \(-1/m\). The cohomology sequence is consequently the displayed quotient. All its arrows are relative arrows of sheaves, so the presentation holds over \(S\), not only over geometric fields. The inverse image of the zero section is precisely \(\underline F\). ◻

Proof. Adjoin all \(p\)-power roots of \(z\). The resulting annulus is perfectoid, and its \(\mathbb Z_p(1)\)-torsor computes the cohomology by \[[\,M\xrightarrow{\gamma-1}M\,].\] Here \(M\) is the tilted Laurent algebra, completed for the two Gauss norms. Its elements have unique expansions \(\sum_{u\in\mathbb Z[1/p]}a_u z^u\); the weighted coefficients form a \(c_0\) family. In this statement \(c_0\) concerns all the fractional exponents: for each positive bound only finitely many weighted coefficients exceed that bound. It is stronger than a condition only as \(|u|\) tends to infinity.

The differential on the \(u\)-th monomial is multiplication by \(\epsilon^u-1\). Write \(u=v p^j\), where \(v\in\mathbb Z\) is prime to \(p\). For \(u\ne0\), characteristic \(p\) and \(|v|_p=1\) give \[|\epsilon^u-1|=\theta^{p^j},\qquad p^j\leq |u|_{\mathbb R}.\] For \(u>0\), the outer-radius estimate is \[\frac{(b'/b)^u}{|\epsilon^u-1|} \leq\theta^{u-p^j}\leq1.\] For \(u<0\), the inner-radius estimate is \[\frac{(r'/r)^u}{|\epsilon^u-1|} \leq\theta^{|u|-p^j}\leq1.\] Thus division by \(\epsilon^u-1\) after restriction defines a continuous contracting homotopy on all nonzero monomials. The \(c_0\) condition is preserved by these inequalities. The zero monomial has zero differential, giving the two stated terms; the second is \(\mathop{\mathrm{Hom}}(\mathbb Z_p(1),R)\) and hence \(R(-1)\).

The two-term model follows by applying continuous Hom to \[0\to\mathbb F_p[[\mathbb Z_p]]\xrightarrow{\gamma-1}\mathbb F_p[[\mathbb Z_p]]\to\mathbb F_p\to0.\] Its comparison with the completed bar resolution is continuous. Affinoid perfectoid acyclicity (Scholze 2026, Proposition 8.8) then identifies it with the cohomology of the annulus. The displayed coefficientwise inequalities are unchanged for the supremum norm on \(C_{\mathrm{cts}}(T,R)\), proving the parameter assertion. ◻

Proof. It suffices to show that the Roos difference map \[\prod_n M_n\longrightarrow\prod_n M_n,\qquad (x_n)\longmapsto(x_n-f_nx_{n+1})\] is surjective. Given \((y_n)\), construct a solution of the first \(j\) equations at stage \(j\). To add the next equation, choose \(x_{j+1}\) so that \(f_jx_{j+1}\) approximates \(x_j-y_j\) to within \(2^{-j}\). Set the new \(x_j\) equal to \(y_j+f_jx_{j+1}\) and propagate this correction backwards through the earlier equations. The transition norm bound makes the change in each earlier coordinate at most \(2^{-j}\). Every fixed coordinate therefore converges, and the resulting tuple solves all the equations. This proves the claim.

Density persists for continuous functions on \(T\): approximate on a finite clopen partition and lift approximations to the finitely many values. Completeness and the norm bound persist as well, so the proof applies unchanged. Fractional Laurent polynomials belong to every algebra in the specified exhaustion and are dense in each of them, verifying its hypotheses. ◻

Proof. We first treat a line. An untilted disc is covered by the perfectoid disc obtained by adjoining all roots of its coordinate. A section of \(\mathcal O^\flat\) on the original disc pulls back to an invariant section on this cover. The monomial computation in 19, now with nonnegative exponents, shows that every such section is constant. Consequently the degree-zero cohomology of a relative disc is \(R\).

Restrictions from sufficiently large discs to a fixed smaller disc kill positive cohomology. To see this without making an assertion about the branched cover at the center, choose a point \(a\) outside the smaller disc and inside the larger one. The inclusion of the smaller disc factors through two nested annuli centered at \(a\), with the radius gaps of 19. The larger disc can be chosen to contain these annuli. Restriction between them kills the nonconstant part and all degrees above one. The remaining degree-one class is the mod-\(p\) Kummer class of \(z-a\). On a sufficiently small disc about zero it vanishes: if \(|z/a|<|p|^{p/(p-1)}\), the binomial series for \((1-z/a)^{1/p}\) converges, and a \(p\)-th root of \(-a\) supplies a first \(p\)-th root of \(z-a\). All choices can be made using constants from \(C\), with bounds independent of \(S\) and \(T\).

Exhaust the line by such discs. For each positive degree the inverse system of cohomology groups is pro-zero, and degree zero is the constant system \(R\). The countable derived-limit sequence therefore proves the first assertion for a line. In particular, no assertion that density alone kills a derived limit is used.

Exhaust the punctured line by annuli whose inner radii tend to zero and outer radii tend to infinity, taking a cofinal subsequence with the stated radius gaps. The homotopies of 19 show that its cohomology is \(R\oplus R(-1)[-1]\). The constants identify the first summand. The fiber of restriction from the line to the punctured line is therefore \(R(-1)[-2]\). Its generator is the boundary of the Kummer class. Multiplication of a coordinate by an invertible constant changes its Kummer class by the class of that constant, which vanishes locally after adjoining a root. Translation simply changes the chosen section. This proves the asserted naturality.

For \(d\) coordinates apply the relative line calculation one coordinate at a time. Successive cohomology with support in their zero loci gives the product of the \(d\) Kummer boundary classes, in degree \(2d\) with twist \(-d\). Invertible changes of coordinates preserve this local fundamental class: elementary additions are checked on the complement by the same nearby-coordinate Kummer calculation, diagonal changes were just checked, and coordinate permutations permute classes of even degree. These operations generate the invertible linear changes locally on the base. The constructions and the limiting arguments above are uniform on continuous profinite families, proving the relative statement. ◻

Proof. We describe both the neighborhoods and the limit that computes their support cohomology. On a geometric fiber the map in (ii) is a continuous linear injection from the finite-dimensional \(\mathbb Q_p\)-Banach space \(\mathbb Q_p^2\) into the geometric untilt. Its induced norm is equivalent to the maximum norm, and hence is bounded above and below on the unit sphere. The lower bound can be made uniform on a neighborhood of that fiber. Indeed the parameter unit sphere in \(\mathbb Q_p^2\) is compact; for each of its points the corresponding \(az+b\) is nonzero, and finitely many neighborhoods give a common positive lower bound. Shrink the base by their intersection. Homothety by powers of \(p\) now gives constants \(c_-,c_+>0\) such that \[c_-\max(|a|,|b|)\leq |az+b| \leq c_+\max(|a|,|b|)\] uniformly on that neighborhood. The same assertion for (i) is the norm comparison for the fixed embedding \(F\subset C\).

Write \(V\) for the parameter space and \(d=\dim_{\mathbb Q_p}V\); thus \(d=[F:\mathbb Q_p]\) in (i) and \(d=2\) in (ii). For the compact open restriction on \(a\), use \(V=\mathbb Q_p^2\) and start at a level at which the restricted strip is a union of lattice cosets; compact openness permits this. Choose a lattice adapted to the norm, as follows. At the chosen geometric point, the norm of the injection is locally constant and nowhere zero on the compact parameter unit sphere, so it takes only finitely many values there. Choose a radius in a gap and let \(\Lambda\) be its closed parameter ball. This is a \(\mathbb Z_p\)-lattice. After a further rational restriction of the base there are constants \(\rho_-<\rho<\rho_+\) such that \[|\ell(v)|\leq\rho_-\quad(v\in\Lambda),\qquad |\ell(v)|\geq\rho_+\quad(v\notin\Lambda),\] where \(\ell\) denotes the indicated injection. Only finitely many inequalities are needed for this restriction: use a \(\mathbb Z_p\)-basis of \(\Lambda\) for the upper bound and representatives \(w_j\) of the nonzero cosets of \((p^{-1}\Lambda)/\Lambda\) for the lower bound. Indeed, for \(v\notin\Lambda\), a nonnegative power of \(p\) takes \(v\) to \(v'=w_j+\lambda\in p^{-1}\Lambda\setminus\Lambda\). The strict gap gives \(|\ell(v')|=|\ell(w_j)|\geq\rho_+\), and scalar rescaling gives the desired bound for \(v\). These are inequalities on the rational base neighborhood and hence hold on every perfectoid test over it.

For every \(n\) take discs of radius \(|p|^n\rho\) about the images of representatives modulo \(p^n\Lambda\). Distinct centers are separated by at least \(|p|^n\rho_+\), so the discs are disjoint and nested under refinement. Only finitely many meet any bounded relative affine disc: the lower norm bound restricts their parameters to a bounded, hence compact, subset of \(V\), which has finitely many cosets modulo \(p^n\Lambda\). Thus the family is locally finite.

We verify its limiting support on an arbitrary affinoid perfectoid test \(S'\to S\). Choose two buffered radii between \(\rho_-\) and \(\rho_+\). A section \(w\) belonging to every smaller closed tube system belongs to every larger open tube system. The inverse images of the latter’s disjoint components are open and closed in \(S'\). They define compatible locally constant maps \(c_n:S'\to V/p^n\Lambda\). Quasicompactness gives finitely many values at each level. Since \(V=\varprojlim_n V/p^n\Lambda\) with its locally profinite topology, these maps give a continuous parameter \(c:S'\to V\). Choose representatives of the \(c_n\). Their images under \(\ell\) converge uniformly, since the successive differences have norm at most \(|p|^n\rho_-\). The tube condition gives \(\|w-\ell(c_n)\|\leq |p|^n\rho_{\mathrm{outer}}\to0\); completeness therefore gives \(w=\ell(c)\). Conversely a continuous parameter \(c\) belongs to all the smaller tube systems by the upper bound on \(\Lambda\). The buffered neighborhoods consequently have the required intersection as relative v-sheaves, including analytic boundary points. Their complements exhaust the desired complement, and support cohomology is their derived inverse limit.

The exterior of a disc is exhausted by annuli. By 20 its cohomology is the two-term Kummer complex of Laurent series converging on that exterior. Its degree-zero invariants are \(R\). Taking the fiber from the affine line consequently puts the tube support entirely in degree two. Constant-monomial extraction gives a split surjective residue map in that degree to \(R(-1)\). Its kernel is the nonconstant cokernel of the difference operator.

There is a canonical description of this residue: map to the direct limit of support cohomology over all larger enclosing discs. The annulus homotopy identifies that limit with \(R(-1)\). Recentring gives the same identification, since sufficiently far out the ratio of the two coordinates has a first \(p\)-th root. Inclusions of discs therefore commute with residues to this common target. Excision identifies disjoint children with a direct sum, so the residue of their merger is literally the sum of their residues. Surjectivity is also witnessed by the class of any section inside the tube, which has residue one.

The kernels of the residue maps form a pro-zero system with a fixed level gap. Choose \(j\) such that \(|p|^j<\theta\), with \(\theta\) as in 19; increase \(j\) if necessary for the two buffered radii. The ratio of a level-\((n+j)\) child’s radius to its level-\(n\) parent’s radius is \(|p|^j\), independently of \(n\) and of the parent coset. Recenter the parent disc at this child’s center. Its exterior is unchanged because the center difference is strictly smaller than the parent’s radius. On restriction between these exteriors the inverse of the difference operator is bounded on negative monomials by the fixed radius gap. For positive monomials use a source outer radius at least the target outer radius divided by \(\theta\), which is always available on an exterior. Thus the entire child residue kernel maps to zero in the parent’s support cohomology.

Each parent has exactly \(p^{dj}\) children. Finite summation shows that the map from all level-\((n+j)\) residue kernels to all level-\(n\) kernels is zero. For infinitely many parents the homotopies operate coordinatewise in the product topology; they require no common bound on the coefficient values. For a profinite \(T\) the same inequalities apply in each \(C_{\mathrm{cts}}(T,R)\) supremum norm. Thus one level gap annihilates the kernels on every profinite test, and not merely on geometric fibers.

This proves that the actual system of tube support cohomology is pro-isomorphic to the system of finite sums \(R(-1)\) under summation. It does not require that the classes of different sections in a fixed tube be equal. Those classes can differ by a residue-zero class before passage to the limit. The residue system has split surjective transitions, while the discarded systems are pro-zero. Both their \(\varprojlim^1\) terms consequently vanish. Locally finite disjoint components give products by excision. This proves the formula [h3:ana:distribution-definition] with its product topology.

All norm estimates and all residue maps were made over the base neighborhood. Applying them to \(C_{\mathrm{cts}}(T,R)\) proves the parameter assertion, and descent glues the neighborhoods of the base. The product statement follows by performing the two relative support calculations successively; its partition system consists of rectangles and is cofinal in the finite clopen partitions of the product. ◻

Proof. Choose a lattice \(K\simeq\mathbb Z_p^m\) in \(F\). On its distributions the finite Koszul complex is the Koszul cochain complex for \[R[[X_1,\ldots,X_m]],\qquad X_i=\gamma_i-1.\] The power-series notation here means the inverse limit of finite group algebras, as in [h3:ana:distribution-definition], hence a product of coefficient copies. Multiplication by a variable is injective and its quotient is obtained by setting that variable to zero. Successively applying these statements shows that the Koszul cochain complex has just its top cohomology, the augmentation quotient \(R\), in degree \(m\). The coefficient splitting by powers of \(X_i\) makes these statements exact also with continuous parameters.

There is an identification of translation modules \[\mathcal D_{\mathrm{lf}}(F,R) =\operatorname{Coind}_{K}^{F}\mathcal D(K,R).\] Indeed \(F/K\) is discrete, and both sides are the product of one copy of \(\mathcal D(K,R)\) for each coset, with the same translation rule. Evaluation on \(K\), and the bar contraction using representatives for these open cosets, give continuous cohomological Shapiro for this coinduced module. This proves the second formula. Changing a lattice basis acts on the top cochain generator by the inverse determinant of its change matrix. Scalar multiplication by \(a\) has determinant \(a^m\), proving the asserted character.

For the first formula, exhaust \(F\) by \(p^{-j}K\). In every cochain degree, restriction between these open-and-closed subgroups is surjective by extension by zero. The inverse limit is the continuous cochain complex on \(F\), since these open subgroups cover \(F\); the same statements hold with the parameter \(T\). On trivial characteristic-\(p\) coefficients the Koszul differentials are zero, and compact-lattice cohomology is the exterior algebra on the continuous linear forms of the lattice. Restriction from \(p^{-j-1}K\) to \(p^{-j}K\) multiplies each degree-one form by \(p\) and is therefore zero in every positive degree. In degree zero restriction is the identity. The derived inverse limit is consequently \(R\). The same finite complexes and the same vanishing transition maps compute the parameter versions. ◻

Proof. Use the support fiber sequence for \(\underline F\subset\mathbb A^1_C\) in 18. Its middle term is \(R\) by 21, and its first term is \(\mathcal D_{\mathrm{lf}}(F,R)(-1)[-2]\) by 22. Take the translation cohomology of 23. The resulting support term has degree \(m+2\); rotation gives the stated cofiber in degree \(m+1\). The map from \(R\) throughout this construction is the coefficient unit. Scalar multiplication preserves the normal Kummer class and contributes exactly the translation orientation computed in that lemma. ◻

Proof. The preimage of the double origin under \(\mathbb A^2_C\to N^2\) is \(\mathbb Q_p^2\). The product support calculation gives its distributions in degree four. Taking the two translation directions adds degree two and inverse determinant orientation, giving degree six and scalar weight \(-2\).

On a slope chart, lift the first coordinate to \(z_1\notin\mathbb Q_p\). The support in the second coordinate is \[z_2=a z_1+b,\qquad a,b\in\mathbb Q_p,\] with \(a\) in a compact open slope patch when desired. 22 gives distributions in \((a,b)\) in normal support degree two. Taking translations in the second coordinate is the \(b\)-translation calculation of 23. It leaves distributions in \(a\) and adds degree one with inverse orientation. First-coordinate translation replaces \(b\) by \(b-ac\) and leaves \(a\) fixed. It consequently preserves the resulting orientation line. The remaining base is \(N^*\); 24 adds its degrees zero and two. We obtain degrees three and five, and weights \(-1\) and \(-2\) with the same sign. To justify this computation with distributions in the slope, first use a finite clopen slope partition. It gives a finite sum of the relative punctured calculation. Then take the partition limit. The summation maps are split and the discarded residue kernels are uniformly pro-zero by 22; hence this limit introduces no additional cohomology. This also specifies the topology of the resulting distribution rows.

For equivariance, a point mass at a direction is the support map from that direction’s tubes. These maps commute with linear changes of the pair. Finite sums of point masses are dense in the finite-partition topology: on any finite partition every prescribed collection of values is realized by such a sum. Transport of the point masses therefore determines transport on the full distribution module. At the direction \((1,0)\) the tangential coordinate is changed by \(a\) and the normal coordinate by \(d\). The inverse translation orientations just computed are \(\bar d^{-1}\) and \(\bar a^{-1}\bar d^{-1}\); the upper-right entry acts trivially on both fibers. The quotient frame \(v\) scales both coordinates by \(v^{-1}\), giving the stated \(P_1\) characters. These are finite characters. The calculations of tubes and translations were relative, so this transport argument also applies with all the indicated parameters. ◻

Proof. By [h3:ana:affine-purity,h3:ana:translations], constants compute the cohomology of \(N^2\). Remove the double origin and then the proportional locus. These two support fiber sequences are equivariant for \(G_f\) and their maps from constants are the coefficient unit.

The central subgroup \(\mu_{p-1}I_2\subset\mathop{\mathrm{GL}}_2(\mathbb Z_p)\) acts on their support cohomology by the characters of weights \(-1\) and \(-2\) in 25. Neither character is trivial when \(p\geq5\). Its averaging idempotent is defined in characteristic \(p\), so its derived invariants are exact and annihilate both support complexes. Taking the remaining frame-group cohomology therefore turns the constants arrow into an equivalence. The arrow was induced by the structure map and coefficient unit, which are multiplicative. Its equivalence on underlying complexes proves the assertion for derived algebras as well. ◻

Proof. Write \(g=\dim G\). Analytic group duality identifies cohomology with homology in complementary degree: \[H^a_{\mathrm{cts}}(G,M) \simeq H^{\mathrm{cts}}_{g-a} (G,\operatorname{or}_G\otimes_k M).\] The orientation module is retained in this formula. One obtains it by dualizing a finite projective completed group-ring resolution of \(k\) and reversing its degrees. Such resolutions exist for the stated groups by (Ardakov and Wadsley 2006, Proposition 3.3); their dualizing module is the top inverse adjoint orientation (Beaudry, Goerss, Hopkins, et al. 2022, Proposition 4.16, Remark 4.23, and Proposition 4.40). Thus the formula applies to compact measure modules, not just to finite coefficients.

Completed homological Shapiro gives \[H^{\mathrm{cts}}_{g-a}(G,\operatorname{or}_G\otimes M) \simeq H^{\mathrm{cts}}_{g-a} (B,\operatorname{or}_G|_B\otimes V).\] For completeness, local sections identify \(k[[G]]\), as a right completed \(k[[B]]\)-module, with a completed free module on \(G/B\). Inducing a completed projective \(B\)-resolution and then taking \(G\)-coinvariants is therefore the same complex as taking \(B\)-coinvariants. This proves the displayed Shapiro identification. The last homology is finite-dimensional, since a finite projective \(B\)-resolution has finite-dimensional terms after tensoring with the finite-dimensional fiber. This argument includes the duality shift; it makes no unshifted cohomological Shapiro assertion for \(M\).

To justify extension to \(R\), start with compact \(k\)-modules and apply the exact functor \[A\longmapsto\mathop{\mathrm{Hom}}_k(A^\vee,R),\qquad A^\vee=\mathop{\mathrm{Hom}}_{\mathrm{cts},k}(A,k).\] After a choice of a basis of \(A^\vee\) its value is a product of copies of \(R\). Exactness follows by splitting sequences of the discrete vector spaces \(A^\vee\). Finite projective resolutions use only finite powers and idempotent summands, so their comparison, duality, and Shapiro maps commute with this functor. Apply the same argument to \(C_{\mathrm{cts}}(T,R)\), or use (33). ◻

Proof. The ambient constants and the origin-support line have finite-dimensional fibers with finite-character action. The two remaining support rows in 25 are completed measure induction from \(B\times P_1\) to \(G_f\). To check the local-section hypothesis, a primitive vector over \(\mathbb Z_p\) can, on either standard slope chart, be completed continuously to an invertible matrix. These sections also show that \(\mathop{\mathrm{GL}}_2(\mathbb Z_p)\) acts transitively on \(\mathbb P^1(\mathbb Q_p)\). Its stabilizer is the displayed upper triangular group. Both \(G_f\) and \(B\times P_1\) are compact analytic without \(p\)-torsion when \(p\geq5\): a matrix of order \(p\) in dimension two would require the degree-\(p-1\) cyclotomic polynomial, and \(\mathbb Z_p^\times\) has no \(p\)-torsion. 27 proves finiteness of their support cohomology. The two support sequences prove the claim for the independent locus.

On the connected Honda marking the periodicity line is framed by the leading derivative of the formal-group isomorphism. At height one its change of frame is the reduction \(\mathbb Z_p^\times\to\mathbb F_p^\times\). The determinant identity of 13 expresses a tame middle determinant character through the two frame characters. The indicated coefficient lines therefore have exactly the finite-character property used above. ◻

Proof. Use the expansions in fractional monomials on exhausted closed smaller polyannuli. An invariant expansion has each coefficient in \(\mathbb F_q=k\), and hence is precisely an element of \(M_k\). For a coefficient \(a\in C^\flat\), choose a root \(b\) of \(b^q-b=a\) of least absolute value. The Newton polygon, or the convergent series \(b=-\sum_{j\geq0}a^{q^j}\) when \(|a|<1\), gives \[|b|=|a|\quad (|a|<1),\qquad |b|\leq1\quad (|a|=1),\qquad |b|=|a|^{1/q}\quad (|a|>1).\] In particular \(|b|\leq|a|\). Solving for each coefficient preserves the \(c_0\) convergence conditions in every smaller polyannulus norm. This proves surjectivity and also supplies solutions arbitrarily small whenever the input is sufficiently small in any finite collection of these norms. The Frobenius map itself is continuous in the full analytic topology: for a Gauss poly-radius \(\rho\), \(\|\varphi f\|_\rho=\|f\|_{\rho^{1/q}}^q\). The slightly larger poly-radius on the right belongs to the same open domain.

For continuous parameter families, first approximate a given function on a finite clopen partition of \(T\) and lift the finitely many values. The remaining error is small. Solve for its coefficients with the small branch just described, refine the partition, and repeat with errors tending to zero in successive defining norms. The estimates give a uniformly convergent sum of continuous corrections; its limit is a continuous lift. This argument proves exactness on parameters directly. For the spaces defined by an exhaustion, the continuous Laurent-polynomial approximation and the complete norm topologies meet the hypotheses of 20. Thus these analytic inverse limits have no additional derived term.

The maps \(M_k\to M_C\) and \(\varphi-1\) themselves are natural for changes of variables defined over \(k\). Their auxiliary coefficient choices need not be natural. The exact sequence therefore identifies the original coefficient complex with the fiber of \(\varphi-1\), equivariantly and with parameters. If a finite marking chart has residue field \(k'/k\), geometric base change gives a product indexed by the embeddings of \(k'\). The canonical base Frobenius permutes these factors. On a cycle of length \(e=[k':k]\), eliminating successive coordinates reduces its difference equation to coefficient Frobenius \(q^e\) minus identity. The preceding estimates and continuous lifting apply with \(q^e\). Thus finite products and finite constant-field descent retain the claimed exact sequence, with the actual permutation of factors. ◻

Proof. First compute over \(C^\flat\). For the ordinary locus, 26 is precisely the desired map on the quotient-stack presentation in 13. The perfected polyannular exhaustion has affinoid perfectoid charts. Its coefficient complex is the function complex \(B_{\mathrm{pk}}\), since the successive polynomial approximation in 20 gives the required vanishing of the derived analytic limit. Thus this is also the ordinary continuous \(P_3\)-cochain comparison with the stated topology.

Take derived invariants of base-field Frobenius using 29. On the constants side the fiber of \(\varphi-1\) is \(k\). This proves the untwisted comparison over \(k\) with its actual coefficient map. For a twist, 28 gives a bounded complex with finite-dimensional geometric cohomology. Frobenius acts semilinearly and bijectively on these groups. On a finite-dimensional \(C^\flat\)-space a bijective \(q\)-semilinear operator \(F\) has \(F-1\) surjective and a finite-dimensional fixed space over \(k\). Indeed write \(F(x)=A x^{[q]}\) with \(A\) invertible. The system \(A x^{[q]}-x=b\) is equivalent to monic equations \(x_i^q=(A^{-1}(x+b))_i\). Their pairwise coprime leading monomials show that the coordinate algebra is free over the \(b\)-coordinate algebra on the \(q^n\) monomials with each exponent less than \(q\). The derivative is \(-I\), so the map is finite étale, surjective, and has kernel of order \(q^n\). The kernel is a \(k\)-vector space and therefore has dimension \(n\). This proves the asserted semilinear calculation. Hence Frobenius descent preserves the finite-dimensional conclusion. All these operators commute with the frame actions and with the coefficient maps.

For (ii), define Frobenius before choosing an untilt-divisor presentation. The space \(N_2\) and its \(P_2\) and residual norm actions are defined over \(k\). Consequently \(\mathrm{id}_{N_2}\times\varphi_q\) induces a \(q\)-semilinear automorphism \(\Phi\) of \[\mathcal K_C=R\Gamma(N_{2,C}^*,\mathcal O^\flat),\qquad N_{2,C}=N_2\times_{\operatorname{Spd}k} \operatorname{Spd}C^\flat,\] commuting with both actions. The presentation \(N_{2,C}^*\simeq[(\mathbb A_C^1\setminus F)/F]\) computes \(\mathcal K_C\). It is not required to commute with \(\Phi\): transporting \(\Phi\) through it can change the chosen untilt divisor. Put \(\mathcal K_k=\mathop{\mathrm{fib}}(\Phi-1:\mathcal K_C\to\mathcal K_C)\).

Write \(X_B=\operatorname{Spa}(B^{\mathrm b})\) and \(M_{B,C}=\Gamma(X_B\times_k\operatorname{Spd}C^\flat, \mathcal O^\flat)\). The perfected punctured-ball exhaustion and 20 give no higher cohomology on this geometric chart. On its \(k\)-defined coordinates the canonical base Frobenius is coefficient Frobenius, with any finite residue-field factor permutation retained. Thus 29, applied in every bar degree with parameter \(H^a\), identifies \(R\Gamma_{\mathrm{cts}}(H,B^{\mathrm b})\) with the fiber of \(\Phi-1\) on \(R\Gamma_{\mathrm{cts}}(H,M_{B,C})\).

The actual marking torsor of [h3:geom:finite-level-stack] gives a \(k\)-defined equivalence \([X_B/H]\simeq[N_2^*/U]\). Base-changing this equivalence supplies the vertical maps in the commutative diagram \[ \begin{tikzcd}[column sep=large] R\Gamma_{\mathrm{cts}}(H,M_{B,C}) \arrow[r,"\Phi-1"]\arrow[d,"\simeq"'] &R\Gamma_{\mathrm{cts}}(H,M_{B,C})\arrow[d,"\simeq"]\\ R\Gamma_{\mathrm{cts}}(U,\mathcal K_C)\arrow[r,"\Phi-1"] &R\Gamma_{\mathrm{cts}}(U,\mathcal K_C). \end{tikzcd} \tag{34}\] These maps come from the quotient stacks over \(k\). The diagram is natural for \(P_2\)-transport between marking levels and is equivariant for the full residual norm and the coherent Frobenius actions. Taking fibers, and commuting the two derived limits, gives the natural identity \[R\Gamma_{\mathrm{cts}}(H,B^{\mathrm b}) \simeq R\Gamma_{\mathrm{cts}}(U,\mathcal K_k).\]

There are no additional rows in this fiber. 24 gives geometric cohomology only in degrees zero and three. In every degree the fiber sequence gives \[0\longrightarrow \operatorname{coker}(\Phi-1\mid H^{i-1}\mathcal K_C) \longrightarrow H^i\mathcal K_k \longrightarrow \ker(\Phi-1\mid H^i\mathcal K_C) \longrightarrow0.\] The semilinear calculation above makes the two potentially nonzero cokernels vanish. Thus \(\mathcal K_k\) has the two stated \(k\)-rows, with degree zero equal to \(k\) and trivial \(P_2\) action. Smoothness of the upper action can also be seen directly: if \(e\) is a basis, \(\Phi(e)=Ae\), and \(u(e)=c(u)e\), commutation gives \(c(u)^q=c(u)\). Hence \(c(u)\in k^\times\). The scalar norm character \(\bar a^{-2}\) survives this descent since it already takes values in \(\mathbb F_p^\times\).

A compact analytic \(U\subset P_2\) has finite completed resolutions: it has no \(p\)-torsion, because an order-\(p\) element in its degree-two division algebra would require a subfield of degree \(p-1>2\). The quotient spectral sequence with the two finite \(k\)-rows therefore proves finite-stage finiteness and identifies degree zero with \(k\). The resolutions consist of finite powers and idempotent summands. The continuous Artin–Schreier lifting above and these finite resolutions identify the actual parameter cohomology with \(C_{\mathrm{cts}}(T,-)\) of the point-valued finite groups. For a line framed on a finite connected-marking cover, the same argument has a finite-character \(U\)-fiber. The tautological connected frame trivializes the periodicity line under \(H\), and a tame determinant character is trivial on \(H\). This proves the additional finite-stage assertion.

Finally choose a cofinal sequence of open uniform subgroups \(U\), small enough to act trivially on these two finite rows. Restriction to a sufficiently deep \(p\)-power subgroup is zero in positive cohomological degrees: on the mod-\(p\) exterior cohomology of a uniform group it is zero in degree one and hence in every positive degree. Filtered colimits of the quotient spectral sequences therefore retain precisely the two rows themselves. The constants arrow gives the stated fiber sequence. Its scalar character was computed on \(N_2^*\) in 24. By determinant normalization in 13, this scalar is the full reduced norm of an element of \(P_3\). The induced quotient action on \(H\)-cohomology is well defined because inner conjugation by \(H\) acts trivially there. Exact averaging in the prime-to-\(p\) norm quotient gives the final assertion. ◻

Proof. The groups needed here satisfy the hypotheses by 6; their open subgroups inherit the absence of \(p\)-torsion and have the same Lie dimension. For the general assertion, use the Noetherian finite-global-dimension theorem for the completed group ring (Ardakov and Wadsley 2006, Proposition 3.3) and the fact that its integral dualizing cohomology is a free rank-one module in degree \(d\) and vanishes in the other degrees, with the adjoint orientation action (Beaudry, Goerss, Hopkins, et al. 2022, Proposition 4.16 and Remark 4.23). These statements give a finite projective resolution over \(\mathbb Z_p[[G]]\). Noetherianity makes its terms finitely generated. Give each term its compact topology as a summand of a finite power of the group ring. The maps and their images are continuous and closed. Its syzygies are \(p\)-torsion-free, so reduction modulo \(p\) is exact. The integral completed bar resolution is projective in compact modules: the completed free \(\mathbb Z_p\)-module on a profinite set is compact and torsion-free, hence projective, since its Pontryagin dual is divisible and therefore injective. Induction to \(\mathbb Z_p[[G]]\) preserves that projectivity. The continuous projective comparison therefore shows that the dual of the finite integral resolution computes the quoted continuous dualizing cohomology. Its reduction modulo \(p\) does so as well, because both its terms and its unique integral cohomology module are \(p\)-torsion-free. The reduced dual complex has cohomology only in degree \(d\). If the reduced resolution extends beyond \(d\), its last dual differential surjects onto a projective module; splitting and dualizing removes a contractible pair. Repetition gives the stated length-\(d\) resolution.

Here is why the resolution computes the asserted topological cochains. For a profinite space \(Z\), continuous maps \(Z\to M\) identify with continuous linear maps \(\mathbb F_p[[Z]]\to M\). Modulo an invariant open subspace of \(M\), the map has finite image and its linear extension factors through a finite clopen partition of \(Z\). Taking the inverse limit over these subspaces gives the claimed extension into \(M\). The completed bar resolution is projective in compact modules: a surjection of compact \(\mathbb F_p\)-vector spaces admits a continuous linear section, obtained by dualizing and splitting the corresponding injection of discrete vector spaces. Such a section lifts the profinite set of bar generators; the lift extends group-ring linearly. The projective comparison construction therefore gives continuous chain maps and homotopies between the completed bar resolution and the finite resolution. Applying continuous Hom into \(M\) proves the assertion. Continuity for the uniform cochain topologies can be checked after quotienting by each invariant open subspace.

For compact coefficients, the finite complex has compact terms and closed boundaries, which proves the compact Hausdorff assertion. Now take \(G=H\) or \(G=P_3\). The adjoint orientation of each is trivial by 6: conjugation has determinant one on \(D_3\), and also on its trace-zero summand. Thus the dualizing module has the trivial right action, in degree \(d_G=8\) or \(9\) respectively.

Let \(Q_\bullet\) be the finite resolution for this \(G\), and set \(Q_i^*=\mathop{\mathrm{Hom}}_{\mathbb F_p[[G]]}(Q_i,\mathbb F_p[[G]])\). The reversed group-ring dual resolves the trivial right module, by the dualizing calculation above. Regard \(Q_i^*\) as a left module through the involution \(g\mapsto g^{-1}\). Finite-projective evaluation gives \[\mathop{\mathrm{Hom}}_{\mathrm{cts},\mathbb F_p[[G]]}(Q_i,M)^\vee \cong \mathop{\mathrm{Hom}}_{\mathrm{cts},\mathbb F_p[[G]]}(Q_i^*,M^\vee).\] For a finite free module this is evaluation coordinate by coordinate; passing to a projective summand preserves the identification. It commutes with both differentials and coefficient maps. Continuous \(\mathbb F_p\)-duality is exact on compact vector spaces, as follows by extending a functional on a finite-dimensional quotient. Reversing degrees about \(d_G\) and taking cohomology proves (38) and its naturality for both groups. For a discrete continuous representation, each element has a finite \(G\)-orbit; finite sets therefore lie in finite \(G\)-stable submodules. Their annihilators give invariant open neighborhoods in the compact dual. Dualizing the compact formula again proves the discrete version. Finally, the finite-resolution complex has finite terms when \(M\) is finite. ◻

Proof. The cycles form a closed Polish group. The boundary subgroup is an analytic subset of it, being the continuous image of a Polish space, and hence has the Baire property (Kechris 1995, Theorem 21.6). Its finite index implies that it is nonmeagre. A nonmeagre set with the Baire property is comeagre in some nonempty open set; intersecting two sufficiently small translates of that open set shows that its difference set contains a neighborhood of zero. Applied to the subgroup \(B^a\), this proves openness, and an open subgroup is also closed.

We also recall the open-mapping step. If \(q:E\twoheadrightarrow F\) is a continuous surjective homomorphism of Polish groups and \(V\) is a neighborhood of zero in \(E\), choose open \(V'\) with \(V'-V'\subset V\). Separability gives a countable covering of \(E\) by translates of \(V'\). Thus \(q(V')\) is an analytic nonmeagre subset of \(F\), and its difference set contains a neighborhood of zero. This difference set lies in \(q(V)\). Hence \(q\) is open. Apply this to the differential with target \(B^a\), which is already open in \(Z^a\). ◻

Proof. For the first assertion, choose complete translation-invariant metrics on the groups and successively smaller neighborhoods of zero in \(E\) whose diameters tend to zero and whose sums converge. Openness provides corresponding neighborhoods in \(F\). Compactness and zero-dimensionality give a finite clopen partition on which the given map is approximated by finitely many chosen images. Lift those values to \(E\). Repeat on the error, refining the partition and choosing the correction inside the next prescribed neighborhood of zero in \(E\). The successive locally constant lifts converge uniformly to a continuous lift; their images converge to the original map. This construction uses no metrizability assumption on \(T\).

A continuous family of cycles gives a continuous family of cohomology classes. If that family is zero, the first assertion and 34 lift it continuously through \(A^{a-1}\twoheadrightarrow B^a\). Conversely, a continuous map from \(T\) to the finite discrete cohomology group has finite image; choose a cycle for each value to obtain a continuous family. This proves the second assertion. For the third, a continuous function to a discrete filtered colimit has finite image. Its finitely many values and equality relations are represented at one stage. Filtered colimits of vector spaces are exact, which proves the claimed compatibility. ◻

Proof. The Cartier dual of the height-two, dimension-one Honda group is again connected of height two and dimension one. Indeed, on the Honda model \(\Phi^2=[p]\), so duality gives \(V^2=[p]\) on its dual. Together with \(\Phi V=[p]\), cancellation on the divisible group gives \(\Phi=V\); thus each finite dual torsion kernel is killed by a Frobenius iterate and is connected. The dimension formula and formal-group construction in (Tate 1967, sec. 2.2, Proposition 1, and Section 2.3, Proposition 3) give dimension one and a power-series coordinate, already over \(\mathbb F_p\). No identification with a chosen coordinate law on the original Honda group is needed. A parameter on the dual identifies the finite distribution algebra with \[\mathcal O(\Gamma_2[p^l])^* =k[D]/(D^{p^{2l}}).\] The duals of the torsor transition maps are the inclusions of the dual torsion kernels. Their restriction maps give the inverse limit \(k[[D]]\).

After a faithfully flat trivialization of a torsor, its functions form the dual regular module of this truncated polynomial ring. The pairing given by the coefficient of \(D^{p^{2l}-1}\) in a product is perfect, so that module is one regular nilpotent block. Ranks descend under faithful flatness. Consequently the kernel of \(D^e\) on \(R_l\) has rank \(e\) over \(L\) for \(1\leq e\leq p^{2l}\).

For \(q=p^i\), the formal-group coproduct satisfies \[\Delta(D^q)\in(D^q\otimes1,1\otimes D^q).\] The module-algebra identity therefore makes \(\ker D^q\) a subalgebra. On a field factor it is an intermediate field of degree \(q\). The element \(y\) is a \(p\)-basis of \(K\) and remains a \(p\)-basis after separable extension: if \(E/K\) is finite separable, then \([E:E^p]=[K:K^p]=p\), and \(y\notin E^p\) because a separable extension contains no nontrivial purely inseparable subextension of \(K\). An intermediate field of degree \(q\) in \(E^{1/p^{2l}}/E\) lies in \(E^{1/q}\), since every element has purely inseparable degree at most \(q\); equality follows from the degrees. Applying this on every factor and then passing to \(L\) proves the kernel formula. The block calculation also proves nonvanishing of the top operator. In a larger block the old kernel is contained in the image of \(D\), which proves surjectivity on \(R\).

By 15, a full reduced-norm unit \(u\) rescales the normalized étale generator by \(u\) or \(u^{-1}\). On the fixed connected framing cover this conjugates translations by the corresponding scalar automorphism. Cartier duality preserves scalar multiplication, so the dual tangent character is \(\chi(u)=u^{s_{\mathrm{dist}}}\) with \(s_{\mathrm{dist}}=\pm1\). Since \(H\) acts trivially on translations, this is an action of the quotient \(H'/H\), whether or not that quotient has a subgroup section. Starting with a parameter \(D_0\), replace it by \[D=\frac1{p-1}\sum_{u\in\mu_{p-1}} \chi(u)^{-1}u(D_0).\] Its linear term is the linear term of \(D_0\), and it is an eigenparameter. This proves the second assertion, including the inverse twist on the target of (41).

For the Frobenius assertion, the Honda model over \(\mathbb F_p\) satisfies \(\Phi^2=[p]\). On the full divisible group, \(V\Phi=[p]=\Phi^2\); cancellation of the faithfully flat epimorphism \(\Phi\) gives \(V=\Phi\). Cartier duality exchanges the two operators. For the stated even \(i\), their \(i\)th iterates are \([p^{i/2}]\). To make the scalar and duality conventions explicit, put \(A_l=\mathcal O(\Gamma_2[p^l])\) and pair it with its finite dual by \(\langle b,\xi\rangle=\xi(b)\). Since \(q\) fixes \(k\), Cartier-dual naturality for the \(i\)th relative Frobenius reads \[ \langle b^q,D\rangle =\langle b,(V^i)^*D\rangle =\langle b,D^q\rangle. \tag{43}\] For the second equality, choose an \(\mathbb F_p\)-coordinate \(t\) on the Cartier dual, which descends to \(\mathbb F_p\). Tame averaging leaves \(D=\sum_n c_nt^n\) defined over \(k\), and hence \((V^i)^*D=\sum_n c_nt^{nq}=D^q\) because \(c_n^q=c_n\). Here multiplication in the distribution algebra corresponds to composition of operators. If a finite torsor coaction is written \(\rho(f)=\sum_j f_j\otimes b_j\), (43) therefore gives \[D(f^q)=\sum_j f_j^qD(b_j^q) =\sum_j f_j^qD^q(b_j) =\left(\sum_j f_jD^q(b_j)\right)^q.\] The last equality uses \(q\)-power invariance of the constants. The coefficients \(f_j\) retain their \(q\)th powers throughout; this finite Hopf-coaction calculation requires no splitting of the torsor. This proves (42). If \(a\in L^{1/q}\), apply it to \(af\) and use \(a^q\in L\) and \(L\)-linearity of \(D\) to obtain \(D^q(af)=aD^qf\). The cancellation above was on a divisible group; no cancellation on a finite torsion kernel has been used.

On a fixed root level only finitely many powers of \(D\) act. In the basis of successive fractional powers of the \(p\)-basis \(y\), their matrices have finitely many coefficients in \(L\), and these lie in one finite separable stage. The resulting matrices between finite-dimensional spaces over complete local fields are continuous. To construct a right inverse on a stage, choose a \(D\)-preimage of each member of its finite field basis and put those preimages in one larger complete separable/root stage. Linear extension gives a continuous additive section. The kernel at a finite stage has the required subspace topology; intersection with the separable union is that stage’s separable part, by uniqueness of purely inseparable roots. The cochain-exactness criterion preceding 34 proves the last assertion. All of these arguments are factorwise for a finite product, with finitely many choices at each stage. ◻

Proof. The functional \(I_i(f)=D^{q-1}(f^{1/q})\) takes values in \(L\), is nonzero on every factor, is \(H\)-equivariant, and satisfies \(I_i(a^qf)=aI_i(f)\). The same semilinearity and equivariance hold for \(\mathcal C^i\). The Cartier pairing gives an isomorphism \[ \mathrm{Fr}^i_*L\longrightarrow \mathop{\mathrm{Hom}}_L(\mathrm{Fr}^i_*L,L),\qquad b\longmapsto\bigl[f\longmapsto\mathcal C^i(bf)\bigr]. \tag{50}\] Here \(a\) acts on \(\mathrm{Fr}^i_*L\) as multiplication by \(a^q\). For a field factor, both sides have dimension \(q\); the map is injective, since for \(b\ne0\) one can choose \(z\) with \(\mathcal C^i(z)\ne0\) and evaluate at \(b^{-1}z\). This proves the isomorphism on products and separable filtered unions as well; equivalently it is the perfect coefficient-extraction pairing in the common \(p\)-basis. Thus there is a unique multiplier \(b_i\) with \(I_i(f)=\mathcal C^i(b_if)\). It is a unit because both functionals are nonzero on every field factor. Equivariance and uniqueness give \(b_i\in L^H=k\), by (44). Semilinearity gives (46), with \(c_i=b_i^{1/q}\).

This comparison applies in particular when \(i=1\); it does not require the even-iterate identity (42). Since \(D\) kills \(L\), it yields \(c_1\mathcal C(f^p)=D^{p-1}f=0\), proving (47). The latter is therefore a consequence of the invariant torsor and differential, not an identity for an arbitrary normalization of Cartier.

On a factor \(\kappa((s))\) the logarithmic Laurent formula is \[ \mathrm{Car}\!\left(\sum_n a_ns^n\frac{ds}{s}\right) =\sum_n a_{pn}^{1/p}s^n\frac{ds}{s}. \tag{51}\] It proves continuity and surjectivity. The kernel consists of the forms whose coefficients in indices divisible by \(p\) vanish; a continuous additive primitive is \(\sum_{p\nmid n}(a_n/n)s^n\). The kernel of \(d\) is \(B^p\), and its root map is continuous. A continuous section of Cartier sends \(\sum_n a_ns^n ds/s\) to \(\sum_n a_n^ps^{pn}ds/s\). Multiplication or division by the fixed nonzero \(\eta\) transports these sections to (48). The maps themselves are \(H\)-equivariant because \(\eta\) is invariant and Cartier is intrinsic.

Residue is invariant under change of uniformizer. The sum of finite field traces is preserved by residue-field automorphisms and factor permutations. This proves \(H\)-invariance of (49). Finally, \(\mathop{\mathrm{Res}}\mathrm{Car}(\alpha)=(\mathop{\mathrm{Res}}\alpha)^{1/p}\) and \(g\mathrm{Car}(\alpha)=\mathrm{Car}(g^p\alpha)\). The finite-field trace is unchanged by \(p\)th roots. Applying these identities to \(\alpha=f\eta\) proves the transposition formula. ◻

Proof. The nilpotent block on \(L^{1/p}\) puts \(1\) in the image of \(D\), so choose \(\theta\) there. The finite data \(\theta^p\) and \(\eta\) lie at a finite separable stage. The action of \(J\) on the finite set of jets \(k[[D]]/(D^{p+2})\) is continuous. Intersect their open stabilizers with any prescribed open subgroup and pass to a normal core. This proves cofinality and (52). For \(f\in B^{1/p}\), \(D^pf=0\) and \[\sigma(Df)-Df=(\sigma(D)-D)f=0.\] The kernel filtration and uniqueness of roots therefore put \(Df\) in \((L^{1/p})^U=B^{1/p}\). Its finite matrix over the complete stage proves continuity.

The condition defining \(P_B\) says that \(f\eta\) is an integral ordinary differential on each factor. That condition is independent of the uniformizer and is preserved by \(H\). Formula (51) preserves strictly positive logarithmic order, proving Cartier stability. The annihilator statements follow coefficient by coefficient: an integral \(g\) pairs trivially with all strictly positive logarithmic coefficients, while every pole of \(g\) is detected by a suitable positive coefficient of \(f\eta\). Conversely every nonpositive coefficient of \(f\eta\) is detected by a monomial in \(\mathcal O_B\). Nondegeneracy of the finite-field trace makes these detections nonzero when needed. ◻

Proof. The first assertion is the finite-stage statement of 30. Let \(z\) be a continuous positive-valued \(a\)-cocycle. Compactness of \(H^a\) supplies \(\varepsilon>0\) with \(v(z)\geq\varepsilon\) on the entire cochain domain. Hence \(z^{p^n}\to0\) uniformly. For \(a>0\), apply 34 to the complete coefficient complex, prescribing the open neighborhood of positive-valued \((a-1)\)-cochains. A sufficiently high Frobenius power has a primitive \(b\) in that neighborhood. Inverse Frobenius is a continuous additive automorphism of the completed perfect algebra and preserves positivity, so \(b^{1/p^n}\) is a positive primitive of \(z\). For \(a=0\), the finite invariant group is discrete; the invariant elements \(z^{p^n}\) tend to zero and are eventually zero. Injectivity of Frobenius then gives \(z=0\).

The quotient of \(\mathcal O_{B^b}\) by its strictly positive ideal is the finite product of the residue fields of \(B\). The two valuation-cutoff sequences are exact on cochains because their quotients are discrete. Their long exact sequences, the vanishing just proved, and finite cohomology of finite coefficients establish the last assertions. ◻

Proof. The residue pairing and 38 give topological identifications \[ P^\vee=B/\mathcal O_B,\qquad (B/P)^\vee=\mathcal O_B. \tag{55}\] A continuous functional on \(P\) uses only finitely many Laurent coefficients and is represented by a finite principal part in the first quotient. An arbitrary functional on the discrete \(B/P\) is represented by a power series, which proves the second identification and its topology. The trace transposition in 37 and (38) identify the dual of the inverse system \((M_a,\mathcal C)\) with \[\bigl(H^{8-a}_{\mathrm{cts}}(H,B/\mathcal O_B),\mathrm{Fr}\bigr).\] Its coefficient colimit is \[ \mathop{\mathrm{colim}}_{\mathrm{Fr}}B/\mathcal O_B =B^{\mathrm{perf}}/\mathcal O_{B^{\mathrm{perf}}} =B^b/\mathcal O_{B^b}. \tag{56}\] At index \(n\) the first map sends \(f\) to \(f^{1/p^n}\). The second equality follows by approximating any completed element with integral error. All these quotients are discrete, so their continuous cochains commute with the colimit. By 40, the direct limit of the displayed cohomology system is finite. Compact–discrete duality shows that \(\varprojlim_{\mathcal C}M_a\) is finite.

Its zeroth projection has image \(I_a\). To prove surjectivity onto that intersection, fix \(x\in I_a\). For each \(n\) there is a compatible length-\(n\) chain of preimages ending at \(x\). These conditions define nested nonempty closed subsets of the compact product \(\prod_{n\geq0}M_a\). Their intersection is an infinite compatible chain. Thus \(I_a\) is an image of the finite inverse limit. ◻

Proof. For \(x\in I\), the sets \(C^{-1}(x)\cap C^nM\) are nested nonempty compact sets, so they meet. This proves \(C(I)=I\). The compact images \(C^nQ\) have intersection zero and shrink uniformly to zero: otherwise their intersections with the complement of a fixed open neighborhood would be nested nonempty compact sets. Therefore \(\sum_{n\geq0}a_n C^n x\) converges for every formal power series \(\sum a_nt^n\), uniformly in the tails, and defines an \(\mathbb F_p[[t]]\)-action on \(Q\).

Lift a finite basis of \(Q/CQ\) to \(e_1,\dots,e_s\). Repeatedly expressing an element modulo \(CQ\) gives \[x=\sum_{i=1}^s\sum_{j=0}^{N-1}a_{ij}C^je_i+C^Nx_N.\] The remainder tends uniformly to zero. Taking the limit expresses \(x\) as a sum of \(s\) power-series multiples of the \(e_i\). The structure theorem over the discrete valuation ring \(\mathbb F_p[[t]]\) now writes \(Q\) as a finite free module plus finitely many modules \(\mathbb F_p[[t]]/(t^e)\). Its \(C\)-power torsion and \(C\)-kernel are finite. The kernel on \(M\) has kernel in \(I\) and image in the kernel on \(Q\), so it is finite too.

The image in \(Q\) of an element with finite forward orbit both tends to zero and belongs to a finite set; it is eventually zero. Thus the image of \(N\) is in the finite group \(Q[C^\infty]\), and its kernel lies in finite \(I\). Hence \(N\) is finite and closed. A coset meeting every \(C^nM\) meets their intersection, by compactness of its nested intersections with those images. This proves (57). ◻

Proof. For \(a=0\), Frobenius is injective. For \(a>0\), its kernel is the kernel of the inclusion into the root-stage cohomology under the \(H\)-equivariant homeomorphism \(B^{1/p}\to B\). Consequently each kernel class has a presentation \([\partial b]\), where \[b\in\mathcal B_a :=\{b\in C^{a-1}_{\mathrm{cts}}(H,B^{1/p}): \partial b\in C^a_{\mathrm{cts}}(H,B)\}.\] Put \(w=Db\). It is a \(B^{1/p}\)-valued cocycle because \(D\) kills \(B\). The quotient map \[\mathcal B_a\longrightarrow C^{a-1}_{\mathrm{cts}}\bigl(H,B^{1/p}/(B^{1/p})^{++}\bigr),\qquad b\longmapsto Db\bmod ++,\] has countable target. Its kernel \(\mathcal B_a^+\) consists of presentations with positive-valued \(w\) and has countable index. It suffices to put the classes presented by this kernel into the finite intersection in the hypothesis.

Fix such a \(b\), and let \(i\) range through arbitrarily large positive integers divisible by two and by \([k:\mathbb F_p]\); put \(q=p^i\). For \(\theta\) in 38, define \[b'=\theta^{1/q}w\in C^{a-1}_{\mathrm{cts}}(H,B^{1/(pq)}).\] By (42), \(D^q\theta^{1/q}=1\), and \(D^q\) is \(L^{1/q}\)-linear. As \(w\in B^{1/p}\subset L^{1/q}\), this proves \(D^qb'=w\). Taking differentials and using the kernel formula gives \[ \partial b'\in C^a_{\mathrm{cts}}(H,B^{1/q}). \tag{58}\] Here \(b'\) is fixed by \(U\), and \((L^{1/q})^U=B^{1/q}\) by uniqueness of roots.

We must also descend \(D^{q-1}b'\). For \(\sigma\in U\) write \(\sigma(D)=D(1+h_\sigma)\), with \(h_\sigma\in D^{p+1}k[[D]]\). Then \[\sigma(D)^{q-1}-D^{q-1} \in D^{q+p}k[[D]].\] The whole error kills \(b'\), since \(D^{q+p}b'=D^pw=D^{p+1}b=0\) and \(D^pb=0\). Local nilpotence makes each error series finite on the relevant stage. Thus \(D^{q-1}b'\) is \(U\)-fixed, and \(D(D^{q-1}b'-b)=0\) proves that \(D^{q-1}b'-b\) is a \(B\)-valued cochain. For each \(q\) all the operators act continuously after one finite-stage enlargement. The descended subspaces carry their valuation subspace topologies, so these descended cochains are continuous as well.

Let \(\alpha_i=(\partial b')^q\). By (58), it is a \(B\)-valued cocycle. The descended difference and 37 give \[ [\partial b]=[D^{q-1}\partial b'] =c_i\mathcal C^i[\alpha_i]\quad\text{in }X_a. \tag{59}\] Compactness gives a common \(\varepsilon>0\) with \(v(w)\geq\varepsilon\). Because \(H\) preserves valuations, \[v(\alpha_i)\geq q\varepsilon+v(\theta).\] This eventually exceeds the finitely many thresholds defining \(P\). Hence \(\alpha_i\) is a \(P\)-valued cocycle for all sufficiently large admissible \(i\). The scalar \(c_i\) can be absorbed into the lattice: \(c_i\mathcal C^i(\alpha_i)=\mathcal C^i(c_i^q\alpha_i)\) and \(c_i^qP=P\). Equation (59) puts our class in a cofinal set of the decreasing images \(f_a(\mathcal C^iM_a)\), hence in their intersection. The subgroup \(\mathcal B_a^+\) has finite image and countable index, so the entire Frobenius kernel is countable. ◻

Proof of 39. Suppose some \(X_a\) were uncountable and choose the largest such \(a\). It exists by the degree-eight bound in 33. Split (48) at \(V=\mathop{\mathrm{im}}(d/\eta)=\ker\mathcal C\): \[0\longrightarrow B\xrightarrow{\mathrm{Fr}}B\longrightarrow V \longrightarrow0, \qquad 0\longrightarrow V\longrightarrow B\xrightarrow{\mathcal C}B \longrightarrow0.\] The first sequence places \(H^{a+1}_{\mathrm{cts}}(H,V)\) between a quotient of \(X_{a+1}\) and a subgroup of \(X_{a+2}\), so this group is countable. The second embeds \(X_a/\mathcal C X_a\) into it. By (54) and the observation after 42, \(M_a/\mathcal C M_a\) is countable. It is compact Hausdorff since \(\mathcal C M_a\) is compact and closed. A countable compact Hausdorff group is finite: Baire applied to its singleton cover gives an isolated point, and a discrete compact group is finite.

Apply 42 using the finite stable image from 41. To treat \(N_a=\ker f_a\), use its connecting presentation from \(H^{a-1}_{\mathrm{cts}}(H,B/P)\). A representing cochain has finite image, hence a uniform lower bound on the logarithmic orders of its differential coefficients. The identity \[\mathcal C^n(f)\eta=\mathrm{Car}^n(f\eta)\] and (51) show that for sufficiently large \(n\) all negative indices in that finite image disappear modulo \(P\). Its iterate is therefore a cocycle in the finite, \(H\)- and \(\mathcal C\)-stable module \[P(0)/P,\qquad P(0)=\prod_jh_j^{-1}\kappa_j[[s_j]].\] Finite cohomology of this module and naturality of the connecting map show that every element of \(N_a\) has finite forward Cartier orbit. For \(a=0\) the kernel is already zero. The compact-module lemma makes \(N_a\) finite and gives \[\ker(\mathcal C:X_a\to X_a)\text{ countable},\qquad \bigcap_n f_a(\mathcal C^nM_a)=f_a(I_a)\text{ finite}.\] By (47), the Frobenius image in \(X_a\) is countable. An uncountable group with countable image has uncountable kernel, contradicting 43.

All \(X_a\) are therefore countable. Equation (54) makes the compact \(M_a\) countable, hence finite. Any \(H\)-stable order lattice is commensurable with \(P\); intersecting the two lattices gives finite quotients. The associated long exact sequences prove finite cohomology for every such lattice, including \(\mathcal O_B\). By the second identification in (55) and (38), \[H^a_{\mathrm{cts}}(H,B/P)^\vee \cong H^{8-a}_{\mathrm{cts}}(H,\mathcal O_B)\] is finite. Thus the quotient cohomology is finite, and the lattice sequence now proves finiteness of \(X_a\) itself. Other lattice quotients follow by commensurability.

The homeomorphism \(B^{1/p^e}\to B\), \(f\mapsto f^{p^e}\), is \(H\)-equivariant and \(\mathbb F_p\)-linear and carries order lattices to order lattices. It proves the assertion for finite root stages. Finally, an \(H\)-invariant frame identifies a coefficient line at a sufficiently deep finite stage with \(B\) as an \(H\)-module, and its lattices with order lattices. The connected marking supplies such a frame for every periodicity power; the full norm kernel fixes every determinant twist. Enlarging the good stage to contain finitely many frame coefficients proves the stated twisted assertion. ◻

Proof. Fix a good \(B\). The positive order lattice \(B^{++}\) has finite cohomology by 39. A positive-valued cocycle has a uniform positive valuation bound, so its Frobenius powers tend uniformly to zero. 34 makes each cohomology class eventually zero under Frobenius. Since the cohomology is finite, a single power in each degree kills every class.

The homeomorphisms \(B^{1/p^e}\to B\) identify inclusion of one positive root stage into the next with Frobenius. Their cochain colimit is therefore acyclic. The positive ideal in \(B^b\) is acyclic by 40. Density gives an isomorphism of discrete coefficient modules \[B^{\mathrm{perf}}/(B^{\mathrm{perf}})^{++} \cong B^b/(B^b)^{++}.\] A continuous cochain to either quotient has finite image, whose values can be lifted at one root stage. Thus this is also an isomorphism of the prescribed quotient cochain complexes. Comparing the two valuation-cutoff exact sequences proves the quasi-isomorphism for this \(B\). Passing through the cofinal good stages proves (60).

Each finite root stage has finite cohomology by 39, and each completed stage has finite cohomology by 40. Apply 35 to their Polish cochain complexes and pass to the filtered colimits. It proves the assertion with any profinite parameter space. All comparisons used inclusions and valuation quotients, so their naturality retains all the indicated actions. ◻

Proof. 44 transports the actual residual actions in (36) to \(R\). The quotient \(H'/H\) has order prime to \(p\), so its invariants functor is exact; the continuous Hochschild–Serre complex thus computes \(H'\)-cohomology by taking those invariants on \(H\)-cohomology. The untwisted characters are \(0\) and \(2\epsilon\). Only the first is trivial modulo \(p-1\). After twisting by \(\chi^{-1}\) the exponents are \[-s_{\mathrm{dist}},\qquad 2\epsilon-s_{\mathrm{dist}},\qquad \epsilon,s_{\mathrm{dist}}\in\{1,-1\}.\] They belong to \(\{1,-1,3,-3\}\) and are nonzero modulo \(p-1\geq4\). Both target rows therefore vanish. In the source the invariant row is exactly the given constants row.

The distinction from the center can be seen at \(p=7\). A central \(a\in\mu_6\subset P_3\) has norm \(a^3\), so pulling a norm-weight-two character to that central subgroup would make it trivial. Here the normalized étale generator transforms by the full norm unit \(u\); the source weight is \(u^{\pm2}\) and the twisted exponents are \(\pm1,\pm3\), nontrivial modulo six. The reduced norm surjects on tame units, as follows from the residue-field norm \(\mathbb F_{p^3}^\times\to\mathbb F_p^\times\). Since \(H\) acts trivially on the translation group and by inner automorphisms trivially on its own cohomology, all residual actions used here factor through the quotient itself. Neither the eigenparameter construction nor the invariants calculation requires it to split inside \(P_3\). ◻

Proof of 32. The distribution sequence (41) is exact on the specified continuous cochain complexes. By 45, its last term has zero \(H'\)-cohomology, and its middle term is computed by constants. Thus its first term is also computed by the actual constants map, proving (37). The finite-stage \(H'\)-cohomology is finite by 39 and exact tame invariants. 35 proves the parameterized assertion. Although a choice of \(D\) was used to prove the comparison, the resulting quasi-isomorphism is the original constants inclusion. Its naturality therefore retains the connected-frame action and the remaining height-three action. ◻

Proof. For fixed normal open \(U\), set \(W_U=C(T,B_U)\) with its uniform topology. The action on it factors through \(J/U\). Compare locally constant \(J\)-cochains, viewing \(W_U\) as discrete, with continuous \(J\)-cochains for its given topology. A fixed finite-projective \(\mathbb F_p[[J]]\)-resolution from 33 computes both. Their Hom complexes are the same underlying finite summands of powers of \(W_U\): every module map factors through the finite-dimensional quotient of a resolution term by the augmentation ideal of \(U\), and is continuous with either topology. The canonical inclusion of the two bar-cochain complexes is consequently a quasi-isomorphism.

After taking the colimit over normal open subgroups, every locally constant cochain, its finite clopen partition, and its finitely many values are represented at one finite quotient. Refining \(U\) if necessary gives an equality of complexes \[\mathop{\mathrm{colim}}_U C^\bullet_{\mathrm{lc}}(J,W_U) =\mathop{\mathrm{colim}}_U C^\bullet(J/U,W_U).\] A finite topological \(K\)-basis of \(B_U\) gives \(W_U=B_U\otimes_K C(T,K)\). The normal-basis theorem for the finite Galois étale algebra \(B_U/K\) makes this a coinduced \(J/U\)-module. The finite-group cochain complex of a coinduced module has zero positive cohomology: its bar contraction is evaluation in the induced function coordinate. Its invariants are exactly \(C(T,K)\). Thus the canonical augmentation is a quasi-isomorphism at each finite quotient. Exactness of the filtered colimit proves (61).

For a pulled-back coefficient line, tensor the finite-basis and normal-basis argument with that line over \(K\). All choices were used only to prove exactness. The comparison maps themselves are the canonical augmentations, hence are natural and retain the commuting action. This proof makes no claim about cochains into the metric union \(L\). ◻

Proof. Choose a sufficiently deep good normal stage \(B=B_U\) containing the frame and put \(Q=J/U\). Let \(\mathcal O_B\) be its integral closure lattice and set \[M=\mathcal O_B\otimes_{k[[y]]}\Lambda.\] The actions of \(H\) and \(Q\) commute and \(M^Q=\Lambda\). In the \(H\)-invariant frame, \(M\) is an \(H\)-stable order lattice at the stage \(B\), so 39 gives finite \(H\)-cohomology for it.

For \(b>0\), \(H^b(Q,M)\) is a finite group. To see this without averaging, the finite-group cochain terms are finitely generated modules over \(k[[y]]\). Their cohomology is finitely generated as well. Inverting \(y\) commutes with that cohomology and gives zero in positive degrees by normal-basis descent on the field algebra. Hence \(H^b(Q,M)\) has finite length over \(k[[y]]\), whose residue field is finite. The finite group boundaries are compact and closed, so these finite cohomology groups have their discrete quotient topology and continuous \(H\)-action.

Use the bounded finite-projective \(H\)-resolution together with the finite-\(Q\) bar complex on \(M\). Taking \(H\)-cohomology first gives finite groups \(H^i(Q,H^j_{\mathrm{cts}}(H,M))\) in every bidegree, hence finite total cohomology in every degree. Taking \(Q\)-cohomology first gives \[E_2^{a,b}=H^a_{\mathrm{cts}}(H,H^b(Q,M)).\] All rows with \(b>0\) are finite by 33. The bottom term \(E_2^{a,0}=H^a_{\mathrm{cts}}(H,\Lambda)\) has no outgoing differential and only finitely many incoming differentials, each with finite image. Its surviving quotient is finite because the total cohomology is finite. Thus it is finite. These two spectral sequences compute the same first-quadrant complex: finite projective Hom is exact on the compact \(Q\)-cycle and boundary sequences, and the finite \(Q\)-cochain functors are finite products. No strictness assertion about the still unknown base-lattice cohomology was used.

Finally \(P_3/H=\mathbb Z_p^\times\) has finite cohomology on finite discrete modules: take exact invariants for \(\mu_{p-1}\) and the two-term resolution for \(1+p\mathbb Z_p\). The continuous Hochschild–Serre sequence for \(H\subset P_3\) now proves finite \(P_3\)-cohomology of \(\Lambda\). The residual action on its finite \(H\)-cohomology is continuous, since the coefficient lattice is compact and the finite-resolution cohomology has the quotient topology.

The connected marking trivializes all periodicity powers and the full norm kernel fixes determinant characters. The canonical-line and conormal identifications express the other indicated lines as products of these. An adjacent \(x\)-pole strip is a power-series lattice in \(y\) with precisely one such conormal and coefficient-line twist. It therefore satisfies the hypotheses just proved. ◻

Proof. Here \(\Delta_{\leq s}\) is the full simplex category on \([0],\ldots,[s]\), so its limit includes the codegeneracies. Let \(\mathcal P_s^\times\) be the finite poset of nonempty subsets of \(\{0,\ldots,s\}\). The functor \[\theta_s:\mathcal P_s^\times\longrightarrow\Delta_{\leq s}, \qquad J\longmapsto[|J|-1],\] sends an inclusion to the order-preserving injection between the ordered subsets. It is initial for limits. To verify the cofinality criterion, fix \([q]\), where \(q\leq s\). An object of \(\theta_s\downarrow[q]\) is a nonempty subset \(J\) together with a weakly increasing map \(J\to\{0,\ldots,q\}\); its morphisms are extensions of these partial maps. Thus this comma category is the nonempty face poset of the simplicial complex \(P(s,q)\) whose simplices are the lists \[(i_0,j_0),\ldots,(i_b,j_b),\qquad 0\leq i_0<\cdots<i_b\leq s, \quad 0\leq j_0\leq\cdots\leq j_b\leq q.\] Its nerve is the barycentric subdivision of \(P(s,q)\).

The complex \(P(s,q)\) is contractible whenever \(s\geq q\). The base case \(P(0,0)\) is a point. For the induction, the simplices containing the vertex \((s,j)\), together with their faces, form the cone on \(P(s-1,j)\). Start with the cone with apex \((s,q)\) on \(P(s-1,q)\); this includes all simplices with first coordinates less than \(s\) and is contractible even if its base is not. Attach the cones with apices \((s,j)\), for \(j=q-1,\ldots,0\). The intersection of each such cone with the preceding union is exactly its base \(P(s-1,j)\): a simplex cannot contain two vertices with first coordinate \(s\). Every attaching base is contractible by induction, since \(j<q\leq s\) implies \(j\leq s-1\). These are inclusions of finite simplicial subcomplexes, hence cofibrations; attaching a cone along a contractible subcomplex preserves contractibility. This proves the claim, including the endpoint \(q=s\), where no assertion about the contractibility of the initial base \(P(s-1,s)\) was used. The comma categories are therefore contractible, proving initiality of \(\theta_s\).

It follows that \(\operatorname{Tot}_s C^\bullet\) is the limit of the punctured finite \((s+1)\)-cube \[J\longmapsto\bigotimes_{i=0}^s \begin{cases}E,&i\in J,\\ \mathbf1,&i\notin J,\end{cases} \qquad \varnothing\ne J\subseteq\{0,\ldots,s\},\] whose maps insert units. Successive fibers and exactness of tensor product identify its augmentation fiber as the arrow in (62). Omitting the last cubical direction gives the stated transition. Thus the model and its transition refer to full partial totalizations, not a coface-only truncation. An exact functor preserves this finite cubical limit. The finite-cube formula appears in (Mathew et al. 2017, Proposition 2.14); the comparison with full partial totalizations and the augmentation fiber have been proved here in the stated setting.

For a descendable \(E\), the partial Amitsur tower represents the constant pro-object with value the unit (Mathew 2016, Proposition 3.20). Its limit is preserved by every finite-limit-preserving functor (Mathew 2016, Proposition 3.10). Together with the finite cubical comparison, this proves the asserted identification of the transformed totalization. There is also an integer \(N\) such that every composite of \(N\) maps which become null after tensoring with \(E\) is null (Mathew 2016, Proposition 3.27). One can read this bound from a finite thick-ideal construction of the unit: it is one for \(E\), is unchanged by retracts and tensor products, and adds across a cofiber sequence. The arrow \(I_E\to\mathbf1\) becomes null after tensoring with \(E\): its tensor is the composite through the null map \(I_E\to E\), followed by multiplication. Thus the error transitions in (62) become null after tensoring with \(E\), and their \(N\)-fold composites are null. An exact functor preserves those null composites, supplying the same convergence bound after applying the functor. ◻

Proof. We begin with the standard coefficient field, keeping the scalar extension separate. Set \(R_n=L_{K(n)}S\), \(E_F=E_n(F)\), \(\Gamma_n=\mathop{\mathrm{Gal}}(F/\mathbb F_p)\), and \(G_n^0=P_n\rtimes\Gamma_n\). The Hopkins–Ravenel theorem in the form of (Mathew 2016, Theorem 4.18) says that \(E_F\) generates the unit \(L_n\mathbb S_{(p)}\) as a thick tensor ideal in the \(E(n)\)-local category. Applying the symmetric monoidal localization \(L_{K(n)}\) preserves this finite generation statement (Mathew 2016, Corollary 3.21). The map \(\mathbb S_{(p)}\to S\) is a mod-\(p\) equivalence, hence a \(K(n)\)-equivalence for \(n\geq1\), so the resulting unit is \(R_n\). The same localized base case is stated directly in (Mathew 2016, Proposition 10.10) for Morava theory attached to a perfect residue field; the relative assertion still needs the following scalar step. Now base change along \(R_n\to R_{n,k}\) inside the \(K(n)\)-local category. Thus \[B=E_F\widehat\otimes_{R_n}R_{n,k}\] is descendable over \(R_{n,k}\).

We identify one factor of this base change and prove that it is itself descendable. A \(\mathbb Z_p\)-basis of \(W(k)\) gives an equivalence \(R_{n,k}\simeq R_n^{\vee [k:\mathbb F_p]}\) of underlying \(R_n\)-modules. More precisely, the natural map \(R_n\otimes_S S_k\to R_{n,k}\) is an equivalence: its source is a finite sum of \(K(n)\)-local copies of \(R_n\), and localizing the finite free decomposition of \(S_k\) gives the same map. The natural scalar map therefore identifies the graded coefficient ring of \(B\) with \(E_{F,*}\otimes_{\mathbb Z_p}W(k)\); no infinite limit is interchanged with scalar extension here. The finite étale algebra of constants splits as \[ W(F)\otimes_{\mathbb Z_p}W(k) \xrightarrow{\ \cong\ } \prod_{\sigma:F\hookrightarrow k}W(k), \qquad a\otimes b\longmapsto(\sigma(a)b)_\sigma. \tag{64}\] Let \(e_\sigma\) denote its embedding idempotents. Idempotent localization splits \(B\) into the finite product of \(B_\sigma=B[e_\sigma^{-1}]\): on homotopy groups the map to this product is the displayed orthogonal-idempotent decomposition, hence is an equivalence. Each factor is even periodic, with degree-zero ring \(W(k)[[u_1,\ldots,u_{n-1}]]\). Its orientation is the universal Honda deformation after the indicated unramified base change. We may therefore take the factor for the chosen inclusion \(\iota:F\subseteq k\) as the model \(E_n(k)\) used here.

For \(\gamma\in\Gamma_n\), the Galois automorphism of the standard factor \(E_F\), base changed with the identity on \(R_{n,k}\), is an \(R_{n,k}\)-algebra automorphism of \(B\). It permutes the idempotents by \(e_\sigma\mapsto e_{\sigma\gamma^{-1}}\). In particular it identifies all the factors \(B_\sigma\) as \(R_{n,k}\)-algebras while the second scalar action stays fixed. This is the factor equivalence being used; coefficient Frobenius on the chosen \(E_n(k)\) alone is semilinear over \(W(k)\). The underlying module of \(B\) is consequently a finite sum of modules equivalent to \(E_n(k)\). The thick tensor ideal generated by \(E_n(k)\) contains \(B\), and the ideal generated by \(B\) contains the unit. This proves descendability of the chosen factor.

Apply 49 in the \(K(n)\)-local category of \(S_k\)-modules, with its local tensor product and unit \(R_{n,k}\). The inclusion into spectra is exact, as are the ordinary tests in the statement. The lemma therefore proves the asserted totalization and uniform error-tower bound for the relative Amitsur object and for each of those tests. We now keep the same \(B\) and its embedding idempotents to identify the terms and maps of that object.

It remains to prove (63) with its maps. The completed-cooperations theorem of Devinatz–Hopkins applies to the standard \(E_F\) and the standard extended group \(G_n^0\): (Devinatz and Hopkins 2004, Proposition 2.2 and formulas (2.5)–(2.7)) identifies \[\pi_*\bigl(E_F^{\widehat\otimes_{R_n}(q+1)}\bigr) \cong C_{\mathrm{cts}}((G_n^0)^q,E_{F,*}).\] The tensor on the left is the \(K(n)\)-local tensor; with its local unit \(R_n\) it is the completed smash appearing in that theorem. Base change gives a canonical equivalence of actual terms \[B^{\widehat\otimes_{R_{n,k}}(q+1)} \simeq E_F^{\widehat\otimes_{R_n}(q+1)} \widehat\otimes_{R_n}R_{n,k}.\] Because \(R_{n,k}\) is finite free over \(R_n\), the preceding coefficient identification becomes \[ \pi_*\bigl(B^{\widehat\otimes_{R_{n,k}}(q+1)}\bigr) \cong C_{\mathrm{cts}}((G_n^0)^q,E_{F,*})\otimes_{\mathbb Z_p}W(k) \cong C_{\mathrm{cts}}((G_n^0)^q,B_*). \tag{65}\] The last isomorphism uses a finite basis of \(W(k)\), so it only commutes continuous functions with a finite direct sum.

We spell out the coordinate convention because it controls the first coface and the field component. Write the action of \(G_n^0\) on \(B\) for its action on \(E_F\) tensored with the identity on \(R_{n,k}\). In inhomogeneous coordinates the evaluation of a class in the left side of (65) at \((g_1,\ldots,g_q)\) is induced by the actual multiplication map \[ \mu\circ\bigl(1\widehat\otimes g_1\widehat\otimes (g_1g_2)\widehat\otimes\cdots\widehat\otimes (g_1\cdots g_q)\bigr): B^{\widehat\otimes_{R_{n,k}}(q+1)}\longrightarrow B. \tag{66}\] For \(q=0\) this is the identity. For \(q\geq1\), the evaluation in Devinatz–Hopkins applies \(h_i^{-1}\) to the first \(q\) factors and leaves the last factor fixed. Put \(h_i=g_i\cdots g_q\) and then apply \(h_1\) to the value. This continuous change of variables gives exactly (66). Their natural evaluation formulas apply to every homotopy class, so this argument does not assume that products of factor coefficients generate the completed term.

The projection \(B\to E_n(k)=B_\iota\) is a unital algebra map. In degree \(q\), the induced projection to \(\mathcal N_n^q\) is the all-\(\iota\) idempotent summand of the tensor product. Under (66), its projector acts on a function by multiplication by \[e_\iota\,g_1(e_\iota)\,(g_1g_2)(e_\iota)\cdots (g_1\cdots g_q)(e_\iota).\] The action of \(\Gamma_n\) on the embedding idempotents is free and transitive, and its kernel in \(G_n^0\) is \(P_n\). This product is \(e_\iota\) precisely when every \(g_i\) lies in \(P_n\), and is zero otherwise. Since \(P_n^q\) is clopen in \((G_n^0)^q\), the selected summand is exactly \(C_{\mathrm{cts}}(P_n^q,(B_\iota)_*)\). The \(P_n\)-action on the selected coefficient ring fixes its \(W(k)\) scalars. This proves (63).

The same evaluation maps identify all cobar maps. For a coefficient function \(f\) in degree \(q\), their formulas are \[\begin{aligned} (d^0f)(g_1,\ldots,g_{q+1}) &=g_1\bigl(f(g_2,\ldots,g_{q+1})\bigr),\\ (d^if)(g_1,\ldots,g_{q+1}) &=f(g_1,\ldots,g_i g_{i+1},\ldots,g_{q+1}) &&(1\leq i\leq q),\\ (d^{q+1}f)(g_1,\ldots,g_{q+1}) &=f(g_1,\ldots,g_q),\\ (s^if)(g_1,\ldots,g_{q-1}) &=f(g_1,\ldots,g_i,1,g_{i+1},\ldots,g_{q-1}) &&(0\leq i<q). \end{aligned}\] They follow by inserting a unit or multiplying adjacent factors in (66). The unital projection from the \(B\)-Amitsur object to the \(E_n(k)\)-Amitsur object commutes with these maps. Thus its selected cofaces are obtained by projecting the actual cofaces and then restricting the displayed formulas to \(P_n\). Their higher compatibilities are those of that actual Amitsur object. The augmented unit evaluates to the constant unit. The first coface is \(R_{n,k}\)-linear and has the displayed varying stabilizer action on coefficients; the other cofaces and all codegeneracies are linear for the first-factor \(E_n(k)\)-module structures. This proves the claims about maps and naturality. ◻

Proof. For each power \(\mathfrak m^a\) of \(\mathfrak m=(p,x,y)\), coefficient reduction gives \[C_{\mathrm{cts}}(Q,\widetilde A)/\mathfrak m^a \cong C_{\mathrm{lc}}(Q,\widetilde A/\mathfrak m^a).\] Surjectivity follows by lifting on a finite clopen partition. For the kernel, partition the monomials in \(p,x,y\) among the finitely many monomial generators of \(\mathfrak m^a\); the resulting division operations on coefficients are continuous and write a kernel function as a sum of these generators times continuous functions. The right side is the filtered colimit of finite free modules associated to finite clopen partitions. It is flat over the Artinian local ring \(\widetilde A/\mathfrak m^a\).

Choose a basis of the residue module over \(k\), lift the basis elements, and let \(F\) be the free \(\widetilde A\)-module on those lifts. An element of \(\widehat F\) is a family of coefficients for which, modulo each \(\mathfrak m^a\), only finitely many are nonzero. Consequently \(\widehat F/\mathfrak m^a\) is the free \(\widetilde A/\mathfrak m^a\)-module on the chosen basis; continuous monomial division gives the kernel equality used here. The map \(\widehat F\to C_{\mathrm{cts}}(Q,\widetilde A)\) is an isomorphism modulo \(\mathfrak m\). It is an isomorphism modulo every \(\mathfrak m^a\): both modules there are flat, and successive nilpotent-ideal reduction proves injectivity and surjectivity. Taking the inverse limit proves pro-freeness. For the ordinary flatness conclusion, \(\widetilde A=W(k)[[x,y]]\) is a regular complete Noetherian local ring, and \(\mathfrak m=(p,x,y)\) is generated by the minimal regular sequence \(p,x,y\). The module \(F\) is free, hence flat, and \(F/\mathfrak mF\) is free over \(\widetilde A/\mathfrak m\). Thus (Barthel and Stapleton 2016, Proposition 3.13) applies and makes the ordinary \(\mathfrak m\)-adic completion \(\widehat F\) flat. The pro-free characterizations are also described in (Hovey and Strickland 1999, Theorem A.9 and Proposition A.13).

Now use the ordinary module tensor identity \[E_2(k)\wedge N_Q \simeq (E_2(k)\wedge E_3(k))\otimes_{E_3(k)}N_Q.\] The module Künneth spectral sequence has no positive Tor terms by the flatness just proved. Periodicity reduces graded modules over this coefficient ring to parity-graded modules over the regular local ring \(W(k)[[x,y]]\), of global dimension three. A projective resolution of length at most three therefore gives bounded convergence of this Künneth calculation. This proves (67). ◻

Proof. Before imposing the residue ideal, both Morava theories are Landweber exact over \(BP\): the sequence \(p,u_1,\ldots,u_{n-1}\) is regular and the height-\(n\) coefficient is a unit, and these facts are retained by finite unramified scalar extension. Landweber exactness gives the homology base-change formula for all spectra, as recalled in the introduction and Example 0.1(d) of (Hovey and Strickland 2005). Applying it first to \(E_2(k)_*(E_3(k))\), then to \(E_3(k)_*(BP)\) and using the symmetry of ordinary smash, gives \[ (E_2(k))_*E_3(k) \cong (E_2(k))_*\otimes_{BP_*}BP_*BP \otimes_{BP_*}(E_3(k))_*. \tag{68}\] The two tensor products use the two unit maps of \(BP_*BP\). The strict \(p\)-typical formal-law groupoid represented by \((BP_*,BP_*BP)\), including its units and composition, is recalled in (Bhattacharya and Egger 2019, sec. 2, Lemma 2.9 and the following discussion, p. 7). Naturality of the two multiplicative orientations identifies these base-changed operations with those of the oriented formal isomorphisms. This is an absolute ordinary cooperation calculation, which is precisely the first factor in (67).

We also record the needed flatness over each coefficient factor. For a Landweber-exact \(BP_*\)-algebra \(B\), (Hovey and Strickland 2005, Lemma 2.2) makes \(B\otimes_{BP_*}BP_*BP\) flat for its other \(BP_*\)-action. Hopf-algebroid conjugation gives the analogous assertion for \(BP_*BP\otimes_{BP_*}B\). Applying these two assertions with \(B=(E_2(k))_*\) and \(B=(E_3(k))_*\), and then base changing the remaining unit, proves that (68) is flat over either coefficient factor. Hence the sequence \((p,u_1)\) on the height-two factor acts regularly on these cooperations. Tensoring over the height-three factor with the flat module in 51 preserves that regularity. The two residue cofibers therefore create no odd homotopy in the absolute smash product.

We next pass from this absolute residue calculation to the relative test in the statement. Let \[C=\mathop{\mathrm{cofib}}(\mathbb S_{(p)}\longrightarrow S).\] The completion map is a mod-\(p\) equivalence, so \(C/p=0\). Thus multiplication by \(p\) is invertible on \(C\), and the \(p\)-local spectrum \(C\) is rational. The spectrum \(\overline E_2\) is rationally acyclic, already after its first residue cofiber. Hence \(\overline E_2\wedge C=0\). The unit map \(\overline E_2\to\overline E_2\wedge S\) is an equivalence of spectra. Its composite with the action map \(\overline E_2\wedge S\to\overline E_2\) is the identity. The action map is therefore an equivalence of right \(S\)-modules. Consequently, for every \(S\)-module \(N\), there is a natural equivalence \[ \overline E_2\wedge N \simeq(\overline E_2\wedge S)\otimes_S N \simeq\overline E_2\otimes_S N. \tag{69}\]

For \(S_k\)-modules, the two scalar actions on the last object give an action of \[S_k\otimes_S S_k \simeq\prod_{\sigma\in\mathop{\mathrm{Gal}}(k/\mathbb F_p)}S_k.\] The multiplication map \(S_k\otimes_S S_k\to S_k\) is projection to the identity factor. Indeed, balancing the two scalar actions is \[(\overline E_2\otimes_S N) \otimes_{S_k\otimes_S S_k}S_k \simeq \overline E_2\otimes_{S_k}N.\] Thus the identity idempotent selects \(\overline E_2\otimes_{S_k}N\). On the absolute residue coefficient algebra already calculated, this is the equal-embeddings summand of \(k\otimes_{\mathbb F_p}k\simeq\prod_\sigma k\). It is a retract and therefore remains even. Applying this to \(N=C_{\mathrm{cts}}(P_3^q,E_3(k))\) gives the required relative term. The comparison (69) and the identity idempotent are natural for \(S_k\)-linear maps. Every \(P_3\)-cobar coface is \(S_k\)-linear, so these identifications commute with all cofaces and with the constant unit. Regularity was needed only for the absolute calculation before the residue cut.

In this identity summand, the coefficients are described by the formal-group-isomorphism Hopf algebroid over \(k\). Under an isomorphism of formal groups the ideals \((p,v_1)\) agree on the two sides. Thus this cut is \(p=x=0\) on the height-three factor. The height-two \(v_2\) is a unit, so \(y\) is a unit as well. After fixing the test periodicity generator, the conversion from graded to ungraded coordinates retains the invertible leading coefficient of the isomorphism (Bhattacharya and Egger 2019, (2.10)–(2.11) and (2.15)–(2.16), pp. 7–9). For the oriented isomorphism used here, write \(aT^{p^2}\) and \(bT^{p^2}\) for the first nonzero terms of the two \(p\)-series and \(rT\) for the linear term of the isomorphism. The equation intertwining the \(p\)-series gives \(ra=br^{p^2}\), or \(r^{p^2-1}=a/b\). Both \(a\) and \(b\) are units, and \(p^2-1\) is prime to \(p\), so this is the finite étale equation identifying the nonzero height-two coefficients. The remaining equations are precisely the coefficients of an isomorphism from the specialized law to the Honda law.

The finite connected-marking stages of [h3:thm:coordinate-comparison,h3:prop:integral-frames] represent these truncated equations. Their union represents the full isomorphism algebra; every polynomial in the cooperation coordinates belongs to a finite such stage. Thus the ordinary algebraic tensor product gives this algebraic union, without its valuation completion.

It remains to check the parameter topology. Reduction of continuous power-series functions modulo \((p,x)\) gives continuous functions into \(k[[y]]\): coefficient lifting proves surjectivity, and continuous monomial division by \(p,x\) identifies the kernel with \((p,x)C_{\mathrm{cts}}(Q,\widetilde A)\). There is an equality \[ C_{\mathrm{cts}}(Q,k[[y]])[y^{-1}] = C_{\mathrm{cts}}(Q,k((y))). \tag{70}\] Indeed, the image of compact \(Q\) in the local field is bounded, so multiplication by one power of \(y\) puts that image in \(k[[y]]\). The reverse inclusion in (70) is immediate. A finite étale algebra over \(k((y))\) is a finite product of finite-dimensional complete vector spaces over it; a basis gives the identical assertion at each finite marking stage. Consequently the final union is a colimit on continuous cochain complexes with a common stage for every compact family.

Lastly, all the identifications were made in the isomorphism Hopf algebroid. Its composition law transports the tautological marking and is exactly the coface action. The identity arrow gives the constant test-unit. This proves the assertions about the maps, not just the coefficient modules. ◻

Proof. The unit and the integral class just constructed are maps out of \(S\). Thus they define \(w\) by applying \(L_2\); no extension of a map defined only on \(R_2\) is being asserted.

Use [h3:prop:exact-descent,h3:des:residue-cooperations] over \(S_k\). By 32, the actual coefficient augmentation gives \[R\Gamma_{\mathrm{cts}}(H',L)\simeq k.\] It retains compact parameters and the action of \(P_3/H'\). This quotient is \(\mathbb Z_p\), with trivial action on \(k\). The continuous group-ring resolution \[0\longrightarrow k[[\mathbb Z_p]] \xrightarrow{\gamma-1}k[[\mathbb Z_p]] \longrightarrow k\longrightarrow0\] therefore shows that the residue-tested spectral sequence has only columns zero and one. They are generated by the constant unit and the normalized logarithm. There is no possible differential, and the specified maps \(1,\zeta\) detect these two generators by the coface identification in 52.

Because \(L_2\) is smashing and \(\overline E_2\) is \(E(2)\)-local, the ordinary relative tests satisfy \[\overline E_2\otimes_{S_k}(W\otimes_S S_k) \simeq \overline E_2\vee\Sigma^{-1}\overline E_2, \qquad \overline E_2\otimes_{S_k}(X\otimes_S S_k) \simeq \overline E_2\otimes_{S_k}R_{3,k}.\] The preceding two detected classes therefore prove that the test of \(w\otimes_S S_k\) is an equivalence. By associativity this test is \(\overline E_2\otimes_S w\), and (69) identifies it with the absolute ordinary residue test of \(w\). The residue theory \(\overline E_2\) has Bousfield class \(K(2)\); its coefficient ring is a graded field after a finite field extension. Consequently \(w\) is a \(K(2)\)-equivalence. Applying \(L_{K(2)}\) to \(w\) therefore gives an equivalence with source \(R_2\vee\Sigma^{-1}R_2\) and target \(L_{K(2)}T\). The scalar passage preserved the two original \(S\)-maps by the naturality of the test and the unit-compatible finite-field descent. Its components are consequently the specified unit and \(\zeta\). ◻

Proof. By 17, the finite scheme of splittings of the ordinary connected–étale sequence at level \(p^\ell\) is a torsor under \[\underline{\mathop{\mathrm{Hom}}}(\mathcal E[p^\ell],\mathcal C[p^\ell]),\] a form of \(\mu_{p^\ell}^2\). Its algebra is \(B_0^{1/p^\ell}\) over \(B_0\). This includes the divisor \(y=0\). A unit \(p\)-basis on this open is \(x,1+y\); adjoining a root of \(1+y\) also adjoins the corresponding root of \(y\). Thus multiplicative torsor coordinates can be units throughout the open. The additive monomial basis \(x^iy^j\) used below only expresses the matrix of the resulting operator; its equivariance comes from the splitting torsor. We recall why the construction supplies a retraction without a choice of a splitting. After an étale cover the group is diagonalizable, and a comodule is graded by its character group. Projection onto the degree-zero summand is linear over the invariant algebra. Every comodule map preserves this summand. Changing the étale frame permutes characters and fixes zero, so the projections descend by faithfully flat descent. The invariant subalgebra of a torsor is the base algebra, and its inclusion is fixed by this projection. This proves the required retraction. It is an additive, base-linear projection; no multiplicativity or preservation of the ideal \((y)\) is used. The construction commutes with automorphisms of the ordinary sequence, hence with \(G\) and with coefficient-field automorphisms. It uses no division by the order of \(\mu_{p^\ell}^2\).

For the untwisted coefficients choose \(q=p^e\) with \(q\) acting trivially on \(k\), and let \(\rho_1:B_0^{1/q}\to B_0\) be the corresponding retraction. Set \(\mathcal P(f)=\rho_1(f^{1/q})\) and \(\mathrm{Fr}(f)=f^q\). Then \(\mathcal P(g^qf)=g\mathcal P(f)\) and (72) holds. On the level \(B_0^{1/q^n}\) define \[ \rho_n(f^{1/q^n})=\mathcal P^n f. \tag{73}\] These maps are \(B_0\)-linear. They agree on overlapping levels because \(\mathcal P^{n+1}(f^q)=\mathcal P^nf\). Thus a single natural root retraction already supplies the whole compatible system.

Here is also the construction for the actual line twists. If \(\omega\) denotes the periodicity line, the invariant height-one section \(v_1\) trivializes \(\omega^{p-1}\) on this open. In a free frame it is \(x\) times that frame to the power \(p-1\), with the exponent reversed if the dual convention for \(\omega\) is used. Adjoin a \((p-1)\)st root of this section. This is a finite étale frame cover on the open, and the resulting horizontal frame has finite tame transition characters. Include the finite determinant character and choose \(q\) to fix all these characters and the constants. The \(q\)-power map on coefficients in that frame then descends to the line, as does the base-changed root retraction. Projection onto the required tame eigensummand is exact, since its order is prime to \(p\). On the \(n\)th root level of a line, the intrinsic version of (73) is \(\rho_n=\mathcal P^n\mathrm{Fr}^n\); thus its notation does not require taking a root of a chosen line generator.

In the original rational frame the coefficients in a horizontal frame have \(x\)-exponents in a fixed translate \(\lambda+\mathbb Z\), with \((q-1)\lambda\in\mathbb Z\). More explicitly, write the horizontal frame as \(h=x^{-\lambda}e_0\), where \(e_0\) is the original line frame, and put \(\kappa=(q-1)\lambda\). Then \[\mathrm{Fr}(fe_0)=f^q x^\kappa e_0.\] Thus Frobenius multiplies normalized order by \(q\). For line coefficients we use these normalized Gauss valuations: relative to the original frame they add the fixed number \(a\lambda\) to \(v_{a,b}\). They define the same topology, and satisfy \(v_{a,b}(\mathrm{Fr}f)=qv_{a,b}(f)\) exactly. Choosing \(r+\lambda\geq0\) makes \(x^r\mathcal L_A\) Frobenius-stable. This also shows explicitly that passing to the tame frame introduces only a fixed shift of pole orders. The continuity, extension to completion, and assertion about \(A_D\) follow from the estimates proved next; these estimates apply to the descended eigensummand as well. ◻

Proof. Let \(e_0\) be the original free line frame, and temporarily write \(w_{a,b}\) for the unshifted scalar Gauss valuation. Every scalar coefficient has the unique decomposition \[f=\sum_{0\leq i,j<q}x^iy^j f_{ij}^{\,q}.\] The inverse-semilinear operator is consequently given by a finite matrix, here a single row: \[\mathcal P(fe_0)=\left(\sum_{0\leq i,j<q} b_{ij}f_{ij}\right)e_0, \qquad b_{ij}\in B_0.\] The terms of the decomposition have disjoint exponent residue classes, so \[w_{a,b}(f)=\min_{i,j}\{ai+bj+qw_{a,b}(f_{ij})\}.\] Put \[c^0_{a,b}=\max\left(0, \max_{i,j}\left\{\frac{ai+bj}{q}-w_{a,b}(b_{ij})\right\}\right).\] The scalar output has valuation at least \(q^{-1}w_{a,b}(f)-c^0_{a,b}\). Since \(v_{a,b}(fe_0)=w_{a,b}(f)+a\lambda\), its normalized valuation is at least \[q^{-1}v_{a,b}(fe_0)-c^0_{a,b}+a\lambda(1-q^{-1}).\] Thus (75) holds, for example, with \(c_{a,b}=c^0_{a,b}+a|\lambda|(1-q^{-1})\). All \(b_{ij}\) have one common finite pole bound. For \(f\in A_L\), its components have pole at most \(L/q+(q-1)/q\). This proves (74) with a constant independent of \(L\). Each iteration divides the previous loss by \(q\), giving the displayed geometric sums and proving (76) and (77).

The component decomposition and finite matrix are continuous on \(B_{\mathrm{hol}}\): taking an exponent residue class and then a root preserves convergence in every defining norm. On a root level the last inequality reads \[v_{a,b}(\rho_n h)\geq v_{a,b}(h)-C_{a,b}.\] Thus the compatible maps extend continuously from the dense root union to \(B_{\mathrm{pk}}\), with values in the complete space \(B_{\mathrm{hol}}\). Their common restriction is the identity on \(B_{\mathrm{hol}}\). Finally \(D>C\) implies \(D/q+c<D\), proving stability of \(A_D\). The proof on the tame frame cover uses its finitely many character components and the same finite matrix calculation. Descent therefore retains both estimates for every specified twist. ◻

Proof. The retraction induces a split injection on cohomology into the finite groups of 30. Hence holomorphic cohomology is finite before any bounded-pole assertion is used. The identity \(\mathcal P\mathrm{Fr}=1\) makes Frobenius injective on these finite groups, hence bijective. Both cochain complexes are Polish, so their finite cohomology groups are discrete by 34.

For completeness, consider a completed class, rather than merely a class in the algebraic root union. On perfected coefficients define \[R_\ell=\mathrm{Fr}^{-\ell}\rho\mathrm{Fr}^{\ell}: B_{\mathrm{pk}}(\mathcal L)\longrightarrow B_{\mathrm{hol}}(\mathcal L)^{1/q^\ell}.\] This is a projection onto the indicated root level and satisfies \[ v_{a,b}(R_\ell f)\geq v_{a,b}(f)-q^{-\ell}C_{a,b}. \tag{78}\] It converges to the identity uniformly on compact subsets in every defining norm. Indeed, approximate a compact subset, at any prescribed accuracy in finitely many norms, by finitely many elements at a common finite root level. For every later \(\ell\), \(R_\ell\) fixes these approximants. The bound (78) controls the errors uniformly; increasing the accuracy proves the assertion.

If \(z\) is a continuous completed cocycle, its image is compact. Thus \(R_\ell z\to z\) in the cochain topology. These are cocycles because the projections are equivariant. Discreteness of completed cohomology gives \([R_\ell z]=[z]\) for all sufficiently large \(\ell\). Every completed class therefore comes from a finite root level. Under Frobenius transport, the image of this level is the image of holomorphic cohomology, because Frobenius is already invertible on that image. This proves surjectivity and hence the comparison.

The finite-cohomology lifting assertion of 35 now applies to these two Polish complexes. It proves the comparison with any profinite parameters. All the maps used to define the comparison are the natural inclusions; the retractions prove their properties. In particular the comparison preserves products and the original constants maps. Naturality of the torsor construction also proves its coefficient-field equivariance. ◻

Proof. Choose \(0<\delta<D-C\) and set \(L_n=\lfloor\delta q^n\rfloor\). The pole estimate implies \[\mathcal P^n(1-T_{L_n})f\in A_D(\mathcal L),\qquad T_D\mathcal P^nf=T_D\mathcal P^nT_{L_n}f.\] Fix \(a,b>0\). For any \(R>0\), compactness gives a finite lower bound \[m_R:=\inf_{f\in K}v_{a+R,b}(f)>-\infty.\] A monomial with pole \(i>L\) has normalized \(v_{a,b}\)-value equal to its \(v_{a+R,b}\)-value plus \(R(i-\lambda)\). Consequently \[\inf_{f\in K}v_{a,b}(T_{L_n}f)\geq m_R+R(L_n-\lambda).\] Truncation cannot lower a Gauss valuation. By the iterated norm bound, \[ \inf_{f\in K}v_{a,b}(T_D\mathcal P^nf) \geq q^{-n}(m_R-R\lambda)+Rq^{-n}L_n-C_{a,b}. \tag{79}\] The lower limit is at least \(R\delta-C_{a,b}\). Since \(R\) can be arbitrarily large, it is \(+\infty\). This includes \(\lambda=0\) for the untwisted line. ◻

Proof. Compactness, Hausdorffness and the asserted cohomology topology follow from the finite completed resolution in 33. The continuous map \(\phi\) has finite discrete target by 56, so \(K=\ker\phi\) is compact. For \(a=0\) it is zero, since the coefficient inclusion is injective. For \(a>0\), represent \(z\in K\) by an \(A_D\)-valued cocycle \(z_0\) and write \(z_0=\partial b\) with a holomorphic continuous cochain \(b\). Set \[b_n=(1-T_D)\mathcal P^nb,\qquad e_n=\mathcal P^nz_0-\partial b_n =\partial(T_D\mathcal P^nb).\] The first expression shows that \(e_n\) is an \(A_D\)-valued cocycle. The image of \(b\) is compact, so 57 makes \(T_D\mathcal P^nb\) tend uniformly to zero. The group differential preserves this convergence in total-degree order: every group element preserves \((x,y)\), sends \(x\) to \(x\) times a unit and sends a line generator to a unit multiple. Thus its action does not decrease total-degree order in a fixed rational frame. The finite sum defining \(\partial\) has the same property. It follows that \(e_n\to0\) in the compact lattice topology, and \(\mathcal P^nz=[e_n]\to0\) in \(M\).

We give the uniformity step explicitly. For an open additive subgroup \(U\) of \(K\), let \[K_N(U)=\bigcap_{n\geq N}\{z\in K:\mathcal P^nz\in U\}.\] These are increasing closed subgroups covering \(K\) by the convergence just proved. The Baire theorem makes one of them have interior; as a subgroup it is open and has finite index. Choose representatives for its finitely many cosets. Each representative has all sufficiently late iterates in \(U\). Taking the largest of these finitely many bounds gives \(\mathcal P^nK\subseteq U\) for all sufficiently large \(n\). Hence \[ \bigcap_{n\geq0}\mathcal P^nK=0. \tag{80}\]

The operator \(\mathcal P\) is bijective on holomorphic cohomology: it is the inverse of Frobenius there by finiteness and (72). If \(z\in M_\infty\cap K\), choose, for each \(n\), an \(m_n\in M\) with \(\mathcal P^nm_n=z\). Applying \(\phi\) and using this bijectivity gives \(m_n\in K\). Equation (80) therefore forces \(z=0\). Thus \(M_\infty\) injects into the finite holomorphic cohomology group, which proves its finiteness. ◻

Proof. Each adjacent quotient \(A_j(\mathcal L)/A_{j-1}(\mathcal L)\) is a free \(k[[y]]\)-line. Its action is the restriction of \(\mathcal L\) tensored with the appropriate power of the conormal of \(x=0\). That conormal is a periodicity power. Thus 47 says that every such quotient has finite \(G\)-cohomology. Finite strips of adjacent quotients have finite cohomology as well, by their finite filtration. The sequences are exact on continuous cochains: the finite Laurent-coordinate truncations give continuous nonequivariant sections.

Choose \(A_{-r}(\mathcal L)\subseteq A_D(\mathcal L)\) with the former Frobenius-stable and the latter \(\mathcal P\)-stable, as in 54. If \(z\in H^a(G,A_{-r}(\mathcal L))\), then its image in \(M\) satisfies \[\iota(z)=\mathcal P^n\iota(\mathrm{Fr}^nz) \quad\hbox{for every }n.\] It therefore belongs to the finite subgroup \(M_\infty\) of 58. The kernel of this map is a quotient of \(H^{a-1}(G,A_D/A_{-r})\), which is finite by the strip calculation (and is zero for \(a=0\)). Hence \(H^a(G,A_{-r})\) is finite. The same strip sequence then proves finiteness for \(A_D\); further finite strips reach every \(A_d\), in either direction. All twists have been retained throughout. ◻

Proof. The height-three local term \(T/p\). Apply the finite Moore quotient to the relative object of 50. Its abutment is \(\pi_*(R_{3,k}/p)\), and its coefficient in internal degree \(2j\) is \(A\otimes\omega^j\); odd coefficients vanish. These compact modules have finite \(P_3\)-cohomology by 59. They are complete and separated with invariant open neighborhoods, so the reduced \(\mathbb F_p[[P_3]]\)-resolution of 33, which has length nine, computes their continuous cochains. Hence only columns \(0\leq s\leq9\) occur. The bounded convergence from 50 gives finitely many finite associated-graded groups in each total degree, so \(\pi_*(R_{3,k}/p)\) is degreewise finite. The underlying equivalence \(R_{3,k}/p\simeq(T/p)^{\vee [k:\mathbb F_p]}\) then gives finiteness for \(T/p\) over \(S\).

The monochromatic term \(M_3S/p\). The chromatic fracture square gives natural equivalences \[M_3S\simeq M_3T\simeq\mathop{\mathrm{fib}}(T\longrightarrow L_2T).\] The first identity is also (Hovey and Strickland 1999, Theorem 6.19). Put \(E=E_3(k)\), \(\widetilde A=\pi_0E=W(k)[[x,y]]\) and \(I=(p,x,y)\). The module-spectrum form of chromatic localization is needed here. For every ordinary \(E\)-module \(N\), including a completed continuous-function term, there is a natural fiber sequence \[\operatorname{Kos}_E(I)\otimes_E N\longrightarrow N \longrightarrow L_2N, \qquad \operatorname{Kos}_E(I)= \bigotimes_{z\in\{p,x,y\}}\mathop{\mathrm{fib}}(E\longrightarrow E[z^{-1}]).\] This is (Barthel and Stapleton 2016, Proposition 3.4(1) and Remark 3.5), with its support functor described in Lemmas 2.3 and 2.5 of that paper. The statement is about the underlying chromatic localization of arbitrary module spectra; it imposes no flatness or finite-generation condition. Since \(E\) is \(E(3)\)-local and \(L_3\) is smashing, every \(E\)-module is \(E(3)\)-local. Thus the first term is \(M_3N\). In particular it applies to the actual completed Amitsur spectrum \(N_Q=\mathcal N_3^q\) for \(Q=P_3^q\), equipped with (63) and viewed as an ordinary \(E\)-module; no completed tensor product is replaced by an ordinary one.

The spectrum \(N_Q/p\) has zero \(p\)-inversion, so its \(p\)-support factor is the whole spectrum. The remaining two factors give the finite Čech support complex for \((x,y)\). Its homotopy spectral sequence is \[H^s_{(x,y)}(\pi_t(N_Q/p)) \Longrightarrow \pi_{t-s}(M_3(N_Q/p)),\qquad 0\leq s\leq2.\] By 51, the even coefficient modules of \(N_Q/p\) are \(C_{\mathrm{cts}}(Q,A)\otimes\omega^{t/2}\), where \(A=k[[x,y]]\), and are flat over \(A\); the odd ones vanish. The regular sequence \(x,y\) therefore makes the displayed support cohomology vanish except for \(s=2\). For \(Q\) a point it is \[H^2_{(x,y)}A =A[x^{-1},y^{-1}]/(A[x^{-1}]+A[y^{-1}]).\] Thus the shift is exactly two. The finite support complex introduces no further convergence issue.

These identifications also respect the action-cobar maps. All faces other than the first, and all degeneracies, are \(E\)-linear; the first face uses the continuous stabilizer action. The invariant ideal \(I\) is preserved even for this varying family: in \(C_{\mathrm{cts}}(P_3^{q+1},\widetilde A)\) one has \((p,g_1(x),g_1(y))=(p,x,y)\). Indeed, the continuous monomial-division operations used in 51 express any element of \(I\) as a continuous sum of \(p,x,y\) multiples. Apply them to \(g_1^{-1}(x)\) and \(g_1^{-1}(y)\), then apply \(g_1\); joint continuity gives continuous coefficients expressing \(x,y\) in the image ideal. The reverse containment follows in the same way. For precision, let \(I_q\) be the image of \(I\) in \(\pi_0N_Q\). Base change of the displayed finite fiber construction from \(E\) to \(N_Q\) identifies the ordinary \(E\)-support term with \(\operatorname{Kos}_{N_Q}(I_q)\). Along an actual coface ring map \(N_Q\to N_{Q'}\), its further base change is the construction for the image generators in \(\pi_0N_{Q'}\). The support functor on ordinary \(N_{Q'}\)-modules is the canonical right adjoint for modules supported at that finitely generated ideal, so it depends on the ideal, not on the chosen generators. The equality of ideals just proved identifies the first-face image with the target support ideal. Uniqueness of this right adjoint makes the resulting comparisons compatible with the first face and with compositions. No individual generator is assumed fixed by the action. The calculation therefore applies to the whole completed Amitsur object, and 50 permits its termwise use before totalization.

For a continuous function term the same quotient is \[ H^2_{(x,y)}C_{\mathrm{cts}}(Q,A) =C_{\mathrm{lc}}(Q,H^2_{(x,y)}A), \tag{81}\] where the target coefficient module is discrete. To verify this, express \(H^2_{(x,y)}A\) as the colimit of \(A/(x^a,y^a)\), with transition multiplication by \(xy\). At each stage, quotienting continuous functions gives continuous functions to the finite discrete quotient, using coefficient truncation to lift them. A locally constant map from compact \(Q\) has finite image, so one common stage represents it. This proves (81) with the required denominator and parameter bounds.

Residue pairing, followed by \(\mathop{\mathrm{Tr}}_{k/\mathbb F_p}\), identifies the continuous dual of \(H^2_{(x,y)}(A\otimes\omega^j)\) with the compact module \[M=A\otimes\omega^{-j}\otimes {\textstyle\bigwedge^2}\Omega^1_{A/k,\mathrm{cts}}.\] Concretely the pairing is the coefficient of \(x^{-1}y^{-1}\) in a product with a two-form. It is perfect on finite principal-parts quotients and hence on their colimit against the inverse-limit power-series module. Its change-of-variables formula proves equivariance. The canonical-line identity of 16 makes this an admissible twist in 59. The principal-parts module is discrete and \(M\) is compact. The reversed compact/discrete duality (38), applied to \(P_3\) in dimension nine, therefore gives \[H^s_{\mathrm{cts}}\bigl(P_3,H^2_{(x,y)}(A\otimes\omega^j)\bigr) \cong H^{9-s}_{\mathrm{cts}}(P_3,M)^\vee.\] The right side is finite by 59, so the principal-parts cohomology is finite in every degree. The relative descent spectral sequence for \(M_3(R_{3,k}/p)\) is bounded in columns by the same reduced length-nine resolution and has finite entries in every total degree. It therefore has degreewise finite abutment. Its underlying spectrum is a sum of \([k:\mathbb F_p]\) copies of \(M_3(T/p)\), because the exact functor \(M_3(-/p)\) preserves the finite free decomposition of \(R_{3,k}\). Using \(M_3T\simeq M_3S\) proves finiteness of \(\pi_*(M_3S/p)\).

The overlap \(X/p\). The fiber sequence \(M_3S/p\to T/p\to X/p\) gives finiteness for \(X/p\).

The height-one localization \(L_1X/p\). By 53, \(L_{K(2)}X/p\) is the sum of two shifts of \(R_2/p\). The completion map identifies \(S/p\) with the finite torsion Moore spectrum \(\mathbb S_{(p)}/p\), and exactness of \(L_{K(2)}\) gives \[R_2/p\simeq L_{K(2)}(S/p).\] For \(p\geq5\), (Hovey and Strickland 1999, Theorem 15.1) applied at height two to this finite torsion spectrum says that every integer-graded homotopy group is finite; see also (Barthel and Beaudry 2020, 49(c)).

For the other height-two boundary put \(E=L_1R_2\). The completion map from the \(p\)-local sphere to \(S\) has rational cofiber, so it is a \(K(2)\)-equivalence. Thus \(R_2\) is the \(K(2)\)-local sphere in the group calculation of (Behrens 2012, Remark 7.8). We use the following features of that formula. Write \(d_v=|v_1|=2(p-1)\). Besides finitely many shifted copies of \(\mathbb Z_p\), \(\mathbb Q_p\), and \(\mathbb Q/\mathbb Z_{(p)}\), the formula contains finite cyclic \(p\)-groups with generators \(1_{sp^n/(n+1)}\), where \(n\geq0\), \(p\nmid s\), and the internal degree is \(d_vsp^n\). Its exterior factor on the degree-minus-one class denoted here by \(\zeta_{\mathrm B}\) adds only two shifts. This notation distinguishes that height-two class from the map \(\zeta\) constructed above.

For a fixed nonzero internal degree, division by \(d_v\), when possible, determines at most one integer of the form \(sp^n\) with \(p\nmid s\): \(n\) is its \(p\)-adic valuation and \(s\) its remaining \(p\)-unit integer. The cyclic family therefore contributes at most one summand in each such degree before the two exterior shifts. Multiplication by \(p\) has finite kernel and cokernel on each listed group: these pairs are \((0,\mathbb F_p)\), \((0,0)\), and \((\mathbb F_p,0)\) on \(\mathbb Z_p\), \(\mathbb Q_p\), and \(\mathbb Q/\mathbb Z_{(p)}\), respectively, and they are finite on a finite cyclic \(p\)-group. Hence both \(\pi_j(E)[p]\) and \(\pi_j(E)/p\) are finite for every integer \(j\). The Moore cofiber sequence gives \[0\longrightarrow\pi_j(E)/p\longrightarrow\pi_j(E/p) \longrightarrow\pi_{j-1}(E)[p]\longrightarrow0.\] It follows that \(\pi_j(E/p)\) is finite in every degree. Exactness identifies \(E/p\simeq L_1(R_2/p)\), the boundary needed here. The local height-two equivalence then identifies \(L_1L_{K(2)}(X/p)\) with two shifts of \(E/p\), so that corner is degreewise finite. Finally, apply the height-two fracture square to \(X/p\). Its other three corners are degreewise finite, so its long exact sequence proves finiteness of \(L_1X/p\). No bounded-pole comparison has been used in this argument. ◻

Proof. Apply \(L_1(-/p)\) to the completed relative descent object. We first identify this exact test on an actual term \(N_Q=\mathcal N_3^q\). For odd \(p\), an Adams self-map of the finite Moore spectrum has degree \(2(p-1)\) and acts nontrivially in \(K(1)\)-homology (Ravenel 1992, Example 2.4.1(ii)). Its action on \(BP_*(S/p)=BP_*/p\) is multiplication by an element of degree \(2(p-1)\), and that graded group is \(\mathbb F_p v_1\). This even-degree coefficient detects the entire \(BP\)-module map: applying maps into \(BP/p\) to \(BP\xrightarrow p BP\to BP/p\) shows that the only possible extra term is \(\pi_{2(p-1)+1}(BP/p)=0\). If the coefficient were zero, the \(BP\)-module map would be null, and so would its base change to \(K(1)\), contrary to the chosen Adams map. Its coefficient is therefore a unit multiple of \(v_1\). Rescaling the self-map by an integer unit makes its \(BP\)-Hurewicz action exactly \(v_1\). The height-three orientation sends this element to \(v_1=xU^{p-1}\) modulo \(p\). This unit rescaling does not change the telescope: powers of the unit give an isomorphism between the two defining towers.

The height-one telescope theorem for this finite Moore spectrum is the consequence of (Miller 1981, Theorem 4.11 and Corollary 4.12) identified as the \(n=1\) case in (Ravenel 1992, sec. 7.5, after Conjecture 7.5.5). Since \(L_1\) is smashing, it gives, for the already formed ordinary module \(N_Q\), \[L_1(N_Q/p)\simeq N_Q\wedge L_1(S/p) \simeq N_Q\wedge\operatorname{Tel}(v_1:S/p\to\Sigma^{-2(p-1)}S/p).\] Thus the test is a termwise ordinary telescope of the same finite Moore self-map. Homotopy groups commute with that telescope. In internal degree zero the resulting coefficient is exactly \[\mathop{\mathrm{colim}}_d x^{-d}C_{\mathrm{cts}}(P_3^q,A).\] Its other coefficients are the periodic translates. The actual action preserves every bounded-pole lattice modulo \(p\), because \(g(x)/x\) is a unit in \(A\). This is the prescribed cochain complex of \(B_0\); no completion of the algebraic localization is taken.

The central tame units act trivially on \(A\) and by their scalar powers on \(U\). Averaging over \(\mu_{p-1}\) kills internal degrees not divisible by \(2(p-1)\). Modulo \(p\), \(v_1\) is invariant, so multiplication by its powers identifies all the remaining coefficient complexes. For each compact lattice \(x^{-d}A\), the reduced finite \(\mathbb F_p[[P_3]]\)-resolution of 33 computes its continuous cochains and has length nine. Taking the exact filtered colimit of these complexes proves the same column bound for this prescribed bounded-pole union; it does not apply a compact-module theorem directly to the union. Thus the spectral sequence has \[E_2^{a,\,2b(p-1)} =H^a_{\mathrm{cts}}(P_3,B_0)\,v_1^b,\qquad 0\leq a\leq9,\] and no other nonzero internal degrees. The abutment is the relative one, \(\pi_*L_1(R_{3,k}/p)\). If \(m=[k:\mathbb F_p]\), the underlying finite free decomposition gives \[L_1(R_{3,k}/p)\simeq L_1(T/p)^{\vee m} \simeq L_1(X/p)^{\vee m}.\] Here \(L_1L_2=L_1\), and all functors involved preserve finite sums. Thus this relative abutment is degreewise finite by 60. The error-tower bound in 50 and the finite column range give the bounded filtration whose associated graded is \(E_\infty\).

The differential has bidegree \((r,r-1)\), so a nonzero differential requires \(2(p-1)\mid r-1\). For \(p\geq7\), no such \(r\geq2\) fits the column range; the finite abutment then gives the result directly. For \(p=5\), the only possible differential is \[d_9:E_9^{0,t}\longrightarrow E_9^{9,t+8}.\] Its source is finite. Indeed, coefficient inclusion gives \[H^0(P_3,B_0)\lhook\joinrel\longrightarrow H^0(P_3,B_{\mathrm{hol}}),\] and the target is finite by 56. All earlier differentials vanish by the displayed sparsity, and hence the \(d_9\)-image is finite. Column zero is already finite by this invariants argument. Columns one through eight are unchanged and are finite from their \(E_\infty\) groups. In column \(9\), \(E_2^{9,t}=E_9^{9,t}\) is an extension of its finite \(E_\infty\)-quotient by that finite image, and is finite. This proves the proposition also at five, without asserting that \(d_9\) vanishes. ◻

Proof. By 61, the cohomology of the left-hand complex in (82) is finite. The lattice, height-two, and spectral arguments establishing this input have already been completed.

First keep \(D>C\) fixed. The quotient \(Q_D=B_{\mathrm{hol}}/A_D\) is Polish and has the continuous tail section \(T_D\). Hence \[0\longrightarrow C^\bullet_{\mathrm{cts}}(G,A_D) \longrightarrow C^\bullet_{\mathrm{cts}}(G,B_{\mathrm{hol}}) \longrightarrow C^\bullet_{\mathrm{cts}}(G,Q_D)\longrightarrow0\] is exact degreewise. Its long exact sequence, together with [h3:prop:holomorphic-comparison,h3:prop:ordinary-lattice-finiteness], shows that \(H^a(G,Q_D)\) is finite. It is discrete by 34. Since \(\mathcal P\) preserves \(A_D\), it acts on this fixed quotient. Represent a quotient cocycle by its continuous holomorphic tail section \(f\). Its iterates are represented by \(T_D\mathcal P^nf\), which converge to zero by 57, uniformly on the compact cochain domain. These are quotient cocycles. Discreteness therefore implies that the given class is killed by some power of \(\mathcal P\). Thus \(\mathcal P\) is locally nilpotent on the cohomology of every fixed \(Q_D\).

Now form the quotient complex \[Q^\bullet= C^\bullet_{\mathrm{cts}}(G,B_{\mathrm{hol}})\big/ \mathop{\mathrm{colim}}_d C^\bullet_{\mathrm{cts}}(G,A_d) =\mathop{\mathrm{colim}}_{D>C}C^\bullet_{\mathrm{cts}}(G,Q_D).\] The equality follows from the preceding degreewise exact sequences. Filtered colimits of vector spaces are exact, so the fixed-quotient argument makes \(\mathcal P\) locally nilpotent on \(H^a(Q^\bullet)\). These groups are finite, using the long exact sequence for this quotient, 56, and the separately proved 61. Both Frobenius and \(\mathcal P\) preserve the bounded-pole cochain union. Their descended maps on \(Q^\bullet\) satisfy \(\mathcal P\mathrm{Fr}=1\). Thus \(\mathcal P\) is surjective, hence bijective, on its finite cohomology groups. A bijective locally nilpotent endomorphism can act only on the zero group. This proves that the first arrow of (82) is a quasi-isomorphism. The second is 56.

Every \(A_d\), \(B_{\mathrm{hol}}\), and \(B_{\mathrm{pk}}\) has finite point-value cohomology and a Polish cochain complex. The family-lifting result 35 identifies the cohomology with a profinite parameter space \(T\) at each such stage with continuous maps from \(T\) to its finite discrete cohomology. In the filtered union, a continuous map from compact \(T\) to the discrete colimit has finite image, and those finitely many classes occur at one common stage. Thus this identification commutes with the stagewise filtered colimit and gives the comparison for every profinite \(T\). This argument does not assert that \(B_0\), or its quotient cochain complex, is Polish. Finally the arrows being proved to be quasi-isomorphisms are the coefficient inclusions, so they retain products and constants. Write \(\phi_p(f)=f^p\) for this untwisted coefficient Frobenius. It is distinct from the fixed \(q\)-power operator \(\mathrm{Fr}\) used with the line retractions above. The inclusions commute with \(\phi_p\), which is an automorphism on perfected cochains; hence \(\phi_p\) is an isomorphism on each of the compared cohomology groups. 26 identifies their endpoint with the displayed group cohomology. All constructions are natural under the semilinear coefficient-field action, so the comparison also descends to the original constants. ◻

Proof. The group \(\mathop{\mathrm{SL}}_2(\mathbb Z_p)\) has dimension three, no element of order \(p\), and trivial adjoint orientation. Its first mod-\(p\) cohomology vanishes. To see this directly, let \(f:\mathop{\mathrm{SL}}_2(\mathbb Z_p)\to k\) be a continuous homomorphism. Choose \(a\in\mathbb Z_p^\times\) with \(a^2-1\) a unit, possible for \(p\geq5\). Conjugation by \(\operatorname{diag}(a,a^{-1})\) gives \[f\left(\begin{smallmatrix}1&a^2t\\0&1\end{smallmatrix}\right) =f\left(\begin{smallmatrix}1&t\\0&1\end{smallmatrix}\right).\] Additivity on this root subgroup and invertibility of \(a^2-1\) force its restriction to vanish. The lower root subgroup is treated by the same displayed conjugation, with \(a^{-2}\) in place of \(a^2\). These subgroups generate \(\mathop{\mathrm{SL}}_2(\mathbb Z_p)\) by row reduction over the local ring \(\mathbb Z_p\), so \(f=0\). Compact group duality now gives \(H^2=0\) and \(H^3=k\), with the orientation generator, while \(H^0=k\).

The determinant quotient \(\mathbb Z_p^\times\) has cohomology \(\Lambda_k(e)\): average its tame subgroup and use the two-term resolution of its procyclic quotient. Its action on \(H^3\) of \(\mathop{\mathrm{SL}}_2\) is trivial, since conjugation on \(\mathfrak{sl}_2\) has determinant one. The Hochschild–Serre sequence for \[1\longrightarrow\mathop{\mathrm{SL}}_2(\mathbb Z_p)\longrightarrow\mathop{\mathrm{GL}}_2(\mathbb Z_p) \xrightarrow{\det}\mathbb Z_p^\times\longrightarrow1\] has only rows \(0,3\) and columns \(0,1\). It has no possible differential, and its edge map onto the orientation row is surjective. Choose a lift \(d\) of that row. The product \(ed\) is nonzero on the associated graded, and odd squares vanish because \(p\) is odd. This proves the exterior algebra on \(e,d\). The finite-resolution Künneth calculation with \(P_1\) adds \(h\).

Integrally the same finite resolutions have finitely generated \(\mathbb Z_p\)-cohomology. Integral duality gives the invariant \(\mathbb Z_p\)-orientation line in degree three for \(\mathop{\mathrm{SL}}_2\). The preceding elementary-matrix argument also gives \(H^1(\mathop{\mathrm{SL}}_2,\mathbb Z_p)=0\). Reduction embeds \(H^2(\mathop{\mathrm{SL}}_2,\mathbb Z_p)/p\) into \(H^2(\mathop{\mathrm{SL}}_2,\mathbb F_p)=0\); finite generation and Nakayama’s lemma therefore give \(H^2(\mathop{\mathrm{SL}}_2,\mathbb Z_p)=0\). For clarity, vanishing in the integral degree needed for this reduction also follows from the finite coefficient models. The length-three reduced resolution and the finite filtration by powers of \(p\) make \(H^a(\mathop{\mathrm{SL}}_2,\mathbb Z/p^\ell)\) finite and zero for \(a>3\) at every \(\ell\). Cochain reductions are surjective by continuous set-theoretic sections, and the finite cohomology towers are Mittag–Leffler. The inverse-limit cochain sequence therefore gives \(H^4(\mathop{\mathrm{SL}}_2,\mathbb Z_p)=0\). The coefficient reduction long exact sequence now proves that the integral orientation surjects onto its mod-\(p\) class. The integral determinant Hochschild–Serre sequence uses the same two-term completed resolution for the procyclic quotient, after exact tame averaging. It again has no possible differential out of the orientation row, so that orientation lifts to \(\mathop{\mathrm{GL}}_2\). Normalized continuous logarithms give integral \(e\) and \(h\). Reduction of these choices gives the three specified mod-\(p\) classes. ◻

Proof. The group \(\mathop{\mathrm{SL}}_2(\mathbb Z_p)\) has no element of order \(p\): a nontrivial \(p\)-th root of unity in a two-dimensional \(\mathbb Q_p\)-representation has minimal polynomial of degree \(p-1>2\). It is an analytic duality group of dimension three. Its orientation character is trivial, since conjugation on its trace-zero Lie algebra has determinant one. The continuous duality theorem, in the finite completed resolution form used in 6, gives perfect pairings between degrees \(a\) and \(3-a\) with coefficients \(\mathbb Z/p^\ell\).

Its first cohomology is zero by the elementary-matrix argument in 63, now with coefficients \(\mathbb Z/p^\ell\). That argument applies at every length: choose \(b\in\mathbb Z_p^\times\) with \(b^2-1\in\mathbb Z_p^\times\); conjugation invariance forces a homomorphism to vanish on the upper root subgroup because multiplication by \(b^2-1\) is surjective on its parameter group \(\mathbb Z_p\). The same holds for the lower root subgroup using \(b^{-2}-1\), and these two subgroups generate \(\mathop{\mathrm{SL}}_2(\mathbb Z_p)\). Duality gives \(H^2=0\) and identifies \(H^3\) with \(\mathbb Z/p^\ell\). The orientation in the completed resolution gives these generators compatibly in \(\ell\).

The determinant extension \[1\longrightarrow\mathop{\mathrm{SL}}_2(\mathbb Z_p)\longrightarrow\mathop{\mathrm{GL}}_2(\mathbb Z_p) \longrightarrow\mathbb Z_p^\times\longrightarrow1\] has the section \(a\mapsto\operatorname{diag}(a,1)\). Its action on the orientation generator is trivial, again by determinant one on the adjoint Lie algebra. The finite subgroup \(\mu_{p-1}\) has exact invariants, and the remaining \(\mathbb Z_p\) has its two-term continuous resolution. The Hochschild–Serre sequence thus has only quotient columns zero and one, and kernel rows zero and three. It gives the logarithm \(e\), a lift \(d\) of the orientation, and their nonzero product. Their squares are zero: \(e^2=0\) because \(2\) is invertible and \(e\) is odd, and \(d^2=0\) also follows from the cohomological dimension. This proves the exterior formula for \(\mathop{\mathrm{GL}}_2(\mathbb Z_p)\). The factor \(P_1=\mathbb Z_p^\times\) contributes the second logarithm \(h\), by its two-term resolution. Tensoring these finite resolution calculations gives the asserted exterior algebra.

The constructions use determinant and oriented Lie-algebra classes, which are preserved under conjugation and transport of a lattice. They have coefficients in \(\mathbb Z/p^\ell\). Consequently coefficient Frobenius fixes them. Extension to the finite free coefficient ring \(W_\ell(k)\) commutes with the finite resolution and proves the last assertion. ◻

Proof. We first identify the unit direction, since an abstract cohomology isomorphism would not specify it. In the integral \(BP_*BP\) cobar construction use the degree-zero differential \(d=\eta_R-\eta_L\), and write bars for reduction modulo \(p\). The invariance of \(\bar v_1\) makes \[c_{v_1}=\frac{\eta_R(v_1)-\eta_L(v_1)}p\] integral; \(d^2=0\) and the \(p\)-torsion-free cobar terms make it a cocycle. For \(0\to BP_*\xrightarrow p BP_*\to BP_*/p\to0\), the coefficient connecting map is \[\partial_{\mathrm{coeff}}(\bar v_1)=[c_{v_1}] \in\mathop{\mathrm{Ext}}^{1,2(p-1)}_{BP_*BP}(BP_*,BP_*).\] Its mod-\(p\) reduction is represented by \(\bar c_{v_1}\). On the mod-\(p\) height-one open, \(\bar v_1\) is an invariant unit, so \(\bar c_{v_1}/\bar v_1\) is a cocycle of cohomological degree one and internal degree zero. Its naturality is literal cobar naturality: for two composable formal-group arrows, the difference of the two images of \(v_1\) is the sum of the two corresponding differences after transport. Changing a trivialization by one tautological arrow therefore changes the pulled-back cocycle by a coboundary.

Pull the height-one specialized formal law to Honda form on the connected-marking torsor of 13. The preceding coboundary identity identifies \(\bar c_{v_1}/\bar v_1\) with the Honda-height-one class. It is nonzero: on an automorphism \(a\in1+p\mathbb Z_p\) the integral periodic frame changes \(v_1\) by \(a^{p-1}\), and the cocycle is \[\frac{a^{p-1}-1}{p}\pmod p.\] At \(a=1+p\) this is \(p-1\pmod p\), a unit. It is a unit multiple of the normalized logarithm on \(P_1\). We choose \(h\) with the corresponding normalization. The leading coefficient of the formal-group arrow has been retained here through the periodic frame \(U\); discarding that frame would discard this calculation.

The height-one sphere calculation identifies its mod-\(p\) groups with \(\mathbb F_p[v_1^{\pm1}]\otimes\Lambda(h)\). For the Adams self-map normalized in 61, the image of the Moore bottom cell has \(BP\)-Hurewicz coefficient \(\bar v_1\). Its Moore boundary is a unit multiple of \(\alpha_1\). The normalized cobar terms for \(BP\) are even and \(p\)-torsion-free, so the comparison in 7 between the termwise cone and the cone of totalization applies to this cofiber sequence. For the lift \(v_1\), the coefficient connecting cocycle is \(c_{v_1}\), while the actual desuspended boundary has leading detector \(-[c_{v_1}]\). The internal degree \(2(p-1)\) is even, and strict desuspension contributes no further sign. Choose the negative of this boundary class, map it to the Moore quotient, and invert the same self-map. The resulting source generator in degree \(-1\) has detector \(\bar c_{v_1}/\bar v_1\), the logarithm normalized above. The odd-prime Adams-summand fiber is realized with the unit as its first map in (Barthel and Beaudry 2020, sec. 4.2, pp. 30–32). Its logarithm, mod-\(p\) algebra, and unit-induced rational splitting are given in (Barthel and Beaudry 2020, sec. 4.3, Remarks 4.16–4.17 and (4.18)). The maps of the preceding paragraph are the unit maps of the formal-group cobar construction. Hence their \(h\) is precisely the \(R_1\)-action, rather than an independently chosen degree-one direction.

For the second direction use the actual determinant identity in 13: the determinant of the middle height-three frame is the product of the norm of the connected height-one frame and the determinant of the rank-two kernel frame. It is an identity of integral determinant lattices. Applying the normalized logarithm yields the sum of the two logarithms, with the signs introduced by inverse-frame conventions. In particular its coefficient on \(e\) is a unit modulo \(p\). This is the class \(\zeta\), by its determinant construction. The change of basis from \((h,e)\) to \((h,\zeta)\) is invertible.

Finally, [h3:thm:ordinary-comparison,h3:des:frame-cohomology] identify the actual coefficient algebra with this exterior algebra. The comparisons preserve products and the two unit maps, proving all assertions. ◻

Proof. The binomial theorem gives \[a\equiv b\pmod p \quad\Longrightarrow\quad a^{p^j}\equiv b^{p^j}\pmod {p^{j+1}} \qquad(j\geq0).\] For the \(i\)-th summand of (84), changing a lift consequently changes the result by a multiple of \(p^\ell\). This proves independence of all lifts. Lift two Witt vectors to \(W_\ell(D_\ell)\), and use their Witt sum or product to lift the corresponding operation over \(B_0\). The last ghost component on \(W_\ell(D_\ell)\) is a ring homomorphism. Its reduction and the lift independence prove the ring identities for \(\gamma_\ell\). The construction is natural for maps of the reduction \(D_\ell\to B_0\), and hence equivariant for the actual \(P_3\)-action on \(D_\ell\). These are the elementary ghost identities of (Hesselholt 2005, Lemma 1 and Proposition 2).

The map is continuous on each compact pole lattice. Indeed, coefficientwise lifts and the finite polynomial (84) prove continuity on \(W_\ell(A)\); its ring property then gives \[\gamma_\ell(\mathcal L_{\ell,d}) \subset x^{-dp^{\ell-1}}A_\ell \subset\mathcal M_{\ell,dp^{\ell-1}}.\] It therefore acts on the specified cochain colimits.

Filter the source by \(V^iW_{\ell-i}(B_0)\) and the target by \(p^iD_\ell\), for \(0\leq i\leq\ell\). The source filtration iterates the one-coordinate restriction–Verschiebung sequence in (Hesselholt 2005, proof of Lemma 9, p. 6). Their \(i\)-th graded pieces are additively \(B_0\); the induced map is \[a\longmapsto a^{p^{\ell-1-i}}.\] The quotient sequences remain exact on cochains. For a Witt quotient the section is the coordinate section \(a\mapsto V^i[a]\); for the corresponding target quotient use \(a\mapsto p^i\widetilde a\) with coefficientwise lifts. These sections are continuous and send a compact bounded-pole family into some bounded-pole stage of (83). They need not be additive or equivariant: their role is to give degreewise surjectivity of continuous cochains. The kernels and differentials are the equivariant ones.

By 62, the coefficient \(p\)-Frobenius \(\phi_p\) is an isomorphism on \(H^*(P_3,B_0)\). The long exact cohomology sequences and induction on the finite filtration now prove the assertion for \(\gamma_\ell\). For \(\beta_\ell\), use the two Verschiebung filtrations. Their graded maps are the natural comparison \(B_0\to B_{\mathrm{pk}}\), which is a cohomology isomorphism by the same theorem. Witt-coordinate sections on the analytic side are continuous, so the identical finite-filtration argument proves the assertion for \(\beta_\ell\).

In characteristic \(p\), truncated Witt Frobenius is \[R_\ell W_\ell(\phi_p)(a_0,\ldots,a_{\ell-1}) =(a_0^p,\ldots,a_{\ell-2}^p);\] see (Hesselholt 2005, Lemma 8). Substitution into (84) proves (85). Since \(\gamma_\ell\) is unital, it is the identity on \(\mathbb Z/p^\ell\). Its restriction to \(W_\ell(k)\), under the identification with \(W(k)/p^\ell\), is \(\sigma_k^{\ell-1}\); we do not identify that restriction with the identity. ◻

Proof. On the ordinary quotient stack, the comparison of 26 is induced by its map to \(BG_{\mathrm f}\) and by the coefficient unit \(k\to\mathcal O^\flat\). Apply \(W_\ell\) to the coefficient sheaves on this same diagram. It gives the displayed map before taking cohomology, and all the asserted naturalities follow from the natural Witt operations. Its Verschiebung filtration has the length-one constants comparison on each graded piece. The continuous coordinate sections make the associated short exact sequences exact on the computing cochains, including their profinite parameters. Finite-filtration induction from 26 proves the equivalence.

The coefficient map \(\beta_\ell\) preserves constants by definition. For \(\gamma_\ell\), the composite \[C^*_{\mathrm{cts}}(P_3,\mathbb Z/p^\ell) \longrightarrow C^*_{\mathrm{cts}}(P_3,W_\ell(B_0)) \longrightarrow C^*_{\mathrm{cts}}(P_3,D_\ell)\] is exactly the ordinary coefficient inclusion, since \(\gamma_\ell\) is the identity on \(\mathbb Z/p^\ell\). Thus these are statements about the specified maps from constants, not about an abstract isomorphism of their images in cohomology. ◻

Proof. The totalization assertion is 50. Put \(M_\ell=\mathbb S_{(p)}/p^\ell\) and \(E=E_3(k)\). The completion map identifies \(M_\ell\) with \(S/p^\ell\), naturally in the ordinary Moore quotient transitions: its cofiber has zero mod-\(p\) quotient, so multiplication by \(p\) is an equivalence there and every mod-\(p^\ell\) quotient vanishes. Each finite \(M_\ell\) has type one, since its rational homology vanishes and its \(K(1)\)-homology is nonzero. Let \(\psi:\Sigma^{2(p-1)}M_1\to M_1\) be the normalized Adams \(v_1\)-map chosen in the proof of 61; its action on \(E/p\) is multiplication by \(xU^{p-1}\). The periodicity theorem (Ravenel 1992, Theorem 1.5.4(i)–(ii), pp. 9–10) supplies a \(v_1\)-map \(f_\ell:\Sigma^{d_\ell}M_\ell\to M_\ell\) and, for the ordinary reduction \(r_\ell:M_\ell\to M_1\), positive integers \(i_\ell,j_\ell\) such that \(d_\ell i_\ell=2j_\ell(p-1)\) and \[r_\ell\circ f_\ell^{i_\ell} \simeq \psi^{j_\ell}\circ \Sigma^{2j_\ell(p-1)}r_\ell.\] Here the theorem is applied after a common suspension to finite CW-complexes and then desuspended to spectra. Multiplication of the Adams map by the integer unit used in its normalization preserves its \(v_1\)-map property. Fix this iterate \(\varphi_\ell=f_\ell^{i_\ell}\), of degree \(e_\ell=2j_\ell(p-1)\).

The cofiber sequence \(E\xrightarrow{p^\ell}E\to E/p^\ell\) and evenness of \(E/p^\ell\) give an isomorphism \[[\Sigma^{e_\ell}E/p^\ell,E/p^\ell]_E \xrightarrow{\ \simeq\ }\pi_{e_\ell}(E/p^\ell)\] by evaluation on the unit: the possible kernel is a quotient of \(\pi_{e_\ell+1}(E/p^\ell)=0\), and multiplication by \(p^\ell\) vanishes on the group on the right. Thus the \(E\)-module map induced by \(\varphi_\ell\) is determined by a scalar \(\theta_\ell\) in this group. Applying \(E\wedge-\) to the chosen reduction square proves \[\theta_\ell\bmod p=(xU^{p-1})^{j_\ell},\qquad \theta_\ell=U^{j_\ell(p-1)}c_\ell,\qquad c_\ell=x^{j_\ell}+p b_\ell\quad(b_\ell\in A_\ell).\] A lift of \(\theta_\ell\) to \(\pi_{e_\ell}E\) gives the same \(E\)-module map by the evaluation isomorphism. Tensoring with an ordinary \(E\)-module \(N_Q\) therefore makes \(\varphi_\ell\) act on \(\pi_*(N_Q/p^\ell)\) by multiplication by this scalar.

The two localizations of \(A_\ell\) at \(c_\ell\) and at \(x\) agree canonically. Indeed, in \(A_\ell[x^{-1}]\) the factorization \(c_\ell=x^{j_\ell}(1+p b_\ell x^{-j_\ell})\) makes \(c_\ell\) invertible. In \(A_\ell[c_\ell^{-1}]\), the factorization \(x^{j_\ell}=c_\ell(1-p b_\ell c_\ell^{-1})\) makes \(x\) invertible. Both parenthesized factors have finite geometric-series inverses because \(p^\ell=0\). Miller’s odd-prime result for \(M_1\) (Miller 1981, Theorem 4.11 and Corollary 4.12), together with the thick-subcategory argument in (Ravenel 1992, sec. 7.5, after Conjecture 7.5.5, p. 85), identifies the telescope of a \(v_1\)-map with \(L_1M_\ell\) for every finite type-one \(M_\ell\). Taking the chosen iterate does not change the telescope, since it selects a cofinal subsequence. As \(L_1\) is smashing, this is an equivalence under \(N_Q/p^\ell\) between \(L_1(N_Q/p^\ell)\) and the telescope of its \(\varphi_\ell\)-action.

For the actual term \(N_Q=\mathcal N_3^q\), with \(Q=P_3^q\), 51 gives even, \(p\)-torsion-free coefficients. Coefficientwise continuous lifts identify \(\pi_0(N_Q/p^\ell)=C_{\mathrm{cts}}(Q,A_\ell)\). Homotopy groups commute with the telescope, so the scalar calculation and the two localizations above give \[\pi_0L_1(N_Q/p^\ell) \cong C_{\mathrm{cts}}(Q,A_\ell)[x^{-1}] =\mathop{\mathrm{colim}}_d x^{-d}C_{\mathrm{cts}}(Q,A_\ell).\] The odd groups vanish, and the other even groups are its periodic translates. This is the algebraic bounded-pole localization; the stable cofinal lattices \(\mathcal M_{\ell,d}\) specify it in (83).

These coefficient identifications preserve the actual transition maps. Each is the unique extension of the map on coefficients induced by the \(L_1\)-localization unit, by the universal property of localization of a module at \(x\). The ordinary Moore quotient transition is induced by the cofiber square with vertical maps \(p\) and \(\mathrm{id}\); it reduces coefficients and preserves the fixed coordinate \(x\). Naturality of the localization unit therefore identifies its localized coefficient map with ordinary reduction \(D_\ell\to D_{\ell-1}\). The same argument applies to every coface and codegeneracy of the actual descent object: coefficient actions send \(x\) to \(xu_g+ph_g\), which is invertible in \(D_\ell\), and precomposition on continuous functions preserves the bounded-pole colimit. Hence the degree-zero coefficient complex is the stated \(D_\ell\) complex, with its actual cochain maps, ordinary Moore transitions, and maps from constants.

The tame central units kill internal degrees not divisible by \(2(p-1)\). For all remaining degrees the finite \(p\)-filtration has associated graded the mod-\(p\) coefficient complex, trivialized by powers of \(v_1\). By [h3:thm:ordinary-comparison,h3:des:frame-cohomology] that complex has finite cohomology in columns zero through five, and vanishes above five. The long exact sequences of the finite \(p\)-filtration give those same finiteness and vanishing conclusions in every length. There is no assertion that \(v_1\) itself is an integral invariant in length \(\ell\); only the associated-graded trivialization is used.

A differential has \(2(p-1)\mid r-1\), so the first candidate has \(r=2p-1\geq9\). It cannot fit the column range \(0\leq s\leq5\). Thus the spectral sequence collapses, and its finite column range proves degreewise finiteness of the abutment. Strong convergence comes from 50.

Finally, in stem \(-3\) one must have \(t-s=-3\) with \(0\leq s\leq5\) and \(t\in2(p-1)\mathbb Z\). The unique solution is \((s,t)=(3,0)\). There is consequently no additive extension or choice of filtration quotient in (86). The construction is natural in the Moore quotient maps and in coefficient constants. This proves its stated naturality as well. ◻

Proof. Write \(S/p^\infty=\operatorname*{colim}_\ell S/p^\ell\), with the usual inclusions of Moore spectra. Duality of the finite Moore \(S\)-modules identifies \(M/p^\ell\simeq F_S(\Sigma^{-1}S/p^\ell,M)\). The inclusion \(S/p^\ell\to S/p^{\ell+1}\) is induced by the square with vertical maps \(\mathrm{id}\) and \(p\); dualizing gives the ordinary quotient transition induced by \(p\) and \(\mathrm{id}\). Thus these identifications respect the tower. Applying the internal function functor to \[\Sigma^{-1}S/p^\infty\longrightarrow S\longrightarrow S[1/p]\] therefore gives a natural fiber sequence \[F_S(S[1/p],M)\longrightarrow M\longrightarrow\widehat M_p.\] Multiplication by \(p\) is invertible on the first term. Moreover \[F_S(S[1/p],\widehat M_p) \simeq F_S(S[1/p]\otimes_S\Sigma^{-1}S/p^\infty,M)=0.\] Thus \(\widehat M_p\) is local with respect to the Moore quotient: maps from any \(p\)-invertible \(S\)-module vanish, since such a module is an \(S[1/p]\)-module and the displayed function object is zero. Each \(M/p^\ell\) is \(E(1)\)-local, and local objects are closed under limits, so \(\widehat M_p\) is also \(E(1)\)-local.

In the \(E(1)\)-local category the Moore-acyclic objects are exactly the rational objects: the Moore condition makes \(p\) invertible, and conversely rationalization kills every Moore spectrum. The height-one fracture square identifies these with the \(K(1)\)-acyclic \(E(1)\)-local objects. Hence Moore localization and \(K(1)\)-localization have the same acyclic objects and the same local objects in this category. The \(E(1)\)-localization unit is itself a \(K(1)\)-equivalence, so passing first to \(E(1)\)-local objects does not change \(K(1)\)-localization. The displayed natural map is therefore the asserted localization in \(S\)-modules. ◻

Proof. Choose the compatible classes \(o_\ell\in H^3(G_{\mathrm f},\mathbb Z/p^\ell)\) of 64. Write \(\rho_\ell\) for the cohomology isomorphism induced by \(\beta_\ell\), and set \[\nu_\ell=\rho_\ell^{-1}\eta_\ell(o_\ell),\qquad z_\ell=\gamma_\ell(\nu_\ell) \in H^3(P_3,D_\ell).\] Here the constant coefficient extension to \(W_\ell(k)\) is understood. The naturalities in 67 give \(R_\ell\nu_\ell=\nu_{\ell-1}\). Coefficient Frobenius fixes \(o_\ell\), because it fixes \(\mathbb Z/p^\ell\); injectivity of \(\rho_\ell\) then gives \[W_\ell(\phi_p)(\nu_\ell)=\nu_\ell.\] Using (85) we obtain \[\operatorname{red}_{\ell,\ell-1}(z_\ell) =\gamma_{\ell-1}R_\ell W_\ell(\phi_p)(\nu_\ell) =z_{\ell-1}.\] Thus the Frobenius twist in the ghost transition does not obstruct compatibility of these actual integral orientation classes.

By [h3:des:frame-cohomology,h3:des:witt-constants,h3:lem:ghost-comparison], the degree-three cohomology over \(k\) is one free copy of \(W_\ell(k)\). The determinant and orientation constructions are preserved under coefficient Galois action and transport of the integral frames. Normal-basis descent therefore identifies its descended group with \(\mathbb Z/p^\ell\). The elements \(z_\ell\) are generators and their transitions are ordinary reduction. This proves the corresponding assertions for \(\pi_{-3}L_1(T/p^\ell)\) by (86).

Apply 69 to \(L_1T\). Exactness of \(L_1\) identifies its Moore quotients with \(L_1(T/p^\ell)\). Consequently \[L_{K(1)}T\simeq \operatorname*{holim}_\ell L_1(T/p^\ell).\] All finite-length homotopy groups are finite by 68; their inverse systems are Mittag–Leffler, so the Milnor \(\lim^1\)-term vanishes in every stem. The compatible generators \(z_\ell\) therefore give an actual class \(\delta\), and their inverse limit is \(\mathbb Z_p\). The last assertion follows from the equality of the constant cochain maps in 67 and from the natural edge identification in 68. ◻

Proof. The target is a \(K(1)\)-local algebra and hence a module over the local unit \(R_1\). Each homotopy class displayed defines an actual map from the indicated suspension of \(R_1\). The product \(\zeta\delta\) is formed using this algebra multiplication, so no independent choice of a degree-minus-four line is being made.

Modulo \(p\), the source groups are the direct sum of the four shifts of \[\pi_*(R_1/p) =\mathbb F_p[v_1^{\pm1}]\otimes\Lambda(h).\] By 65, the images of \(1,\zeta,d,\zeta d\), together with their \(h\)-multiples, are exactly the eight exterior basis elements in the target coefficient calculation. The class \(\delta\) reduces to \(d\) by 70. Products are detected by the cup action, so these statements specify the leading filtered class of each of the eight actual homotopy images. The spectral sequence has collapsed with finite filtration. In each stem, filter the source homotopy group by assigning these eight periodic basis elements the cohomological degrees of their displayed leading classes. The map is filtered, and its associated-graded map is an isomorphism in every degree. Induction on the finite filtration, using the short exact sequences for successive filtration quotients, shows that the original map of homotopy groups is an isomorphism. This argument does not require an a priori splitting of the target filtration. The displayed map is therefore an isomorphism on mod-\(p\) homotopy in every stem.

Its cofiber is \(K(1)\)-local, hence derived \(p\)-complete by 69. The vanishing of its mod-\(p\) quotient makes multiplication by \(p\) an equivalence, so all its quotients by \(p^\ell\) vanish. Derived completeness then makes the cofiber zero. This proves the equivalence, including the asserted compatibility of its first two components. ◻

Proof. For a locally profinite parameter \(Q\), regard \(A_Q=C_{\mathrm{cts}}(Q,\Lambda_\ell)\) as a discrete ring. Evaluation gives a map of ring sheaves on \(X\times\underline Q\), \[\underline{A_Q}\longrightarrow\underline{\Lambda_\ell}, \qquad (f,q)\longmapsto f(q).\] It is defined on the finitely many clopen fibers of each \(f\); this uses no compactness of \(Q\). The unit of the derived constant-sheaf/global-sections adjunction, followed by derived sections of evaluation, gives \[C_{\mathrm{cts}}(Q,\Lambda_\ell)[0] \longrightarrow R\Gamma_v(X\times\underline Q,\Lambda_\ell).\] Equivalently, this sends a continuous \(\Lambda_\ell\)-valued function on \(Q\) to its ordinary section and then to derived sections. With \(Q=G^a\), the projections of the action-groupoid nerve commute with all bar faces and codegeneracies, including the action face. These maps therefore form a cosimplicial map. Its totalization is (87) by quotient Čech descent.

The map of nerves that forgets \(T\) over \(B\) proves the stated equality with \(c_T^*\mathsf u_{B,G,\ell}\). For a continuous homomorphism \(h:H\to G\), let \(S\to Z\) be an \(H\)-torsor over \(B\) with a specified identification \(S\times^H G\simeq T\) over \(Z\). The resulting map of torsor nerves acts in degree \(a\) by \(h^a\) on the transition elements. Evaluation commutes with these maps, with equivariant maps of \(X\), and with every base change. A base automorphism whose descent commutes with the constant group acts on a torsor nerve by \((t,g_1,\ldots,g_a)\mapsto(\gamma(t),g_1,\ldots,g_a)\), which proves the asserted arithmetic naturality. The construction uses maps of coefficient ring sheaves and the lax monoidal derived sections functor, so it preserves cup products. Composing the same ring-sheaf maps with \(\Lambda_\ell\to\Lambda_{\ell-1}\) or with \(\Lambda_\ell=W_\ell(\mathbb F_p)\to W_\ell\mathcal O^\flat\) proves the coefficient naturalities term by term. In particular these are maps of the derived coefficient towers before cohomology is taken.

For (88), use all finite clopen quotients \(Q_i\) of \(Q\). The example after Definition 7.8 and Proposition 7.16 of (Scholze 2026) identify \(B_C\times\underline Q\) with the inverse limit of \(B_C\times Q_i\) and make \(B_C\) a strictly totally disconnected geometric point. Its étale covers split, so its finite constant cohomology is \(A[0]\). Continuity and comparison with v-cohomology in (Scholze 2026, Propositions 14.9–14.10) now give \[R\Gamma_v(B_C\times\underline Q,A) \simeq\mathop{\mathrm{colim}}_i\mathop{\mathrm{Map}}(Q_i,A)[0] =C_{\mathrm{cts}}(Q,A)[0].\] Every map here is induced by the coefficient unit and is natural in the finite partitions, coefficient fiber, and base. This proves the stated naturality, including the arithmetic action on a Tate fiber. Apply the equivalence to \(Q=G^a\) for compact \(G\) and totalize to obtain the last assertion. Over the finite-field base used below we use the finite classifying map itself; this point comparison is only asserted over \(B_C\). ◻

Proof. We first make the continuous comparison on compact groups precise. Each compact group \(K\) under consideration has an open normal uniform pro-\(p\) subgroup \(U\). For \(P_3\) and \(\mathop{\mathrm{GL}}_2(\mathbb Z_p)\) one may use a sufficiently deep principal congruence subgroup. A compact open \(K\subset J_3\) preserves a lattice in \(\mathbb Q_p^3\): the orbit of any lattice is finite, and its sum is stable. A sufficiently deep principal congruence subgroup for this lattice lies in \(K\) and is normal there. If the depth is at least one, the corresponding Lie lattice \(\mathfrak u=p^m\mathcal O\), for the relevant endomorphism order \(\mathcal O\), satisfies \([\mathfrak u,\mathfrak u]\subset p\mathfrak u\). At odd \(p\), logarithm and exponential converge on these lattices, identify the \(p\)th-power map with multiplication by \(p\), and show that the group is finitely generated, powerful, and torsion free. It is therefore uniform.

Give \(U\) its lower-\(p\)-series valuation. For \(p\geq5\) it is saturated and equi-\(p\)-valued, with value denominator \(e=1\) and graded generators in degree one. The trivial finite free coefficient module \(\mathbb Z_p\) has action in \(1+p\operatorname{End}_{\mathbb Z_p}(\mathbb Z_p)\). The continuous comparison of (Huber et al. 2011, Theorem 3.3.3) therefore gives a cup-compatible isomorphism \[H^*_{\mathrm{cts}}(U,\mathbb Z_p)\simeq H^*(\mathfrak u,\mathbb Z_p),\] where \(\mathfrak u\) is the integral Lie algebra of \(U\). Coefficient naturality and rational agreement in (Huber et al. 2011, Theorem 3.1.1(1)–(2)), together with the boundedness of compact rational cochains, identify its rationalization with Lazard’s comparison for \(\operatorname{Lie}(K)=\mathfrak u[1/p]\). Continuous group homomorphisms preserve lower \(p\)-series, so Theorem 3.1.1(3) of the same source gives naturality for restrictions and for the block maps after passing to such small opens.

The rational Hochschild–Serre sequence for \(U\triangleleft K\) collapses to finite quotient invariants: positive cohomology of the finite group \(K/U\) vanishes by averaging in \(\mathbb Q_p\). Here \(\operatorname{Lie}(K)\) is the Lie algebra of a \(\mathop{\mathrm{GL}}_n\)-form, with \(n=2\) or \(3\). The exterior calculation and triviality of its adjoint action are given by (Barthel et al. 2025, Lemma 3.8.2). They give generators in degrees \(1,3,\ldots,2n-1\), without choosing explicit cocycles for all of them. Thus the finite quotient fixes the cohomology, and in particular the low-degree dimensions are \(1,0,1\) in degrees \(1,2,3\).

We identify the particular degree-three generator needed here. For the split Lie algebra put \(q(X,Y)=\operatorname{Tr}(XY)\); for the inner form use reduced trace. Cyclicity gives \(q([X,Y],Z)=\operatorname{Tr}(X[Y,Z])\). After an algebraically closed splitting-field extension the restriction of \(q\) to \(\mathfrak{sl}_n\), for \(n=2,3\), is nonzero and nondegenerate. The degree-three invariant-form calculation of (Koszul 1950, sec. 11, Theorems 11.1–11.2) therefore makes its class nonzero. The finite Chevalley–Eilenberg cochain complex commutes with field extension, so the class already defined on the form over \(\mathbb Q_p\) is nonzero. Comparing restrictions through a common small uniform subgroup shows that the group identifications are independent of the chosen open subgroup. The upper-left block map restricts the displayed degree-three matrix trace form to the identical form on the \(2\times2\) block, and hence restricts it nontrivially.

For \(P_3\), choose \(e_3\) to have exactly this reduced-trace Lie-algebra class under the natural compact comparison. The coefficient-field group \(\Gamma=\mathop{\mathrm{Gal}}(k/\mathbb F_p)\) acts by automorphisms of the Honda division algebra and preserves reduced trace. Naturality of that comparison therefore makes this normalized rational class \(\Gamma\)-invariant.

For the noncompact group \(J_3\), use the reduced affine building in 87 with \(F=\mathbb Q_p\). That lemma supplies a contractible two-dimensional complex on which \(J_3\) preserves types, has compact open face stabilizers fixing their faces pointwise, and has a closed chamber as a fundamental domain. The cellular resolution and continuous Shapiro give, first with finite coefficients and then with the derived integral limit, \[E_1^{a,s}=\bigoplus_{\substack{\sigma\subset\Delta\cr \dim\sigma=a}} H^s_{\mathrm{int}}(K_\sigma) \ \Longrightarrow\ H^{a+s}_{\mathrm{int}}(J_3).\] There are finitely many summands and cellular degrees. After inversion of \(p\), every restriction in a fixed row identifies with the identity on \(H^s(\mathfrak{gl}_3,\mathbb Q_p)\); changing a face representative only introduces an inner conjugation. That row is therefore the cellular cochain complex of the simplex \(\Delta\) with these constant coefficients. It has cohomology only in cellular degree zero. The edge map, and then further restriction to an arbitrary compact open subgroup using intersections, gives the required assertion for \(J_3\).

It remains to account for the unipotent radical of \(D\). Fix \(\ell\) and write \(\mathbb Q_p^2=\bigcup_{r\geq0}p^{-r}\mathbb Z_p^2\). The two-variable Koszul resolution computes \[H^q(p^{-r}\mathbb Z_p^2,\mathbb Z/p^\ell) =\bigwedge^q(\mathbb Z/p^\ell)^2.\] Restriction from stage \(r+1\) to \(r\) multiplies degree \(q\) by \(p^q\). These systems are pro-zero for \(q>0\) and constant for \(q=0\). The restrictions of cochains themselves are surjective: the smaller subgroup is clopen, so a cochain extends by zero. Thus their inverse limit computes the derived limit, and the Milnor sequence gives \(R\Gamma(\mathbb Q_p^2,\mathbb Z/p^\ell)=\mathbb Z/p^\ell[0]\). The calculation remains valid with profinite parameters, by the same finite Koszul complexes and extension of cochains. Thus this equality holds for internal derived condensed invariants. The constants map is equivariant for the Levi action, and its underlying equivalence is therefore an equivariant equivalence. Hochschild–Serre consequently makes inflation from the Levi subgroup an equivalence at each finite level and then integrally. The rational Künneth calculation for \(\mathop{\mathrm{GL}}_2(\mathbb Z_p)\times\mathbb Z_p^\times\) has degree three equal to the degree-three group of its first factor, since the second factor has cohomological dimension one and the first has \(H^2=0\). Restriction along \(A\mapsto\operatorname{diag}(A,1)\) detects the nonzero trace form. This proves (89). ◻

Proof. The kernel of a length-one quotient of a form \(E\) of \(V_3\) is a rank-three bundle \(E^-\) of degree zero. If it had a positive Harder–Narasimhan subbundle of rank \(b<3\) and integral degree \(d\geq1\), its inclusion in \(E\) would contradict semistability: \(d/b\geq1/b>1/3\). Thus \(E^-\) has slope zero. By the relative slope-zero correspondence it is a rational local system (Kedlaya and Liu 2015, Corollary 8.7.10); see 11.

Conversely an upper length-one modification of \(\mathcal O^3\) has, at a geometric point with local parameter \(t\) at \(\infty\), lattice \[(B_{\mathrm{dR}}^+)^3+ B_{\mathrm{dR}}^+t^{-1}\widetilde\ell\] for a direction line \(\ell\). If \(\ell\) lies in a proper rational subspace \(W\), modifying \(W\otimes\mathcal O\) gives a degree-one subbundle of rank less than three, which destabilizes. In the other direction, let \(F'\) be a saturated destabilizing subbundle of the upper modification, of rank \(b<3\) and degree \(d\geq1\). Its intersection \(F''\) with \(\mathcal O^3\) has degree \(d-\epsilon\), where \(0\leq\epsilon\leq1\). Semistability of \(\mathcal O^3\) forces \(d=\epsilon=1\) and \(\deg F''=0\). The saturation of \(F''\) in \(\mathcal O^3\) has degree at least zero, while semistability forces its degree to be at most zero. Its torsion quotient therefore has degree zero, so \(F''\) is already saturated. Thus \(F''\) is a slope-zero subbundle of \(\mathcal O^3\), hence equals \(W\otimes\mathcal O\) for a rational subspace \(W\); maps between trivial bundles are matrices over \(H^0(\mathcal O)=\mathbb Q_p\). The equality \(\epsilon=1\) says that \(\ell\subset W_C\). This proves precisely the Drinfeld condition.

Let \(\mathcal Z\) be the sheaf of injections \(\mathcal O^3\hookrightarrow V_3\) with length-one cokernel at \(\infty\) preserving the chosen determinant lattices. The commuting actions are postcomposition by \(P_3\) and inverse precomposition by \(J_3\). The preceding slope calculation and the relative trivial/basic-stratum theorems (Fargues and Scholze 2024, Theorems III.2.4 and III.4.5) give actual sheaves of frames in families. They identify \(\mathcal Z/J_3=\mathbf P^2_C\). In the other direction they identify \(\mathcal Z/P_3=\Omega_C^2\): the full frame group of the positive bundle is \(D_3^\times\), and its surjective norm valuation lets one match the determinant lattice locally. Taking the remaining quotient proves (90).

The arithmetic field is \(F=W(k)[1/p]\); the coefficient field of the curve remains \(\mathbb Q_p\). Let \(N\) be the fixed Honda isocrystal over \(k\), whose period-ring construction gives \(V_3\). All Honda endomorphisms are defined over \(k\). On the period coefficients tensored over \(F\) with \(N\), define the action of \(\gamma\in G_F\) by \(\gamma\otimes\mathrm{id}_N\). It commutes with Frobenius and so descends to the bundle. Every element of \(D_3\) is an \(F\)-linear endomorphism of \(N\) commuting with its Frobenius; its matrix entries are fixed by \(G_F\). Thus this specified descent commutes with the entire \(D_3\)-action. We have not trivialized a torsor of frames of an arbitrary descended basic bundle.

The untilt ideal \(\ker\theta\) is functorial, so \(\Delta_3\) inherits descent. The relative slope-zero correspondence makes its rational section space a continuous one-dimensional representation of \(G_F\). This representation need not be trivial. Its valuation character has compact subgroup image in \(\mathbb Z\), hence zero, so every lattice in it is stable; no generator is asserted to be invariant. Define the arithmetic action on an injection \(u\) by pullback and the specified descent on source and target. The standard rational coordinate vectors of \(\mathcal O^3\) are fixed, so \[\gamma(guj^{-1})=g\gamma(u)j^{-1} \qquad(g\in P_3,\ j\in J_3).\] The determinant condition is preserved by lattice stability. On the upper side, directions lie in \(\mathbb Q_p^3\otimes(\mathcal O(\infty)/\mathcal O)|_\infty\). The semilinear scalar in the second factor is common to all coordinates and cancels in projective space. This is the usual arithmetic action on \(\Omega_C^2\), commuting with \(J_3\). No arithmetic invariant choice of a frame is required. ◻

Proof. For a profinite set \(Q=\varprojlim_iQ_i\) with finite \(Q_i\), continuity on spatial diamonds and comparison of finite coefficients give \[R\Gamma_v(\mathbf P^2_C\times\underline Q,\Lambda_\ell) \simeq\mathop{\mathrm{colim}}_i\mathop{\mathrm{Map}}\bigl(Q_i, R\Gamma_{\acute et}(\mathbf P^2_C,\Lambda_\ell)\bigr).\] Here one applies (Scholze 2026, Proposition 14.9) to \(\mathbf P^2_C\times Q_i\) and then (Scholze 2026, Proposition 14.10 and Lemma 15.6); these statements allow the coefficient ring \(\Lambda_\ell\) also when its torsion prime is \(p\). The finite-coefficient cohomology of the analytic projective space agrees with its algebraic cohomology: the proper comparison of (Scholze 2013, Theorem 3.12), restating (Huber 1996, Theorem 3.7.2), applies to \(\mathbf P^2_C\to\mathop{\mathrm{Spec}}C\) with \(\mathbb F_p\) coefficients, and the coefficient exact sequences extend it to \(\Lambda_\ell\). Thus the projective-space calculation is valid as the actual condensed geometric complex.

The anticanonical bundle has canonical projective equivariance, including arithmetic descent, and restricts to \(\mathcal O(3)\). Since \(p\geq5\), the class \(h_\ell=3^{-1}c_1(\omega_{\mathbf P^2}^{-1})\) on the quotient restricts to the hyperplane class. Its powers give an actual morphism of equivariant derived coefficient objects \[\bigoplus_{j=0}^2\Lambda_\ell(-j)[-2j] \longrightarrow\mathscr C_{\mathbf P^2_C,\ell}.\] The preceding calculation shows that this is an equivalence on every profinite parameter. Apply it to \(Q=P_3^a\) in the quotient nerve and totalize. The finite direct sum commutes with totalization, giving an equivalence \[\bigoplus_{j=0}^2 C^\bullet_{\mathrm{cts}}(P_3,\Lambda_\ell(-j))[-2j] \xrightarrow{\ \simeq\ } R\Gamma_v(\mathcal M,\Lambda_\ell).\] In its \(j=0\) component, each bar term sends a continuous constant function to its coefficient-unit section. By 73, this component is exactly \(\operatorname{cl}_{c_{3,\mathcal M},\ell}\). The Kummer classes commute with coefficient reduction. Take the derived limit in \(\ell\) and then invert \(p\); the finite direct sum commutes with both operations. Only \(j=0,1\) contribute to degree three, giving (91). Naturality of the Kummer classes gives its arithmetic compatibility and the displayed twists. ◻

Proof. Let \(V=\operatorname{Spa}(R,R^+)\) be affinoid perfectoid, and write \(R^\flat\) for its tilt (equal to \(R\) in characteristic \(p\)). Its product with \(\underline Q\) is affinoid perfectoid with ring \(C_{\mathrm{cts}}(Q,R)\) and tilt \(C_{\mathrm{cts}}(Q,R^\flat)\). Affinoid acyclicity gives \[R\Gamma_v(V\times\underline Q,\mathcal O^\flat) =C_{\mathrm{cts}}(Q,R^\flat)[0]\] by (Scholze 2026, Proposition 8.8). Choose a decreasing countable basis of open additive subgroups \(R^\flat_a\) of the complete additive group \(R^\flat\). Then \(\underline{R^\flat}=\varprojlim_a(R^\flat/R^\flat_a)_{\mathrm{disc}}\) is solid. This limit is also the derived limit: continuous maps from a profinite set to each discrete quotient have finite image, and lifts of their finitely many values give surjectivity at each successive quotient. A v-hypercover by disjoint unions of affinoid perfectoids now expresses the \(\mathcal O^\flat\) complex, with all parameters, as a limit of products of solid modules. Derived solid modules are closed under limits and extensions (Tang 2026, Theorem 4.4(ii)). Artin–Schreier gives the assertion for \(\mathbb F_p\), the coefficient exact sequences give it for \(\mathbb Z/p^\ell\), and the derived limit gives it for \(\mathscr C_Z\). These complexes are bounded below. The finite coefficient complexes are derived \(p\)-complete, so their limit is too. Evaluation at the point is exact and preserves limits; in particular \[H^j(\mathscr C_Z)(*)=H^j(R\Gamma_{\mathrm{int}}(Z)).\]

The compact-resolution argument in 33 gives Noetherianity of \(\Lambda_G\) and projective dimension \(\dim G\) for its trivial profinite module, using (Venjakob 2002, secs. 1.1–1.2, pp. 275–276) together with (Serre 1965, sec. 1, Corollary (1)). Successive finite free presentations therefore give the stated finite resolution. We give the derived comparison needed to use it. Let \(\mathcal C=D(\operatorname{Cond}_{\underline{\mathbb Z_p}})\) and \(\mathcal D=D(\operatorname{Solid}_{\underline{\mathbb Z_p}})\). Solidification is a symmetric monoidal localization \(L:\mathcal C\to\mathcal D\) with fully faithful right adjoint \(I\) (Tang 2026, Theorem 4.4(ii)). For the free condensed group ring \(A=\underline{\mathbb Z_p}[\underline G]\), its solidification \(LA\) is \(\underline{\Lambda_G}\) in degree zero, including multiplication (Tang 2026, sec. 3.6 and the proof of Theorem 4.4). The underived assertion also holds for a free module on any profinite set \(Q\): choose an extremally disconnected hypercover \(Q_\bullet\to Q\) and solidify its free resolution. The resulting augmented complex is \(\mathbb Z_p[[Q_\bullet]]\to\mathbb Z_p[[Q]]\). Its Pontryagin dual is the augmented complex of continuous \(\mathbb Q_p/\mathbb Z_p\)-valued functions. Discrete coefficients are acyclic on a profinite set, by passage to finite clopen partitions, so that dual complex is exact. Pontryagin duality and exact condensation prove the assertion. Functoriality with respect to multiplication on \(G\) gives the ring identification.

For any symmetric monoidal localization there is an induced adjunction \[L_A:\mathop{\mathrm{Mod}}_A(\mathcal C)\ \rightleftarrows\ \mathop{\mathrm{Mod}}_{LA}(\mathcal D):J,\] where \(J\) is inclusion followed by restriction along \(A\to I(LA)\). Its unit has underlying map \(M\to ILM\), and its counit has underlying map \(LIN\to N\). The counit is an equivalence; hence \(J\) is fully faithful, with essential image exactly the modules whose underlying complex is derived solid. This proves the assertion in unbounded degrees. To identify the target, apply this same localization to the already solid ring \(B=\underline{\Lambda_G}\). Its essential image inside \(D(\operatorname{Cond}_B)\) consists of precisely the complexes with derived solid underlying coefficients. By (Tang 2026, Theorem 4.4), the unbounded category \(D(\operatorname{Solid}_B)\) has that same essential image. This identifies the two derived module categories; it is stronger than an equivalence of their hearts. Equivalently, the comparison is computed on free modules \(B[Q]\) for extremally disconnected profinite \(Q\) and their solidifications, which give the projective resolutions in these categories. Thus the actual condensed action on \(K\) belongs to this category, and full faithfulness identifies its external derived invariants with those computed there. For geometric \(K\), the action and its coherent group law come from the maps on \(Z\times\underline Q\) with all group parameters retained; the preceding solidity argument applies to its underlying complex.

Condensation preserves the exact profinite resolution (Tang 2026, Theorem 3.2). Every \(P_a\) is a retract of a finite free \(\Lambda_G\)-module, and \(\mathop{\mathrm{Hom}}_{\Lambda_G}(\Lambda_G,M)=M(*)\) is exact. Thus these terms are projective against arbitrary solid coefficient modules and prove (92). The complex \(\mathop{\mathrm{Hom}}_{\Lambda_G}(P_\bullet,H^j(K))\) consists of retracts of finite sums of \(H^j(K)(*)\). Its retracts commute with \(\gamma\). After exact inversion of \(p\), the entire complex therefore has scalar action \(\lambda_j\), proving the assertion about its cohomology. No assertion about a condensed sheaf is inferred from its point value.

Finally the nerve of \(Z\to[Z/G]\) has terms \(Z\times\underline G^a\). With parameters retained, its descent complex is precisely the continuous bar construction on \(\mathscr C_Z\). At finite level the equivariant coefficient unit \(\underline{\Lambda_\ell}\to\mathscr C_{Z,\ell}\) is the evaluation map of 73 on every profinite parameter. Its map on \(G\)-invariants is therefore exactly \(\mathsf u_{Z,G,\ell}\), since their degree-\(a\) bar maps agree. This identity also holds after the derived coefficient limit and inversion of \(p\). Its source after that limit is \(R\Gamma(G,\underline{\mathbb Z_p})(*)\simeq R\Gamma_{\mathrm{int}}(G)\): the finite reductions of constants are surjective on every profinite parameter, and internal invariants and point evaluation preserve limits. The resolution bounds the group direction by \(\dim G\), giving (93) and the asserted finite filtration before and after rationalization. ◻

Proof. By (91), the class \(e_3\) is nonzero. Arithmetic acts trivially on its first summand and by \(\chi_{\mathrm{cyc}}^{-1}\) on the second. The cyclotomic character of \(G_F\) has open image, so the invariant subspace is precisely \(\mathbb Q_p e_3\).

Choose a compact uniform open subgroup \(U\subset J_3\). The restriction \(\operatorname{res}_U(e_J)\) of the source group class \(e_J\in H^3_{\mathrm b}(J_3)\) is nonzero by 74. Let \(q_U:[\Omega_C^2/U]\to\mathcal M\) be the quotient map and \(c_U:[\Omega_C^2/U]\to\mathrm B_{B_C}U\) its relative classifying map. Extension of torsor structure gives \(c_{J,\mathcal M}q_U\simeq(\mathrm B_{B_C}(U\hookrightarrow J_3))c_U\). Thus 73 identifies the restriction of the geometric \(e_J\) with \(\operatorname{cl}_{c_U,\mathrm b}(\operatorname{res}_U(e_J))\). Apply (93) to \([\Omega_C^2/U]\). The integral Drinfeld computation (Colmez et al. 2021, Theorems 1.1 and 5.1) identifies \(H^j(\mathscr C_{\Omega_C^2})(*)[1/p]\) as a representation on which arithmetic acts by the scalar \(\chi_{\mathrm{cyc}}^{-j}\), with \(j=0,1,2\) and no higher rows. Its integral convention is the derived limit of finite \(p\)-power coefficients followed by inversion of \(p\), exactly the convention here: the finite reduction tower defines \(\widehat{\mathbb Z}_p\) on the pro-etale site, and derived global sections preserve its derived inverse limit. The calculation is for hyperplanes rational over the ground field, here \(\mathbb Q_p\); we merely restrict its arithmetic action to \(G_F\).

In degree zero the same computation gives a one-dimensional rational point value. The finite coefficient units form a map \(\eta:\underline{\mathbb Z_p}\to\mathscr C_{\Omega_C^2}\) after their derived limit: on every profinite parameter the finite reductions of constants are surjective, so their limit is \(\underline{\mathbb Z_p}\). Pullback to any geometric point sends the unit to \(1\). Hence \[H^0(\eta)(*)[1/p]:\mathbb Q_p \xrightarrow{\ \simeq\ } H^0(\mathscr C_{\Omega_C^2})(*)[1/p].\] For each finite projective term \(P_a\) in the \(U\)-resolution of 77, applying \(\mathop{\mathrm{Hom}}_{\Lambda_U}(P_a,-)\) gives a retract of a finite sum of these point maps, and the unit commutes with its defining idempotent. It is therefore an isomorphism after inversion of \(p\) in every term. Naturality of the descent spectral sequence for \(\eta\) maps its single constant row to the bottom row of geometric descent. By the compact bar identity, the \(E_2^{3,0}\) bottom-row map of the spectral-sequence morphism induced by the finite units defining \(\operatorname{cl}_{c_U,\mathrm b}\) is the resulting isomorphism \[H^3_{\mathrm b}(U) \xrightarrow{\ \simeq\ } H^3(U,H^0(\mathscr C_{\Omega_C^2}))(*)[1/p].\] It sends \(\operatorname{res}_U(e_J)\) to the nonzero bottom-row class. This argument uses the rational point value and the natural finite projective complex.

By 77, an element \(\gamma\in G_F\) with cyclotomic character of infinite order acts on the actual \(j\)th descent row by this scalar. The only possible incoming differentials to \((3,0)\) are from \((1,1)\) and \((0,2)\). Their source scalars differ from \(1\), so both differentials vanish. There is no outgoing differential from the bottom row. Thus the nonzero bottom-row image of \(\operatorname{res}_U(e_J)\) survives in \(H^3_{\mathrm b}([\Omega_C^2/U])\), proving that the geometric class \(e_J\) on \(\mathcal M\) is nonzero. The map to \(\mathrm B_{B_C}J_3\) is arithmetic equivariant, so this class is invariant and belongs to the line \(\mathbb Q_p e_3\). This proves the assertion. ◻

Proof. The universal sequence on this stack is \[0\longrightarrow\mathcal L_2\otimes_{\mathbb Z_p}\mathcal O \longrightarrow\mathcal V_3\longrightarrow\mathcal V_1 \longrightarrow0,\] where \(\mathcal L_2\) is the integral rank-two local system of 13. The canonical quotient \(\mathcal V_1\to i_{\infty*}i_\infty^*\mathcal V_1\) defines \(\mathcal V_1^- =\mathcal V_1(-\infty)\) and, by pullback, a lower modification \(\mathcal V_3^-\) of \(\mathcal V_3\). It gives a map \(m:\mathcal Y_{1,C}\to\mathcal M\) and the exact sequence of slope-zero bundles \[0\longrightarrow\mathcal L_2\otimes\mathcal O \longrightarrow\mathcal V_3^- \longrightarrow\mathcal V_1^-\longrightarrow0.\] Apply the relative slope-zero equivalence on the pro-etale site of a perfectoid test object over \(\mathcal Y_{1,C}\). To verify exactness without assuming it, take a common pro-etale cover trivializing the three rational local systems. Full faithfulness identifies the maps with matrices of sections of \(\underline{\mathbb Q_p}\). Write the quotient map as \((b_1,b_2,b_3)\). The open loci \(b_i\ne0\) cover, since its rank is one on every geometric fiber. On such a locus, \(1\mapsto b_i^{-1}e_i\) is a continuous section. The kernel has basis \(e_j-(b_j/b_i)e_i\) for \(j\ne i\). The given map from \(\mathcal L_2\otimes\mathbb Q_p\) to this kernel is a square matrix invertible on every geometric fiber; its inverse is continuous by the determinant formula. Thus the sequence of rational local systems is exact and locally split.

The \(P_3\)-structure gives an integral determinant lattice on the middle local system. Its tensor quotient by \(\det\mathcal L_2\) is an integral lattice on the quotient line. Refine the cover to choose an integral basis of the original \(\mathcal L_2\) and an integral generator of this quotient lattice, and lift the generator by the displayed splitting. The resulting middle frame has determinant generating its prescribed determinant lattice. Its possible changes are exactly \(\bigl(\begin{smallmatrix}A&v\\0&a\end{smallmatrix}\bigr)\) with \(A\in\mathop{\mathrm{GL}}_2(\mathbb Z_p)\), \(a\in\mathbb Z_p^\times\), and \(v\in\mathbb Q_p^2\). Its determinant is a unit, so these adapted frames form exactly the \(D\)-torsor of 74. Forgetting the quotient generator and its lift gives the original integral Tate-frame torsor under \(D\to\mathop{\mathrm{GL}}_2(\mathbb Z_p)\). These constructions commute with all perfectoid base changes and descend on the v-site, giving the actual relative map \(c_D:\mathcal Y_{1,C}\to\mathrm B_{B_C}D\). The identifications of the associated torsors give homotopies of relative classifying maps \[\begin{aligned} c_{3,\mathcal M}m&\simeq c_{3,C},\\ (\mathrm B_{B_C}\iota)c_D&\simeq c_{J,\mathcal M}m,\\ (\mathrm B_{B_C}\rho)c_D &\simeq(\mathrm B_{B_C}\operatorname{pr}_1)c_{o,C}. \end{aligned}\] The first remembers the same structured positive bundle. The second forgets the adapted filtration and retains the middle determinant lattice induced from the same chosen \(\Delta_3\) lattice. The third forgets the quotient generator and its lift, retaining the original integral Tate frame. These forgetful identifications can be checked on the local frames just constructed and then descend; they commute with base change and with the specified semilinear arithmetic descent. Applying the finite classifying naturality and then its rational limit to \(e_3=\kappa e_J\) and (89) therefore gives \(e_3=\kappa b e_2\). Both factors are nonzero. ◻

Proof. For completeness, the Witt extension of the constants calculation uses the natural additive exact sequences \[0\longrightarrow R\xrightarrow{V^{\ell-1}}W_\ell(R) \longrightarrow W_{\ell-1}(R)\longrightarrow0.\] They are exact for the solid coefficient objects and v-sheaves in question: in Witt coordinates, restriction has the continuous set-theoretic section that appends zero. Apply the sequence to both sides of the characteristic-\(p\) constants equivalence. Induction using the resulting morphisms of exact triangles proves the equivalence at every length. Take derived inverse limits and then the residual group invariants. All maps in this construction are induced by constants, so it proves the square with its asserted arrows and their compatibility with cup products.

The group \(G_o\) has no \(p\)-torsion. Choose its resolution in 77 and put \(Q_\bullet=\mathbb Z_p\otimes_{\mathbb Z_p[[G_o]]}P_\bullet\). Each \(Q_j\) is finite projective, hence finite free over \(\mathbb Z_p\). For a trivial coefficient module \(M\), the computing complex is \(\mathop{\mathrm{Hom}}_{\mathbb Z_p}(Q_\bullet,M)\), a fixed bounded complex of finite matrices over \(\mathbb Z_p\) tensored with \(M\). Since \(W(C^\flat)\) is torsion free over the discrete valuation ring \(W(k)\), it is flat, and its fraction field extension is faithfully flat. Consequently \[H^i(G_o,W(k))[1/p]\otimes_{W(k)[1/p]}W(C^\flat)[1/p] \simeq H^i(G_o,W(C^\flat))[1/p].\] This proves the injection in the square. It also proves that the image of \(e_2\) is nonzero: it is the scalar extension of the nonzero trace class of the first factor of \(G_o\).

By (95) and its \(P_3\) counterpart, the two Witt images of the group classes on \(\mathcal Y_{1,C}\) are the coefficient-unit images of the exact classifying images in (94). Their maps over \(k\) base change to these maps before the derived limit is taken. Applying that limit and inversion of \(p\) to the geometric equality, and then using the injection in (96), proves the same equality over \(k\) in Witt cohomology. Its right side is nonzero, as just shown.

It remains to follow this nonzero Witt class through the actual deformation-coefficient map. Put \[\mathsf C^W_\ell=C^\bullet_{\mathrm{cts}}(P_3,W_\ell B_0),\qquad \mathsf C^D_\ell=C^\bullet_{\mathrm{cts}}(P_3,D_\ell),\qquad D_\ell=(W(k)/p^\ell)[[x,y]][x^{-1}].\] By 66, the two maps from \(W_\ell B_0\) to \(D_\ell\) and to \(W_\ell B_{\mathrm{pk}}\) induce cohomology isomorphisms. The graded maps for \(\gamma_\ell\) are powers of \(\phi_p\), while those for \(\beta_\ell\) are the natural maps \(B_0\to B_{\mathrm{pk}}\). Both are cohomological equivalences by 62. This statement is used at each finite length; the two inverse systems have different transition maps.

Choose an integral constant class \(\beta\in H^3_{\mathrm{int}}(P_3)\) with \(\beta[1/p]=p^N e_3\) for some \(N\geq0\), where \(e_3\) denotes the source class in \(H^3_{\mathrm b}(P_3)\). The normalized rational trace is fixed by \(\Gamma=\mathop{\mathrm{Gal}}(k/\mathbb F_p)\), as proved in 74. Thus each difference \(\gamma\beta-\beta\) is killed by a power of \(p\). Since \(\Gamma\) is finite, one common power kills all these differences. Replacing \(\beta\) by that multiple and increasing \(N\) makes the integral cohomology class \(\Gamma\)-invariant while retaining the same nonzero rational trace line. Let \(\bar\beta_\ell\) be its canonical projection to \(H^3(C^\bullet_{\mathrm{cts}}(P_3,\Lambda_\ell))\). These finite cohomology classes are reduction-compatible and \(\Gamma\)-invariant. Let \(a\in H^3(R\varprojlim_{R_\ell}\mathsf C^W_\ell)\) be its coefficient image, where \(R_\ell:W_\ell B_0\to W_{\ell-1}B_0\) is ordinary Witt restriction. The maps to perfected and geometric Witt coefficients commute with these restrictions. The nonvanishing already proved therefore implies \(a[1/p]\ne0\). The groups \(H^2(\mathsf C^W_\ell)\) are finite, by the finite-length additive Witt filtration and [h3:prop:bounded-pole-finiteness,h3:thm:ordinary-comparison]. Their tower is Mittag–Leffler, so the Milnor sequence gives \[H^3(R\varprojlim_{R_\ell}\mathsf C^W_\ell) =\varprojlim_{R_\ell} H^3(\mathsf C^W_\ell).\] Consequently the component classes \(a_\ell\) have unbounded \(p\)-power orders. By naturality of the projections, \(a_\ell\) is the coefficient image of \(\bar\beta_\ell\) in \(W_\ell B_0\) cohomology.

Write \(d_\ell\) for their ghost images in \(H^3(\mathsf C^D_\ell)\). These have exactly the same orders. The ghost maps fix \(\mathbb Z/p^\ell\), so \(d_\ell\) is the direct coefficient image of \(\bar\beta_\ell\) and is compatible with coefficient reduction. More explicitly the transition identity is \[\operatorname{red}_{\ell,\ell-1}\gamma_\ell =\gamma_{\ell-1}R_\ell W_\ell(\phi_p),\] where \(\phi_p(b)=b^p\) is the characteristic-\(p\) Frobenius defined in 66. The class \(a_\ell\) is fixed by \(W_\ell(\phi_p)\) because it comes from \(W_\ell(\mathbb F_p)\). The naturalities of the finite units also make each \(d_\ell\) \(\Gamma\)-invariant. Exact semilinear descent in [h3:lem:framework-semilinear,h3:prop:exact-descent] identifies, by restriction, \[H^3_{\mathrm{cts}}(G_3(k),D_\ell) \simeq H^3_{\mathrm{cts}}(P_3,D_\ell)^\Gamma.\] Here descent is applied after extension to the semilinear \(D_\ell\) coefficients. The descended class restricts to \(d_\ell\), so it has the same order; these identifications also preserve ordinary reduction. Thus the specified descended constant classes have unbounded \(p\)-power order. The natural spectral comparison in [h3:des:minus-three-edge,h3:prop:integral-delta], with its unique bidegree \((s,t)=(3,0)\) in stem \(-3\), identifies this family with the finite projections of a class in \(\pi_{-3}L_{K(1)}T\). Its unbounded finite orders make that class nonzero after rationalization, and the identification preserves the actual maps from constants. ◻

Proof. Reduction followed by the ordinary coefficient localization \(E_{3,0}(k)\to D_\ell\) sends the integral constant class chosen in 80 to its specified finite classes \(d_\ell\). These have unbounded \(p\)-power order and are compatible with ordinary reduction. Hence the coefficient image of the rational trace in \(H^3(P_3,E_{3,0}(k))[1/p]\) is nonzero. The trace construction and its coefficient map commute with the semilinear finite Galois action; their class is invariant and descends by 7.

We explain why this cohomology class is an actual rational homotopy class. A central element \(z=1+p\) acts trivially on \(E_{3,0}(k)\) and by the nontrivial scalar \(z^{-q}\) on the nonzero periodicity twist \(E_{3,2q}(k)\), for \(q\ne0\). This is the action on \(\omega^{\otimes q}\) in the contravariant deformation convention of 16. A central group element acts as the identity on group cohomology when its coefficient action is combined with its trivial inner conjugation. Equivalently, the natural transformation furnished by that element gives a homotopy to the identity on the bar resolution. Thus \(z^{-q}-1\) annihilates \(H^s(P_3,E_{3,2q}(k))\). It is a nonzero element of \(\mathbb Z_p\), so every such row vanishes after inversion of \(p\).

The descent of 50 is exact and strongly convergent. The finite stabilizer resolutions and exact semilinear descent of [h3:rat:solid-resolution,h3:lem:framework-semilinear] bound its cohomological amplitude. Rationalization therefore preserves its finite filtrations. Only internal degree zero remains. After finite coefficient-field descent, the actual coefficient maps fit into the commutative square \[\begin{tikzcd}[column sep=2em] H^3_{\mathrm{cts}}(G_3(k),E_{3,0}(k))[1/p] \arrow[r] \arrow[d,"\simeq"] & \bigl(\varprojlim_\ell H^3_{\mathrm{cts}}(G_3(k),D_\ell)\bigr)[1/p] \arrow[d,"\simeq"] \\ \pi_{-3}L_0T \arrow[r] & \pi_{-3}L_0\operatorname*{holim}_\ell L_1(T/p^\ell). \end{tikzcd}\] The upper arrow is induced by the coefficient reductions and localizations just described. The left isomorphism is the single-row rational descent calculation. The right isomorphism uses the natural, unique \((s,t)=(3,0)\) edge of 68 and the vanishing of the Milnor \(\lim^1\) term in 70. The finite-length model identifies those coefficient maps with the actual Moore quotient and localization maps on every descent term. For commutativity, first choose an integral representative of a \(p\)-power multiple of a rational source cohomology class. Finite source filtration and rational collapse allow a further \(p\)-power multiple to survive to homotopy. Naturality in 50 identifies its finite target edges with that same multiple of the coefficient images, for every \(\ell\). This multiplier is chosen in the source, independently of \(\ell\). Taking the coefficient limit and then inverting \(p\) gives the displayed square.

By 69, the bottom map is the rationalization of \(T\to L_{K(1)}T\). The nonzero rational trace image in the upper left therefore determines \(\alpha_3\in\pi_{-3}L_0T\), and its image in the upper right is the nonzero rational class of the compatible finite constants. The square proves the asserted nonvanishing of its actual localization without an additional arithmetic coefficient splitting. ◻

Proof. The rational exterior calculation (19) makes degree \(-3\) one-dimensional. Choose its generator to be the nonzero constant-origin class of 81. The determinant class is the degree-one group primitive by its construction in 7, and its actual image is nonzero rationally by 72. Thus it may be used as \(\alpha_1\). Choose any remaining exterior generator in degree \(-5\).

By 70, the rational target in degree \(-3\) is precisely \(\mathbb Q_p\delta\). Nonvanishing of the actual image gives \(\alpha_3\mapsto c\delta\) with \(c\ne0\). All coefficient comparisons and localizations used above respect products, and the same actual determinant class occurs on source and target. Multiplication by it gives the second formula with the same scalar. The exterior calculation then supplies the displayed basis. ◻

Proof of 1 and the height-three instance of 4. Put \(T=L_{K(3)}S\), \(X=L_2T\), and \(R_1=L_{K(1)}S\). Recall \(Q=L_0S\simeq H\mathbb Q_p\), and set \[W=W_1\vee W_2=L_2S\vee\Sigma^{-1}L_2S.\] 53 supplies the actual unit and determinant map \(w=(1,\zeta):W\to X\), a \(K(2)\)-equivalence. This verifies (i) of 3. Put \(Y=\mathop{\mathrm{cofib}}(w)\) and write \(q:X\to Y\).

For the middle block, set \[U_{\mathrm{mid}}=U_3\vee U_4=\Sigma^{-3}L_1S\vee\Sigma^{-4}L_1S.\] 72 supplies the equivalence of actual \(R_1\)-module maps \[ (1,\zeta,\delta,\zeta\delta): R_1\vee\Sigma^{-1}R_1\vee\Sigma^{-3}R_1\vee\Sigma^{-4}R_1 \xrightarrow{\simeq}L_{K(1)}T. \tag{98}\] Its first two components are the localizations of the same maps from \(S\) used in \(w\). Their restrictions are the same unit and determinant maps, and the module localization adjunction (1) identifies their extensions to \(R_1\), including the required homotopies. The fourth component is the specified product \(\zeta\delta\). Localizing the cofiber sequence at \(K(1)\) therefore identifies \(L_{K(1)}Y\) with the cofiber of the first two components of (98) using those specified maps. Consequently the last two components of (98), followed by the quotient, give the required equivalence \[ \kappa:L_{K(1)}U_{\mathrm{mid}}\xrightarrow{\simeq}L_{K(1)}Y. \tag{99}\] They are represented by the projections of \(\delta\) and \(\zeta\delta\).

For the rational part of the middle map, 82 gives \[\pi_*L_0T=\Lambda_{\mathbb Q_p}(\zeta,\alpha_3,\alpha_5), \qquad |\zeta|=-1,\quad|\alpha_3|=-3,\quad|\alpha_5|=-5,\] and identifies the actual rationalized localization map by \[ \alpha_3\longmapsto c\delta,\qquad \zeta\alpha_3\longmapsto c\zeta\delta \quad\text{for one }c\in\mathbb Q_p^\times. \tag{100}\] Set \[\beta_3=c^{-1}\alpha_3.\] This normalization is made only in the rational source. No condition \(c\in\mathbb Z_p^\times\) is required. The projections of \(\beta_3\) and \(\zeta\beta_3\) to \(L_0Y\) define a \(Q\)-module map \[r:L_0U_{\mathrm{mid}}=\Sigma^{-3}Q\vee\Sigma^{-4}Q\longrightarrow L_0Y.\] By (100), their images in \(L_0L_{K(1)}Y\) are the projections of \(\delta\) and \(\zeta\delta\). These are also the images, under \(L_0\kappa\), of the two unit generators of \(L_0U_{\mathrm{mid}}\). The maps in (6) therefore agree on the generators of this finite free \(Q\)-module. An identifying homotopy of \(Q\)-module maps exists; choose one as the compatibility datum in (ii) of 3.

It remains to check the final quotient. Since \(L_0L_iS=Q\) for \(i=1,2\), rationalization of \(w\) is the map from \(Q\vee\Sigma^{-1}Q\) represented by \(1,\zeta\). These are two distinct members of the exterior basis in 82, so this map is injective on rational homotopy. Its cofiber \(L_0Y\) has one copy of \(\mathbb Q_p\) in each degree \[-3,\ -4,\ -5,\ -6,\ -8,\ -9.\] There are no suspended kernel terms. The map \(r\) is likewise injective on homotopy and includes the first two lines, represented by \(\beta_3,\zeta\beta_3\).

Put \[\begin{aligned} V&=V_5\vee V_6\vee V_7\vee V_8\\ &=\Sigma^{-5}Q\vee\Sigma^{-6}Q \vee\Sigma^{-8}Q\vee\Sigma^{-9}Q. \end{aligned}\] Define \(b:V\to L_0X\simeq L_0T\) by the four ordered classes \[ \alpha_5,\qquad \zeta\alpha_5,\qquad \beta_3\alpha_5,\qquad\zeta\beta_3\alpha_5. \tag{101}\] They represent the remaining quotient lines in degrees \(-5,-6,-8,-9\). The composite (7) is therefore an isomorphism on all homotopy groups, and hence an equivalence of rational modules. This verifies (iii).

Carry out 3 with this single choice of \(w,\kappa,r,b\) and compatibility homotopy, choosing coherent inverse data for \(L_{K(2)}w\) and for \(L_Ja\) at the steps where those equivalences are established. This gives the same \(h,F_i,a,Z,t,e\) used by 4, with exactly the ordered cofibers in 1. The terminal projection \(e:F_8\to X\) is an equivalence. Its restriction to \(F_1=L_2S\) is the first component of \(w\), namely the canonical map \(L_2(S\to T)\), proving 1. For these same choices, 4 identifies all eight connecting maps by its two signed roofs. This proves its height-three instance without replacing the filtration or choosing new local maps. ◻

Proof. For \(\mathrm{GL}_3\), the semisimple quotient used to define the reduced building is the split group \(\mathrm{PGL}_3\) of rank two. The geometry statement in (Meyer and Solleveld 2010, sec. 1.1.2, p. 6) applies over the local field \(F\): the reduced building is locally finite, has dimension the rank of this quotient, and is equivariantly contractible for every compact subgroup of the quotient. Taking the trivial compact subgroup gives contractibility.

In the lattice-chain model of (Deligne and Flicker 2013, sec. 5.6, pp. 952–953), vertices are homothety classes \([L]\) of \(\mathcal O\)-lattices in \(F^3\), and a chamber can be represented by a strict cyclic chain \[L_0\supset L_1\supset L_2\supset\pi L_0.\] Modulo \(\pi L_0\) this is a complete flag in \(\kappa^3\). Thus a chamber has three vertices and is a two-simplex. An adjacent vertex to \([L]\) has a unique representative \(M\) with \(\pi L\subsetneq M\subsetneq L\). These representatives correspond to the nonzero proper subspaces of \(L/\pi L\cong\kappa^3\). There are \(q^2+q+1\) lines and the same number of planes, giving the stated valency and a direct local-finiteness check. Since the complex is locally finite, its metric polyhedral and weak simplicial topologies agree locally on finite unions of simplices. Its realization is therefore a contractible CW complex. The augmented cellular chain complex over \(\mathbb Z\) is exact, with free direct-sum terms in degrees \(0,1,2\).

Fix the standard lattice \(\mathcal O^3\) and define \[\operatorname{type}([g\mathcal O^3])=\nu(\det g)\pmod 3.\] Changing an integral basis changes the determinant by a unit; scalar homothety changes its valuation by a multiple of three. This type is therefore well defined, and \(J_3(F)\) preserves it. The three vertices of a chamber have all three types, since successive flag steps change the lattice determinant valuation by one.

To send the standard chamber to a given chamber, start at the latter’s unique type-zero vertex. Write its lattice as \(M=g\mathcal O^3\) with \(\nu(\det g)=3m\), and replace \(M\) by \(M_0=\pi^{-m}M\). Reanchor the cyclic chain at \(M_0\). A basis \(f_1,f_2,f_3\) of \(M_0\) adapted to its residue flag gives \[M_i=\pi\mathcal O f_1+\cdots+\pi\mathcal O f_i +\mathcal O f_{i+1}+\cdots+\mathcal O f_3 \quad(i=1,2).\] The map from the standard basis to \((f_1,f_2,f_3)\) has determinant valuation zero, because its image lattice is \(M_0\). It lies in \(J_3(F)\) and sends the standard chamber to the given chamber.

If \(h\in J_3(F)\) stabilizes \([L]\), then \(hL=aL\) for some \(a\in F^\times\). Taking determinant valuations relative to \(L\) gives \(0=\nu(\det h)=3\nu(a)\), so \(a\) is a unit and \(hL=L\). Hence the vertex stabilizer is exactly \(\operatorname{Aut}_{\mathcal O}(L)\), a conjugate of the compact open group \(\mathrm{GL}_3(\mathcal O)\). A setwise face stabilizer fixes each vertex because their types are distinct. It is therefore the finite intersection of these vertex stabilizers, hence is compact open and fixes the face pointwise.

Every face lies in a chamber, and chamber transitivity sends it into the standard chamber. Its set of vertex types picks out a unique face there. Type preservation also preserves the barycentric coordinates labeled by those types, so no distinct points of the closed chamber are identified. This proves the fundamental-domain assertion, including its faces. ◻

Proof. The imported model facts are given in (Grosse-Klönne 2005, secs. 3.6, 3.8, 6.1–6.2, 6.4 and 7.1, and the proof of Proposition 6.5). After ordinary \(p\)-adic completion the local strictly semistable equations and the special fiber are unchanged. The model is separated: each of the increasing weak-formal opens lies in a separated blowup stage, and any two lie in one common such stage; the local closed-diagonal criterion is preserved by completion. The components and their nonempty intersections are products of successive blowups of projective spaces along rational linear subspaces. They are indexed by the vertices and simplices of the locally finite two-dimensional Bruhat–Tits building \(\mathcal B\) of 87. Only the factors in dimensions \(0,1,2\) occur here.

We compute the structure-sheaf cohomology of exactly these factors. For \(\mathbf P^n_k\), \(0\leq n\leq2\), over a field \(k\), use the standard affine cover. On an intersection indexed by a nonempty set \(I\subset\{0,\ldots,n\}\), its degree-zero homogeneous Laurent monomials have exponent vector \((a_0,\ldots,a_n)\) with \(\sum a_i=0\) and \(a_i\geq0\) when \(i\notin I\). The Čech complex decomposes by these monomials. The constant monomial gives the cochain complex of the full simplex and hence \(k\) in degree zero. For any other monomial its negative support \(N=\{i:a_i<0\}\) is nonempty and proper, and the monomial occurs exactly for \(I\supset N\). Adding and deleting a fixed index outside \(N\), with the usual alternating sign, contracts this summand. It follows that \[R\Gamma(\mathbf P^n_k,\mathcal O)=k[0].\]

The only nontrivial blowups needed for a surface factor are blowups at the specified smooth rational points. The calculation for one such point is local on the surface and can be made in étale coordinates \((u,v)\). For the blowup of \(\mathbf A^2_k\) at the origin, the two standard charts and their overlap have rings \[B_u=k[u,t],\qquad B_v=k[v,s],\qquad B_{uv}=k[u,t,t^{-1}],\qquad v=ut,\quad s=t^{-1}.\] In the common Laurent ring a monomial \(u^a t^b\), with \(a\geq0\) and \(b\in\mathbb Z\), belongs to \(B_u\) when \(b\geq0\) and to \(B_v\) when \(b\leq a\). These two conditions cover all monomials. Their intersection is spanned by \(0\leq b\leq a\), where \(u^a t^b=u^{a-b}v^b\). Thus the affine Čech sequence \[0\longrightarrow k[u,v]\longrightarrow B_u\oplus B_v \longrightarrow B_{uv}\longrightarrow0\] is exact. The same sequence remains exact after localization and flat étale base change, which are the local coordinate changes in question. The affine chart calculation therefore proves \(R\pi_*\mathcal O=\mathcal O\) for each required smooth-point blowup \(\pi\). Blowing up an invertible ideal is an isomorphism. This covers the later strict transforms of rational lines on the surface and the point centers on a projective line, all of which are Cartier divisors. Iterating the result, and using the finite affine-cover double Čech complex for products over \(k\), proves \[R\Gamma(Y,\mathcal O_Y)=k[0]\] for every stratum \(Y\) occurring in this height-three model. This argument makes no assertion about other rational varieties. For a profinite parameter \(Q\) the same calculations give \(C_{\mathrm{cts}}(Q,k)[0]\): continuous maps to each discrete section module factor through a finite clopen partition of \(Q\), and the filtered union of the resulting finite powers preserves the exact Čech sequences.

We next justify the resolution by intersections of components. On a standard special-fiber chart its augmented form is the alternating complex \[k[t_0,\ldots,t_d]/(t_0\cdots t_r) \longrightarrow \bigoplus_i k[t_0,\ldots,t_d]/(t_i) \longrightarrow \bigoplus_{i<j} k[t_0,\ldots,t_d]/(t_i,t_j) \longrightarrow\cdots .\] For a nonzero monomial, let \(Z\) be the nonempty set of indices \(i\leq r\) at which its exponent is zero. Its coefficient complex is the augmented cochain complex of the simplex on \(Z\), so it is exact. This proves exactness monomial by monomial. The calculation is unchanged by the torus variables, localizations, and flat étale base changes in the semistable charts. Local finiteness of the components now gives the global resolution by the products of their intersection sheaves. Taking derived global sections with a profinite parameter and using the preceding stratum calculation computes \(R\Gamma(\mathfrak H_{\mathbb F_p}\times\underline Q,\mathcal O)\) by the three product terms below. The arrow from \(C_{\mathrm{cts}}(Q,\mathbb F_p)\) is the separate canonical constants augmentation: \[C_{\mathrm{cts}}(Q,\mathbb F_p)\longrightarrow \prod_{\sigma\in\mathcal B_0}C_{\mathrm{cts}}(Q,\mathbb F_p) \longrightarrow \prod_{\sigma\in\mathcal B_1}C_{\mathrm{cts}}(Q,\mathbb F_p) \longrightarrow \prod_{\sigma\in\mathcal B_2}C_{\mathrm{cts}}(Q,\mathbb F_p),\] with the cellular incidence signs. Products occur because the strata are indexed by all cells of the building.

Here is a contraction valid for these product cochains. By 87, the augmented bounded cellular chain complex \(C_\bullet(\mathcal B;\mathbb Z)\to\mathbb Z\) is exact. Its terms are free abelian, and each cycle subgroup is free because it is a subgroup of a free abelian group. The short exact sequences of cycles therefore split. Choosing splittings gives a chain contraction of the augmented complex. This contraction is not asserted to be equivariant. Each basis cell maps to a finite cellular chain, since a cellular chain group is a direct sum of its cells. Dualizing into \(C_{\mathrm{cts}}(Q,\mathbb F_p)\) consequently gives a contraction of the displayed product cochains: every output coordinate is a finite linear combination of input coordinates and is continuous for the product topology. The contraction is natural in \(Q\).

The canonical constants map into cellular cochains is equivariant for the building action. The contraction proves that its underlying condensed map is an equivalence; the forgetful functor from equivariant derived objects is conservative, so this same canonical map is an equivariant equivalence. In particular \[\underline{\mathbb F_p}\xrightarrow{\ \simeq\ } R\Gamma(\mathfrak H_{\mathbb F_p},\mathcal O).\] Finally the formal structure sheaf is derived \(p\)-complete, and derived global sections preserve its inverse limit of reductions. The cone of the actual scalar map \(\underline{\mathbb Z_p}\to R\Gamma(\mathfrak H,\mathcal O)\) is therefore derived \(p\)-complete. Flatness of the semistable model identifies its reduction modulo \(p\) with the zero cone just computed. Every successive reduction modulo \(p^\ell\) is then zero, and derived \(p\)-completeness makes the cone itself zero. This proves the claimed equivariant equivalence. ◻

Proof of 86. All cohomology complexes below retain their condensed parameters; all inverse limits and group invariants are derived and taken before inverting \(p\).

1. Arithmetic bounds. Let \(K\) be complete discretely valued of mixed characteristic \((0,p)\) with perfect residue field, and let \(C\) be a completed algebraic closure. The scalar class obtained from the cyclotomic logarithm gives a natural map \[\mathcal O_K[\epsilon]\longrightarrow R\Gamma(G_K,\mathcal O_C),\qquad |\epsilon|=1, \quad \epsilon^2=0.\] Its cone is cohomologically nonnegative, and in each degree its cohomology is killed by a power of \(p\). The required bounds can be chosen uniformly after finite tame extensions of \(K\). For the finitely many nonzero twists \(j=1,2\), the same assertion of bounded torsion holds for \(H^i(G_K,\mathcal O_C(-j))\), with \(H^0=0\). Here is the part of the proof that gives uniformity, (Barthel et al. 2025, secs. 4.2–4.4). Choose a finite cyclotomic stage \(B/K\) far enough out that the remaining \(\mathbb Z_p\)-tower is totally ramified and sufficiently ramified in the sense of (Barthel et al. 2025, Definition 4.2.5 and Lemma 4.2.6), generated by \(\sigma\), with \(\chi(\sigma)\) a generator of \(1+p^s\mathbb Z_p\), \(s\geq2\). Write \(M=\mathcal O_{\widehat K_\infty}\) and \(V=\ker(t:\widehat K_\infty\to B)\) for normalized trace. The estimates in (Barthel et al. 2025, (4.2.9), (4.2.15), Lemmas 4.2.12 and 4.2.17, and the proof of Proposition 4.2.18) are \[\|t\|\leq |p|^{-1/(p-1)},\qquad \|(\sigma-1)^{-1}\|_V \leq |p|^{-1-1/[p(p-1)]}.\] For existence of the inverse, at a finite cyclic level \(\sigma-1\) has kernel \(B\) and image the trace-zero subspace. Its inverse there is bounded by the second estimate, so the inverses extend compatibly to the completion. The estimate follows by induction from normalized layer trace \(t_n=p^{-1}\operatorname{Tr}\) and \[x-t_nx=p^{-1}\sum_{a=1}^{p-1} (1-\sigma^{a p^{n-1}})x,\qquad \|t_n\|\leq |p|^{-p^{-n}}.\] Set \(M_0=\mathcal O_B\oplus(V\cap M)\subset M\). The first norm bound gives \(pM\subset M_0\), so \(M/M_0\) is killed by \(p\). For \(j\in\{-2,-1,1,2\}\), put \(\lambda=\chi(\sigma)^{-j}\); then \(v_p(\lambda-1)=s\), since \(p\geq5\). The inequality \(|\lambda-1|\,\|(\sigma-1)^{-1}\|_V<1\) follows from \(s\geq2>1+1/[p(p-1)]\). A Neumann series therefore inverts \(\sigma-\lambda\) on \(V\) with the same bound (Barthel et al. 2025, Lemma 4.4.1), so its cokernel on \(V\cap M\) is killed by \(p^2\). On \(\mathcal O_B\) its cokernel is killed by \(p^s\). Thus \(H^1(\langle\sigma\rangle,M_0(j))\) is killed by \(p^s\), while the invariants vanish. The sequence \(M_0\to M\to M/M_0\) gives a \(p^{s+1}\) bound for \(H^1(\langle\sigma\rangle,M(j))\). The rational decomposition \(B\oplus V\) also shows directly that \(M(j)\) has no invariants under the tail. Restriction from \(\mathop{\mathrm{Gal}}(K_\infty/K)\) to this open tail injects \(H^1\), because the tail has no invariants. The cyclotomic fixed-field calculation and affinoid perfectoid integral acyclicity compare this with \(\mathcal O_C(j)\) (Barthel et al. 2025, Lemmas 4.3.2 and 4.3.5). Before cyclotomic invariants, the perfectoid-tail degree-zero comparison has almost-zero cohomological error. The maximal ideal of the completed valuation ring is flat and idempotent, so this error is annihilated by \(p\) in the equivariant derived category. In the resulting triangle after invariants, the error \(Y\) has \(H^0(Y)=0\) and \(pH^i(Y)=0\) for \(i>0\) (Barthel et al. 2025, Lemma 4.3.5 and Theorem 4.4.3). This gives the conservative bound \(p^{s+2}\) in degree one; the remaining degrees are bounded as in the same triangle. The untwisted calculation retains the constant summand and its cyclotomic-logarithm class, giving the displayed map \(\mathcal O_K[\epsilon]\) and its bounded-torsion cone. Sufficient ramification persists under tame base change. For a finite tame extension \(L/K\), let \(e=e(LB/B)\) be the effective tame ramification index after the chosen initial cyclotomic stage \(B\). If \(d_n\) is the different exponent of a \(p\)-layer above \(B\), then the base-changed exponent is \[d'_n=e d_n-(e-1)(p-1).\] Consequently the same tail, \(s\), and bounds work for all the tame extensions in question. Only existence of these tame-uniform bounds is used; the sharp numerical constants in the preprint are unnecessary.

2. Semistable comparison.

First apply this to a small affine semistable formal \(\mathcal O_K\)-scheme \(\mathfrak U=\operatorname{Spf}(R_0)\) of relative dimension \(d\leq2\), where small means admitting an étale morphism to a standard semistable chart. Write \(U\) for its generic fiber, \(\mathfrak U_C\) for its completed base change to \(\mathcal O_C\), and \(U_C\) for the geometric generic fiber. Use the divisorial log structure and the valuation log structure on the base. The semistable logarithmic comparison of (Česnavičius and Koshikawa 2019, secs. 1.5–1.6 and Theorem 4.11), extending the smooth comparison of (Bhatt et al. 2018, Theorem 8.3), applies on étale charts \[\mathcal O_C\langle T_0,\ldots,T_r, T_{r+1}^{\pm1},\ldots,T_d^{\pm1}\rangle/ (T_0\cdots T_r-a),\qquad 0\ne a\in\mathfrak m_C.\] They identify the cohomology of \(L\eta_{\zeta_p-1}R\nu_*\widehat{\mathcal O}^{+}\) with \(\Omega^j_{\mathfrak U_C,\log}\{-j\}\) for \(0\leq j\leq d\). No properness is required for these local statements. The logarithmic differentials on \(\mathfrak U\) are coherent and flat over \(\mathcal O_K\), and their higher cohomology on this affine vanishes. Their completed base changes are the displayed differentials on \(\mathfrak U_C\). The needed completed projection formula can be checked modulo \(p\): a flat module modulo \(p\) is free over the artinian local ring \(\mathcal O_K/p\), and continuous cochains on a compact group commute with these direct sums, since a map to a discrete direct sum has finite image. Derived \(p\)-completeness then proves the projection formula before reduction. For \(j>0\), the normalized map from the Breuil–Kisin twist to the Tate twist has cokernel killed by \(p\), so the preceding arithmetic bounds apply to each of the finitely many positive graded pieces.

For the comparison with the original complex, put \(C_U=R\Gamma(U_{C,\mathrm{pro\acute et}},\widehat{\mathcal O}^{+})\), \(I=(\zeta_p-1)\subset\mathcal O_C\), and \(A_U=L\eta_I C_U\). The ideal \(I\) is \(G_K\)-stable, although its displayed generator need not be fixed. On the semistable charts above, the perfectoid-cover edge map of (Česnavičius and Koshikawa 2019, sec. 3.3, Equation (3.3.1)) has almost-zero cone. Its source is the continuous cochain complex for \(\Delta\simeq\mathbb Z_p^d\), represented by a \(d\)-variable Koszul complex (Česnavičius and Koshikawa 2019, Lemma 3.7). Thus \(H^q(C_U)\) is almost zero for \(q>d\). Moreover, (Česnavičius and Koshikawa 2019, Theorems 3.9 and 4.11) identifies \(A_U\) with a complex concentrated in degrees \([0,d]\). The degree-zero cohomology of \(C_U\) is \(I\)-torsion-free; indeed, it is the formal structure ring (Česnavičius and Koshikawa 2019, Proposition 4.4).

Set \(B_U=\tau^{\leq d}C_U\). Lemma 6.9 of (Bhatt et al. 2018), together with compatibility of \(L\eta_I\) with truncation, gives maps \[a:A_U\longrightarrow B_U, \qquad b:I^{\otimes d}\otimes B_U\longrightarrow A_U\] whose composites are the natural ideal inclusions. The invariant element \(p^d\in I^d\) gives an equivariant map \(\mathcal O_C\to I^{\otimes d}\); composing it with \(b\) gives \(b_p:B_U\to A_U\) with \(ab_p=p^d\) and \(b_pa=p^d\). Hence the kernel and cokernel of each \(H^q(a)\) are killed by \(p^d\), and the long exact sequence gives \(p^{2d}H^q(\operatorname{cone}(a))=0\). The triangle \[\operatorname{cone}(a)\longrightarrow \operatorname{cone}(A_U\to C_U)\longrightarrow \tau^{>d}C_U\] then shows that \(p^{2d+1}\) kills every cohomology group of the full cone: the last term has almost-zero cohomology, which is killed by \(p\). The full cone is connective. This argument concerns its cohomology and does not assert that it carries a strict \(\mathbb Z/p^{2d+1}\)-module structure.

After \(G_K\)-invariants, the first-quadrant spectral sequence has terms \(H^s(G_K,H^t(\operatorname{cone}(A_U\to C_U)))\). In total degree \(k\) its filtration has at most \(k+1\) pieces, each killed by \(p^{2d+1}\). Therefore \(p^{(2d+1)(k+1)}\) kills the cohomology in that degree. These conservative bounds depend on \(d\) and \(k\), and are independent of the chart and its tame extension. The positive logarithmic graded pieces retain the arithmetic bounds proved in Step 1. Tame descent preserves these bounds, by unramified semilinear descent and averaging over tame inertia. We obtain the natural comparison \[\mathcal O(\mathfrak U)[\epsilon]\longrightarrow R\Gamma(U_{\mathrm{pro\acute et}},\widehat{\mathcal O}^{+})\] with nonnegative cone and degreewise bounds independent of the chosen charts and their tame extensions. For a general semistable affine, choose an étale hypercover refining the Čech nerve of a small-chart cover, with each term a finite disjoint union of small affine charts. Such finite covers of the matching objects exist because the base is quasi-compact and quasi-separated. Apply étale hyperdescent to the actual natural comparison map. Its levelwise cones are connective and have the uniform degreewise bounds just proved. In a fixed total degree the descent spectral sequence has only finitely many cosimplicial and cohomological positions, so the same bounded-torsion conclusion follows. Gluing next by a separated affine formal cover likewise preserves a bound in every total degree, since the relevant Čech diagonal is finite. This proves the part of (Barthel et al. 2025, secs. 5.3–5.6) needed here.

3. The two global calculations. The Lubin–Tate space \(\mathrm{LT}_{\breve K}\) is the open two-dimensional ball. Exhaust it by closed balls of radii \(|p|^{1/\ell}\) for primes \(\ell\ne p\); each has a smooth model after a tame extension. The two-index system of integral functions modulo \(p^j\), indexed by radius and \(j\), is Mittag–Leffler. This coefficient limit is taken over the fixed base \(\breve K\), whose uniformizer is \(p\); the tame extensions were used for the local comparisons. More generally, with uniformizer \(\varpi\), for a fixed inner radius \(r\) and reduction exponent \(j\) take \(M=\lceil(j+1)/\log_{|\varpi|}(r)\rceil\). Restriction from any \(s\geq r^{1/(j+1)}\), with source reduced modulo any \(\varpi^{j'}\) for \(j'\geq j\), kills modulo \(\varpi^j\) all terms of total degree at least \(M\), and the surviving coefficients are integral. Conversely every monomial of degree less than \(M\) is already integral on the larger ball. The image is therefore the fixed module spanned by those monomials. Continuous functions from a profinite parameter preserve this image statement by finite clopen partitions. A cofinal diagonal reduces the two indices to a countable Mittag–Leffler tower. Thus the derived two-index limit is \(\overline A\) in degree zero, as in (Barthel et al. 2025, Lemmas 6.1.2–6.1.3). The preceding local comparisons and the countable inverse-limit spectral sequence give \[\overline A[\epsilon]\longrightarrow R\Gamma(\mathrm{LT}_{\breve K,\mathrm{pro\acute et}}, \widehat{\mathcal O}^{+})\] with nonnegative, degreewise bounded-torsion cone: in any degree the limit spectral sequence uses only a limit in that degree and a \(\lim^1\) from the preceding degree, both with bounds uniform over the radii. No uniform bound over all cohomological degrees is needed.

The other space is \(\Omega^2_{\mathbb Q_p}\), with the formal model \(\mathfrak H\) of 88. That lemma gives the actual equivariant scalar equivalence \(\underline{\mathbb Z_p}\simeq R\Gamma(\mathfrak H,\mathcal O)\), including all profinite parameters. It supplies the structure-sheaf assertion needed from (Barthel et al. 2025, Theorem 6.2.1). The semistable comparison now gives \[\mathbb Z_p[\epsilon]\longrightarrow R\Gamma(\Omega^2_{\mathbb Q_p,\mathrm{pro\acute et}}, \widehat{\mathcal O}^{+})\] with the same degreewise bounded-torsion conclusion.

4. The tower comparison. The bridge between these calculations is an equivalence of families, not merely a bijection of geometric points. The relative Rapoport–Zink/shtuka comparison (Scholze and Weinstein 2020, Theorem 24.2.5 and its proof, pp. 227–229), using the family comparison of its Theorem 22.3.1, identifies the universal integral Tate local system with that in a modification \[0\longrightarrow \mathcal T\otimes_{\mathbb Z_p}\mathcal O \longrightarrow\mathcal E\longrightarrow i_*\mathcal L \longrightarrow0,\] where \(\mathcal T=T_p(\mathcal X)\) is the actual integral Tate local system of the universal \(p\)-divisible group \(\mathcal X\), of rank three, and \(\mathcal L\) is a line at the untilt divisor, and \(\mathcal E\) is fiberwise \(V_3\). The relative basic-stratum frame torsors (Fargues and Scholze 2024, Theorems III.2.4 and III.4.5) identify the two quotient presentations of this moduli problem. Honda endomorphisms are already defined over \(\mathbb F_{p^3}\); their canonical action and the basic-stratum equivalence descend from the geometric residue field as in 11. After the height-component normalization on the Lubin–Tate side, this gives \[[\mathrm{LT}_{\breve K}^{\diamond}/\overline G] \simeq[\Omega^{2,\diamond}_{\mathbb Q_p}/\mathop{\mathrm{GL}}_3(\mathbb Z_p)].\] Pulling back the two atlases gives a common space with commuting pro-étale torsor actions. Their Čech nerves therefore compute the same integral condensed cohomology object; this is the actual descent argument behind (Barthel et al. 2025, Theorem 3.9.1 and (3.9.3)). Taking invariants of the comparison cones remains harmless: they are nonnegative, so each diagonal of the group-cohomology spectral sequence contains finitely many bounded-torsion rows. After inverting \(p\) we obtain \[H^*(\overline G,\overline A)[1/p]\otimes\mathbb Q_p[\epsilon] \simeq H^*(\mathop{\mathrm{GL}}_3(\mathbb Z_p),\mathbb Q_p)\otimes\mathbb Q_p[\epsilon].\] The extended residue-field action is retained here. Witt Artin–Schreier gives the continuous unramified calculation over \(\overline W\) in (Barthel et al. 2025, Lemma 3.8.5(1)); its consequence for the full extended stabilizer is (Barthel et al. 2025, Lemma 3.8.5(2)). Here \(P_3\) fixes \(\overline W\) and the residue-field group acts by Witt Frobenius, exactly as in the chosen Honda semilinear action. The coefficient conventions also agree: for every profinite \(Q\), cochains on \(Q\times\overline G^a\) with values in the discrete \(W_\ell(\overline{\mathbb F}_p)\) have finite image. Lifting those finitely many values makes cochain reduction surjective. Their derived inverse limit is therefore the continuous \(\overline W\) cochain complex. Every continuous cochain on this compact domain with values in \(\overline W[1/p]\) has bounded image, so inverting \(p\) gives the rational cochains used in part (2).

To pass to the prescribed finite field \(k\), put \(\Delta_k=\mathop{\mathrm{Gal}}(\overline{\mathbb F}_p/k)\). Its action fixes \(u_1,u_2\) and acts by coefficient Frobenius. Each quotient \(\overline A/(p,u_1,u_2)^a\) is a finite direct sum of truncated Witt coefficient modules. Witt Artin–Schreier therefore identifies its \(\Delta_k\)-invariants with \(W(k)[[u_1,u_2]]/(p,u_1,u_2)^a\) and makes its positive \(\Delta_k\)-cohomology vanish. The coefficient reductions, including cochains with profinite parameters, are surjective by finite-value lifting. Taking the derived inverse limit gives \[R\Gamma(\Delta_k,\overline A)\simeq W(k)[[u_1,u_2]],\] equivariantly for \(P_3\), since \(k\) contains the Honda endomorphism field. Hochschild–Serre now identifies the algebraically closed residue-field calculation with the finite-field one. The trace-one contraction of 7 supplies the finite semilinear descent used by 50.

5. Constants and rational homotopy. The additive splitting in (Barthel et al. 2025, Proposition 2.5.1) has a direct construction. Compose each power operation with the \(K(3)\)-local transfer and write the resulting operations as \(\beta_m\). Transfer transitivity and the power-operation addition formula give \(\beta_m(a+b)=\sum_{i+j=m}\beta_i(a)\beta_j(b)\). Thus \(a\mapsto\sum_m\beta_m(a)t^m\), followed by reduction to \(\overline\mathbb F_p\) and projection from big to \(p\)-typical Witt vectors, is an equivariant additive map \(\gamma:\overline A\to\overline W\). For continuity, write the localized unit as the filtered colimit of its compact Moore stages \(M_k\). The spectrum \(L_{K(3)}\Sigma^\infty_+B\Sigma_m\) is dualizable for the finite group \(\Sigma_m\) (Hovey and Strickland 1999, Corollary 8.7); its tensor product with the compact object \(M_k\) is therefore compact. Compactness makes the map from \(M_k\otimes L_{K(3)}\Sigma^\infty_+ B\Sigma_m\) factor through some finite tensor stage \((M_j^{\otimes m})_{h\Sigma_m}\); hence each power operation is continuous for the Moore topology (Barthel et al. 2025, Lemma 2.5.4). Transfers are maps of \(K(3)\)-local spectra and are continuous for the same topology (Hovey and Strickland 1999, Proposition 11.1(b)). For \(\overline A=\pi_0\overline E\), where \(\overline E\) is the Morava theory used here, the Moore topology is the \(\mathfrak m=(p,u_1,u_2)\)-adic topology by the following local check. Choose the unit-compatible generalized Moore spectra \(\mathbb S/J_j\) of (Hovey and Strickland 1999, Proposition 4.22). Their \(K(3)\)-local duals \(M_j=D_{K(3)}L_{K(3)}(\mathbb S/J_j)\) are compact stages for the unit by (Hovey and Strickland 1999, Corollary 7.11 and Theorem 8.5). Duality carries restriction along the dual unit exactly to the map induced by \(\mathbb S\to\mathbb S/J_j\). The latter spectrum is finite, so \(\overline E\otimes(\mathbb S/J_j)\) remains \(K(3)\)-local. The unit-normalized Landweber formula (Hovey and Strickland 2005, Introduction and Example 0.1(d)) and the Moore coefficient calculation (Hovey and Strickland 1999, Definition 4.12) identify the restriction as \[\overline A\longrightarrow[M_j,\overline E] \cong\pi_0(\overline E\otimes(\mathbb S/J_j)) \cong\overline A/\overline J_j.\] Here \(\overline J_j\) is the image of the Moore ideal after removing invertible periodicity factors. The normalized images of \(p,v_1,v_2\) form a regular sequence of length three in the three-dimensional regular local ring \(\overline A\). Their ideal is therefore \(\mathfrak m\)-primary. The pure-power ideals \(\overline J_j\), whose exponents tend to infinity, are cofinal with the powers of \(\mathfrak m\), which proves the claimed equality of topologies from the restriction kernels. Thus each composite \(\beta_m:\overline A\to\overline A\) is \(\mathfrak m\)-adically continuous. The coefficientwise Witt-vector construction is therefore continuous. Its restriction \(f\) to \(\overline W\) is the identity modulo \(p\); continuity and additivity make it \(\mathbb Z_p\)-linear. Write \(f=\mathrm{id}+p h\) with a continuous \(\mathbb Z_p\)-linear endomorphism \(h\). The convergent series \(\sum_{n\geq0}(-p h)^n\) is its inverse. Hence \(f^{-1}\gamma\) is an equivariant retraction and \(\overline A=\overline W\oplus\overline A^c\) additively.

The compact comparison used to compute the constants has the following integral hypotheses. Choose a deep normal uniform open in \(P_3\) and in \(\mathop{\mathrm{GL}}_3(\mathbb Z_p)\). At \(p\geq5\) its lower-\(p\)-series valuation is saturated and equi-\(p\)-valued with denominator \(e=1\) and generators in degree one. The trivial finite free module \(\mathbb Z_p\) satisfies the hypotheses of (Huber et al. 2011, Theorem 3.3.3); its continuous comparison is cup compatible. The coefficient and restriction naturality in (Huber et al. 2011, Theorem 3.1.1(1)–(3)) identifies its rationalization with Lazard’s comparison (Lazard 1965, 2.4.9–V.2.4.10) and makes it equivariant for conjugation. Rational Hochschild–Serre then takes invariants under the finite quotient, whose higher cohomology vanishes by averaging. The full exterior calculation for the compact matrix and stabilizer groups, together with the Lie-algebra calculation and trivial adjoint action, is given by (Barthel et al. 2025, Proposition 3.8.1 and Lemma 3.8.2). At height three its generator degrees are \(1,3,5\). The unramified and finite semilinear descent from Step 4, with (Barthel et al. 2025, Lemma 3.8.5(2)) for the full extended stabilizer, therefore identifies both \[H^*(\overline G,\overline W)[1/p] \quad\text{and}\quad H^*(\mathop{\mathrm{GL}}_3(\mathbb Z_p),\mathbb Q_p)\] with that exterior algebra over \(\mathbb Q_p\).

The displayed tower comparison makes the cohomology of \(\overline A\) finite-dimensional in every degree. Its additive decomposition into constants and \(\overline A^c\) is a decomposition of continuous coefficient complexes. If \(d_i=\dim_{\mathbb Q_p}H^i(\overline G,\overline A^c)[1/p]\), comparison with the identical constant algebra, including the common factor \(\mathbb Q_p[\epsilon]\), gives \(d_i+d_{i-1}=0\) (with \(d_{-1}=0\)). Thus every \(d_i\) is zero. The constants inclusion is a ring map, so its resulting cohomology isomorphism is multiplicative. This proves (105).

Apply the local relative Morava descent of 50 to \(T\). The central procyclic subgroup generated by \(1+p\) acts on internal degree \(2t\) through \((1+p)^{-t}\). Its continuous two-term resolution has differential \((1+p)^{-t}-1\) on that coefficient module. For \(t\ne0\) this scalar is nonzero in \(\mathbb Q_p\), so the rationalized two-term complex is acyclic; Hochschild–Serre makes every nonzero internal row vanish. The locally proved relative spectral sequence is strongly convergent with finite cohomological amplitude. Exact rationalization preserves its bounded convergence. Only the internal-degree-zero row remains, with one filtration degree in each total degree. Multiplicativity therefore yields \[\pi_*L_0T=\Lambda_{\mathbb Q_p}(\alpha_1,\alpha_3,\alpha_5), \qquad |\alpha_i|=-i.\] This establishes the rational boundary used in 82; the image of its chosen primitive is determined by the separate map calculation in 10. ◻