Proof. First suppose \(n>1\). We construct successive quotients \(Q_j\) of \(X\), starting with \(Q_0=X\). The first block is \[B_1=L_{n-1}S\ \vee\ \Sigma^{-1}L_{n-1}S.\] Localizing the maps \(z_\varnothing,z_{\{1\}}\) gives a map \(B_1\to Q_0\). Let \(Q_1\) be its cofiber. Applying \(L_{K(n-1)}\), hypothesis (a) identifies this map with an equivalence. The cofiber \(Q_1\) is therefore \(E(n-2)\)-local.

Inductively, assume that \(2\leq j\leq n-1\) and that \(Q_{j-1}\) is \(E(n-j)\)-local. For every \(I\) with \(\max I=j\), compose \(z_I\) with \(D\to X\to Q_{j-1}\). The localization adjunction extends this map uniquely, in the space of such extensions, to \(\Sigma^{-d(I)}L_{n-j}S\). We thus obtain \[ B_j=\bigvee_{\max I=j}\Sigma^{-d(I)}L_{n-j}S \longrightarrow Q_{j-1}\longrightarrow Q_j, \tag{6}\] where the last object is the cofiber.

Here is the compatibility needed to continue the induction. Fix \(t=n-j\). Hypothesis (a) gives a specified direct-sum basis of \(L_{K(t)}D=L_{K(t)}X\), indexed by the subsets of \(\{1,\ldots,j\}\). For each earlier block \(\ell<j\), its sphere localization has height at least \(t\). Its \(K(t)\)-localization is therefore the corresponding shift of \(L_{K(t)}S\), and its map is the image of the original component \(z_I\). Successive cofibers of these components remove exactly the subsets whose maximum is less than \(j\), together with the empty subset. This assertion is inductive: at each step the map into the cofiber is the composite of that same original component with the quotient map, so the cofiber is the quotient by that direct summand.

The remaining summands are precisely the components of \(B_j\). It follows that \(L_{K(t)}(B_j\to Q_{j-1})\) is an equivalence. Consequently \(L_{K(t)}Q_j=0\), and chromatic fracture makes \(Q_j\) \(E(t-1)\)-local. The construction thus reaches a rational object \(Q_{n-1}\). For \(n=2\), this is already the object \(Q_1\) constructed from the first block.

We next determine that rational quotient, keeping track of possible kernels. The elements \[z_I,\qquad I\subseteq\{1,\ldots,n-1\},\] are linearly independent after rationalization in each degree. Indeed, under hypothesis (a) at \(t=1\), their images in \(L_{K(1)}D\) are the units of distinct shifted summands \(\Sigma^{-d(I)}L_{K(1)}S\). The rationalized unit in each such summand is nonzero. A rational relation between the \(z_I\) would therefore give a relation between nonzero elements in different summands, which is impossible. This argument includes subsets with equal values of \(d(I)\).

Since \(L_0L_rS=H\mathbb Q_p\), rationalizing the map from each block gives, in every degree, the injection of the corresponding independent vectors into the quotient by the preceding vectors. The cofiber long exact sequence has no suspended kernel. Subtracting these vectors from the dimensions in hypothesis (b) leaves \[\dim_{\mathbb Q_p}\pi_{-b}Q_{n-1} =\#\{I\subseteq\{1,\ldots,n\}:\max I=n,\ d(I)=b\}.\] The derived category of \(\mathbb Q_p\)-vector spaces splits by homology. Choose a basis in each degree to obtain an equivalence of rational \(S\)-modules \[B_n=\bigvee_{\max I=n}\Sigma^{-d(I)}H\mathbb Q_p \xrightarrow{\;\simeq\;}Q_{n-1}.\] Its cofiber \(Q_n\) is zero.

It remains to pass from these successive quotients to a filtration of \(X\). Set \[G_j=\mathop{\mathrm{fib}}(X\longrightarrow Q_j),\qquad 0\leq j\leq n.\] Then \(G_0=0\) and \(G_n=X\). For each block the square \[ \begin{tikzcd} G_j \arrow[r] \arrow[d] & B_j \arrow[d]\\ X \arrow[r] & Q_{j-1} \end{tikzcd} \tag{7}\] is cartesian: its upper left corner is the fiber of \(X\to Q_j=\mathop{\mathrm{cofib}}(B_j\to Q_{j-1})\). Taking fibers of its horizontal maps yields the cofiber sequence \[G_{j-1}\longrightarrow G_j\longrightarrow B_j.\] If a block is \(B_j=C_1\vee\cdots\vee C_r\), replace \(G_j\) by the intermediate pullbacks along \[0\longrightarrow C_1\longrightarrow C_1\vee C_2 \longrightarrow\cdots\longrightarrow B_j.\] Exactness of pullback in a stable category makes their successive cofibers \(C_1,\ldots,C_r\). In the first block choose the unit summand first. Its pullback identifies the first stage with \(L_{n-1}S\), and its map to \(X\) is exactly the localized \(u\). The other blocks are already ordered by maximum. The number of individual pieces is \[2+\sum_{j=2}^{n-1}2^{j-1}+2^{n-1}=2^n.\] This constructs the required filtration by \(S\)-linear maps.

If \(n=1\), the object \(X=L_0D\) is rational. Hypothesis (b) places one copy of \(\mathbb Q_p\) in degree zero and one in degree \(-1\). The nonzero map \(L_0u:H\mathbb Q_p\to X\) is an isomorphism on degree-zero homotopy, so its cofiber is \(\Sigma^{-1}H\mathbb Q_p\). These are the two required stages. ◻

Proof. The algebra \(\widetilde C_l\) is finite and reduced, and each of its irreducible components dominates \(\mathop{\mathrm{Spec}}A\) by its definition as a generic closure. It has dimension \(m\). Above the closed point of \(A\) all torsion coordinates are nilpotent, so there is a unique closed point and its residue field is \(k\). Its maximal ideal is generated by the height parameters and the \(z_{i,l}\).

Put \(a_i=[p^{l-1}](z_{i,l})\), and for \(v\in\mathbb F_p^m\) let \(a_v\) be the corresponding formal-group linear combination of the \(a_i\). If \(W_p(T)\) is the distinguished polynomial for \([p](T)\), then \[ W_p(T)=\prod_{v\in\mathbb F_p^m}(T-a_v)^{p^t} =\prod_{v\in\mathbb F_p^m}(T^{p^t}-a_v^{p^t}). \tag{11}\] Indeed this is the factorization over every geometric generic field: the distinct étale points are indexed by the labels and each has connected multiplicity \(p^t\). Both sides are monic of degree \(p^n\), so generic density in the reduced algebra proves the identity there.

There is a series \(h_l\) with \([p^{l-1}](T)=h_l(T^{p^{t(l-1)}})\). Consequently \(a_i^{p^t}\) is a power series in \(w_{i,l}\) with coefficients in \(A\) and zero constant term. Taking formal-group linear combinations gives the same assertion for every \(a_v^{p^t}\). Thus (11) is an identity over \(C_l\), and modulo \((w_{1,l},\ldots,w_{m,l})\) it becomes \(T^{p^n}\). The preparation unit for \([p](T)\) is defined over \(A\). All coefficients of \([p](T)\) of degree below \(p^n\) therefore vanish in this quotient, and (8) successively gives \[x_t=x_{t+1}=\cdots=x_{n-1}=0.\] The maximal ideal of \(C_l\) is generated by its \(m\) displayed \(w\)-coordinates. Since \(C_l\) is integral over \(A\), its dimension is \(m\). It is a regular local ring, and the surjection from \(k[[W_1,\ldots,W_m]]\) taking \(W_i\) to \(w_{i,l}\) is an isomorphism: a nonzero kernel would lower the dimension of this power-series domain. The identical maximal-ideal argument with the \(z_{i,l}\) gives the second identification in (9), including the stated inclusion. These finite regular rings are Cohen–Macaulay of dimension \(m\); the parameters of \(A\) form systems of parameters on them. Their Koszul complexes are exact in positive degree, so the finite modules are flat, hence free, over the local ring \(A\).

We next check the finite schemes after localization. Write \[[p^l](T)=g_l(T^{p^{tl}}),\qquad g_l(S)=U_l(S)V_l(S),\] where \(U_l\) is a unit and \(V_l\) is distinguished of degree \(p^{ml}\). The linear coefficient of \(g_l\) is a unit times \(x^{1+p^t+\cdots+p^{t(l-1)}}\). The invariant-differential identity for this Frobenius-divided homomorphism expresses \(g_l'(S)\) as that coefficient times a quotient of unit series. Evaluating in the prepared finite algebra shows that \(V_l'\) is a unit after \(x\) is inverted. Thus the Frobenius-divided finite group is étale there, of rank \(p^{ml}\). Its kernel quotient has removed exactly the connected kernel of rank \(p^{tl}\).

The scheme of bases of this étale group is finite étale over \(B\). The universal labeled points give a surjection from its algebra to \(C_l[1/x]\). It is an isomorphism generically, since the generic closure included every basis. Its kernel is zero: the source is flat over the domain \(B\), whereas a kernel vanishing generically would be \(B\)-torsion. This identifies the basis algebra.

Over this basis algebra, the scheme of lifts of the \(m\) basis vectors is a product of torsors for the connected finite kernel. It is finite locally free of rank \(p^{tlm}\). Its algebra surjects onto \(\widetilde C_l[1/x]\), which is generated by the same lifted coordinates. The latter is free over \(C_l[1/x]\) with basis \[\prod_{i=1}^m z_{i,l}^{e_i},\qquad 0\leq e_i<p^{tl}.\] The two ranks are equal, so the surjection is an isomorphism. This is the assertion about the full lift scheme, rather than its reduction.

A basis at level \(j\) lifts to a basis at level \(l\) over an extension of every geometric generic field. Hence the transition maps on the reduced closures are injective. They are finite because both rings are finite over \(A\). Since \(z_{i,j}=[p^{l-j}](z_{i,l})\), factorization through Frobenius and (9) give (10). ◻

Proof. The quotient \(H\) has the monogenic presentation obtained by dividing the prepared \(p\)-series by relative Frobenius. The full-section condition is a finite set of coefficient identities for its monic polynomial. Indeed equality for the coordinate implies equality of the characteristic polynomial for every polynomial in that coordinate by substitution, and every function in the finite algebra has such a representative. This description commutes with arbitrary base change.

Let \(Q\) be the resulting finite complete \(A\)-algebra of labeled group-homomorphism tuples. It has one closed point, because at the closed point of \(A\) the connected finite group has only the zero section over a field. If all \(m\) labeled coordinates are set equal to zero, the full-section identity makes the divided distinguished polynomial \(T^{p^m}\). Applying (8) before this division successively kills all the height parameters. Thus the maximal ideal of \(Q\) has at most \(m\) generators. There is a surjection \(Q\twoheadrightarrow C_1\) induced by the generic full tuple. Since \(C_1\) has dimension \(m\), the dimension and embedding dimension of \(Q\) are both \(m\). Hence \(Q\) is regular. A quotient of this regular local domain of the same dimension has zero kernel, proving \(Q=C_1\).

All equations in this argument are equations in finite algebras formed before localization. Their subsequent use on an étale component therefore involves polynomial operations in a finite algebra; it does not require substituting an order-one element into an unevaluated formal series. ◻

Proof. At level one let \(a_v\) be the formal-group linear combination of \(z_{1,1},\ldots,z_{m,1}\) with label \(v\in\mathbb F_p^m\), and put \(w_v=a_v^{p^t}\). The coefficient of \(T^{p^t}\) in (11), together with the preparation unit, gives (12). Transport by \(P_n\) is a change of formal coordinate with invertible linear coefficient. Its action on \(w_v\) is \(w_v\) times a unit, proving stability of \((w_v)\). On the conormal only that linear coefficient remains, raised to the \(p^t\)-th power. Equivalently, this is the invariant cotangent line of the Frobenius-divided group restricted to its zero section. This also describes the action without a choice of generator for the line.

Change the labels so that \(v\) is the first basis vector, and put \(R=C_1/(w_1)=k[[w_2,\ldots,w_m]]\). Write \([p](T)=g_t(T^{p^t})\) and let \(Q_t\) be the distinguished polynomial of \(g_t\). Formula (12) gives \(x_t=0\) in \(R\). The labels in each coset of \(\mathbb F_pv\) then give the same point, so \[Q_t(Y)=\prod_{\bar v\in\mathbb F_p^{m-1}} (Y^p-w_{\bar v}^{p}).\] Over \(A/(x_t)\) we have \(g_t(Y)=g_{t+1}(Y^p)\), and uniqueness of preparation gives \(Q_t(Y)=Q_{t+1}(Y^p)\). Therefore \[Q_{t+1}(Z)=\prod_{\bar v\in\mathbb F_p^{m-1}}(Z-w_{\bar v}^p).\] Coefficient Frobenius identifies these labels with the formal-group combinations of \(w_2^p,\ldots,w_m^p\) in the next Frobenius twist. By Lemma 5, the displayed full-section identity gives a map from the starting-height-\((t+1)\) labeled ring, whose labeled coordinates map to \(w_2^p,\ldots,w_m^p\). In the power-series coordinates of Proposition 4, this map is injective with image \(k[[w_2^p,\ldots,w_m^p]]=R^p\), since \(k\) is perfect. Thus \(R\) is exactly its \(p\)-th-root extension, with the indicated \(A/(x_t)\)-structure. Repeating the calculation for \(h\) independent dead labels sends the higher-height labeled coordinates to \(w_{h+i}^{p^h}\) and gives the corresponding \(p^h\)-th-root extension. ◻

Proof. Over \(B\) the lower height coefficients vanish and the height-\(t\) coefficient is invertible. The full isomorphism ring between this law and \(\Gamma_t\) is therefore a filtered union of finite étale extensions, by the exact-height isomorphism theorem (Lurie 2010, Lecture 14, Theorem 1). For clarity, its finite stages record coefficients of a full isomorphism that extend to the next required equations. The first coefficient \(c\) satisfies \(c^{p^t-1}=x^{-1}\) up to the chosen leading unit. The subsequent free coefficients satisfy monic additive equations with invertible linear coefficient; the other coefficients are uniquely determined. The equation constraining a coefficient of degree \(p^j\) can occur at degree \(p^{t+j}\). Thus these are not the schemes of arbitrary finite-bud isomorphisms. Any two full markings differ uniquely by \(P_t\), and quotienting by an open subgroup gives its finite étale torsor stages.

The étale Tate basis tower is finite étale at each level, as proved in Proposition 4. Taking the product of the two frame torsors proves the first assertion. Universal deformation transport carries both markings along the same isomorphism; it commutes with changing the markings. This proves the asserted \(P_n\) equivariance.

At finite level \(r\), two lifts of the marked étale basis differ by exactly one element of \(\Gamma_t[p^r]^m\). Proposition 4 identifies the lift algebra over the étale basis cover with its \(p^{tr}\)-th root ring. Relative Frobenius commutes with finite étale base change, so the same identification holds after any connected marking base change. It is canonical: each element of the lift algebra is the unique \(p^{tr}\)-th root of its power in the base. The identity may be checked on the faithfully flat basis cover and then descended.

The finite torsor maps are faithfully flat and the transition on lifts is multiplication by \(p\). Taking their affine inverse limit gives the torsor identity and faithful flatness for (13). The kernel of the projection \(T\to\Gamma_t[p^r]^m\) is \([p]^rT\): shifting a compatible sequence by \(r\) constructs its unique preimage under \([p]^r\). This proves both the subgroup assertion and (14).

Finally, transport by \(P_n\) carries the connected marking, the étale basis, and its lift simultaneously, so it fixes their difference parameter in the constant Honda group. Changing the markings by \((h,a)\) instead changes that parameter by (15). Each finite kernel and each finite diagram uses finitely many coefficients of the connected marking and a finite étale level, proving finite-stage descent. ◻

Proof. The dual of the dimension-one, height-\(t\) Honda group has dimension \(t-1\) and height \(t\). Since \(t>1\), it is connected; its formal group is smooth of dimension \(t-1\). This proves (16). The Honda relation is \(F^t=[p]\) on \(\Gamma_t\). Duality gives \(V^t=[p]\) on \(\Gamma_t^\vee\), and \(FV=[p]\) then gives \[F^t=[p]^{t-1}\quad\hbox{on }\Gamma_t^\vee.\] Cancellation here is cancellation of an isogeny of the divisible group. Thus, when \(r\) is divisible by \(t-1\), \([p]^r\) on the dual is Frobenius of order \(tr/(t-1)\). Choose \(r_0\) also so that this Frobenius and Frobenius of order \(tr_0\) fix \(k\). Pullback on any system of formal parameters is consequently the stated \(q\)-power map. Its kernel has the displayed truncated power-series ring. The length identity follows directly from \(d=m(t-1)\).

For a finite commutative group scheme \(G\), its function algebra and the dual Hopf algebra are related by the perfect Frobenius pairing \[(f,g)\longmapsto\lambda(fg),\] where \(\lambda\) is a nonzero invariant integral. In our Frobenius kernels, a coordinate description of \(\lambda\) is the highest-monomial coefficient after multiplication by the normalized invariant volume density. The pairing is perfect: its associated graded pairing in augmentation coordinates pairs complementary monomials. More explicitly, let \(v\) be the highest monomial in the original group coordinates. The map \(\Lambda_q\to N_q\), \(a\mapsto a\cdot v\), is an isomorphism of translation representations. Indeed the socle integral of the local Gorenstein algebra \(\Lambda_q\) acts nontrivially on \(v\). A nonzero kernel ideal would meet that socle, a contradiction, and the two dimensions agree. Under an auxiliary change of group coordinates, \(v\) changes by the tangent determinant raised to the \((Q-1)\)-st power; higher coordinate terms have too large total degree to contribute to the socle. This character is trivial, because its values lie in \(k^\times\) and \(Q\)-power fixes \(k\). The displayed regular-module identification is therefore equivariant. With its socle normalization \(\lambda(v)=1\), it satisfies \[\lambda(a\cdot v)=\varepsilon(a)\lambda(v)=\varepsilon(a),\] where \(\varepsilon\) is the distribution augmentation. Thus normalized integration is exactly the quotient \(\Lambda_q\to k\) under this identification. The finite-dimensional algebra is a finite set, since \(k\) is finite. Continuity of the marking action implies that its action factors through a finite quotient. ◻

Proof. Let \(\rho\) be the coaction on \(\mathcal E\) and \(\rho_H\) its restriction to \(H\), transported to \(T\). On \(\mathcal O(T)\), substitution along \([p]^r\) is the \(Q\)-power map, because \([p]^r=F^{tr}\) on the original Honda group. Taking powers in the coaction identity gives \[ \rho\Phi_Q=(\Phi_Q\otimes1)\rho_H. \tag{19}\] The algebra \(\mathcal E\) is perfect, so \(\Phi_Q\) is bijective. It is \(k\)-linear by the choice of \(r\), and has the scalar behavior (18). It identifies \(\mathcal E^H=\mathcal B^{1/Q}\) with \(\mathcal E^T=\mathcal B\).

Finite induction and coinduction for \(H\subseteq T\) are exact. For \(t>1\) this can be seen directly from (16): \(\Lambda\) is finite free over \(\psi_q(\Lambda)\), with the monomials of exponents less than \(q\) as a basis, so both tensor induction and Hom coinduction are exact. After forgetting auxiliary actions their usual Frobenius pairing also identifies these two functors. The equivariance needed here will follow directly from the coaction identity and adjunction below. For \(t=1\) the same statement is the corresponding finite character decomposition. The descent identity for coinduction is \[\mathcal V(\mathcal C_rM) \simeq(\mathcal E\otimes_k M)^H.\] Tensoring (19) with \(M\) and taking invariants gives the claimed comparison.

Every marking automorphism commutes with \([p]^r\), and every characteristic-\(p\) ring automorphism commutes with \(Q\)-powering. Hence these identities are equivariant before taking invariants. They are also natural before invariants, so they apply to entire finite diagrams, not just to their dimensions or individual terms. Finite torsor formulas use finitely many marking coefficients and denominators. They therefore descend, with their natural actions, and respect Definition 8. ◻

Proof. We first verify both the normalization and the absence of a base-dependent scalar. On a root chart write \[R_0=k[[w_1,\ldots,w_m]],\qquad R_1=k[[z_1,\ldots,z_m]],\qquad w_i=z_i^Q,\] with the relevant height parameter inverted and with finite étale scalar extensions permitted. Write the invariant torsor volume as \[\eta_1=a(z)\,dz_1\wedge\cdots\wedge dz_m.\] Its power transport is \(\eta_0=a(z)^Q\,dw_1\wedge\cdots\wedge dw_m\); the coefficient \(a(z)^Q\) belongs to \(R_0\) and is a unit there. For \(f\in R_1\), the Frobenius trace on top forms, divided by \(\eta_0\), is the \(R_0\)-linear functional \[ \tau(f)= \frac{[z_1^{Q-1}\cdots z_m^{Q-1}](f\,a(z))} {a(z)^Q}. \tag{22}\] Brackets mean coefficient extraction in the free \(R_0\)-basis with all exponents between \(0\) and \(Q-1\). This trace is independent of the root coordinates: it is the finite-duality trace on top forms, or, equivalently, the functional characterized by the change-of-variables rule for the residue. Invariant volume and the same rule make it translation invariant.

To determine its normalization, make a faithfully flat extension that supplies a point \(b\) of the torsor. The torsor becomes the finite Honda kernel. In the root chart put \(\epsilon_i=z_i-b_i\), so that \(\epsilon_i^Q=0\). Let \(u_i\) be normalized group coordinates at the identity. Invariance of the volume gives \[\det\left(\frac{\partial u}{\partial\epsilon}\right)(0) =a(b).\] In the truncated polynomial ring the socle therefore transforms as \[u_1^{Q-1}\cdots u_m^{Q-1} =a(b)^{Q-1}\epsilon_1^{Q-1}\cdots\epsilon_m^{Q-1}.\] Only the linear part of the coordinate change contributes in the maximal possible total degree; its action on that socle is the determinant to the \((Q-1)\)-st power. Formula (22) consequently sends this group-coordinate socle to \[\frac{a(b)^{Q-1}a(b)}{a(b)^Q}=1.\] The normalized invariant Hopf integral has exactly this value. The invariant-functional module is a free line, as is also clear from the perfect complementary-monomial pairing. The two functionals are therefore equal over the whole splitting algebra. Faithfully flat descent proves equality on the original torsor. In particular no comparison only on geometric fibers, and no undetermined invertible function on the base, is being used.

Identify the root ring with the original ring by \(Q\)-powering. Formula (22) is then exactly (21). The transitivity of finite-duality traces, or direct iteration of the coefficient-extraction formula, gives \(S^a\) at the \(a\)-th step. This proves stationarity with compatible normalizations. Under the regular-module identifications in Lemma 9, the finite transfers are the quotient maps \(\Lambda_{q'}\to\Lambda_q\). Indeed, in normalized group coordinates the transfer from root exponent \(Q'\) to \(Q\) takes the highest monomial to the highest monomial: extracting the residue classes of exponents modulo \(Q'/Q\) leaves precisely the exponents \(Q-1\). The invariant-volume density and its inverse contribute only their constant terms to these socles. The transfer is \(T\)-equivariant, so its value on this cyclic generator determines it on the whole regular module. For \(t=1\) the same calculation uses the invariant forms \(dy_i/y_i\) in multiplicative coordinates. It extracts the zero character on a finite diagonalizable torsor, so agrees with the constant-character projection described above.

For the transpose, take first a free compact lattice over \(R_0\) and pair it with the negative principal parts of its dual tensored with top forms. The pairing is \[(f,h\eta)\longmapsto \operatorname{res}(fh\eta).\] In coordinates, Cartier selects exponents congruent to \(Q-1\) in every variable and divides their successors by \(Q\). Since \(Q\) fixes \(k\), its residue identity is \[ \operatorname{res}\bigl(S(f)h\eta\bigr) =\operatorname{res}\bigl(fh^Q\eta\bigr). \tag{23}\] This follows as well by applying Cartier to \(fh^Q\eta\) and using its inverse-semilinearity. Thus the transpose is \(h\mapsto h^Q\) in the stationary volume frame. Renaming the root coordinates makes this the root inclusion. The form \(\eta\) is retained once, as in (20).

For a finite étale coefficient algebra the dual is identified by its finite étale trace pairing. Cartier commutes with étale pullback and trace; this can be checked after a faithfully flat étale splitting, where it is the preceding formula on each factor. Hence the transpose is the actual Frobenius on the finite-cover functions, transported through their trace dual, and not an arbitrary semilinear matrix with the same ranks. The same assertion holds for associated modules and their morphisms by finite flat descent and the naturality of the pairing.

Finally, coordinate changes, torsor automorphisms, and finite-level maps preserve finite duality and the normalized invariant-volume construction. Their character changes in source and target agree because the chosen Frobenius fixes \(k\). This proves the equivariance assertions, including those for complete finite diagrams in Lemma 10. All formulas involve finite coefficient data and preserve the common-stage convention of Definition 8. ◻

Proof. The prime in this manuscript is odd. The exterior-power theorem for a height-\(s\) \(p\)-divisible group of dimension at most one applies to the algebraic Barsotti–Tate group, including its finite truncated levels. Its exterior powers have heights \(\binom{s}{r}\) and, on the dimension-one locus, dimensions \(\binom{s-1}{r-1}\); the construction and its universal alternating map commute with base change. Apply this for \(r=s\) to obtain \(\mathcal H_{\det}\) of height and dimension one (Hedayatzadeh 2015, Theorem 3.25). The same functorial construction identifies its first truncated level with the top exterior construction on \(\mathcal G[p]\) (Hedayatzadeh 2015, Lemmas 3.23–3.24 and Theorem 3.25). We use these truncated group schemes before any localization of their coefficient rings.

Here is an explicit justification of the constancy assertion. Let \(H_0\) be the special fibre of \(\mathcal H_{\det}\), with the marking induced from the fixed Honda marking of \(\mathcal G\). The Cartier dual of a height-one, dimension-one group is an étale group of height one. The complete local ring \(\mathcal R_s\) is henselian (The Stacks Project Authors 2026, Lemma 10.153.9, Tag 04GM). For every \(r\), its finite group \(\mathcal H_{\det}^{\vee}[p^r]\) is the unique finite étale lift over \(\mathcal R_s\) of \(H_0^{\vee}[p^r]\), with its specified special-fibre identification. The constant lift along \(k\to\mathcal R_s\) is one such lift. The equivalence of finite étale categories under reduction (The Stacks Project Authors 2026, Lemma 10.153.7, Tag 04GK) identifies the two lifts uniquely, including their group laws, transition maps, and homomorphisms. These identifications are compatible for all \(r\). Dualizing gives the canonical marked identification \[\mathcal H_{\det}\simeq H_0\times_{\mathop{\mathrm{Spec}}k}\mathop{\mathrm{Spec}}\mathcal R_s.\] One can equivalently obtain this uniqueness by height-one deformation theory: its Hodge filtration has a unique lift and its marked morphisms are rigid; see (Zink 2002, Theorem 48, Proposition 40, and Corollary 95). The finite étale argument also shows directly that all the chosen identifications are compatible with nilpotent thickenings. Their pullbacks give the required identifications on every quotient and finite-level base used below.

We identify the action, rather than only its closed-point scalar. On the rational Honda Dieudonne realization the action of the endomorphism division algebra becomes its defining \(s\)-dimensional matrix action after passage to a splitting field. Functoriality of the exterior construction makes the action on the top exterior line its determinant. By the defining identity for the reduced norm this determinant is \(\operatorname{Nrd}(g)\), and the equality descends from the splitting field. The normalized Frobenius in the exterior construction changes the Frobenius of the line, not the determinant action of an automorphism on its underlying line. For \(g\in P_s\) the determinant is an integral unit, so on the height-one group \(H_0\) it is the automorphism \([\operatorname{Nrd}(g)]\). Rigidity of the marked height-one lift then identifies the semilinear transport on \(\mathcal H_{\det}\) with this same constant automorphism over the whole parameter ring. In particular this is not an identification inferred merely by reducing a varying invariant-line cocycle at the closed point.

Under inverse substitution the action on an invariant differential of \(H_0\) is multiplication by \(\overline{\operatorname{Nrd}(g)}^{-1}\). Its dual line therefore has character \(\nu_s\), proving the displayed equivariant identification of \(\mathcal N_s\).

It remains to check the finite-stage admissibility condition. On any exact-height stratum, choose finite connected and étale markings sufficient for the first truncated group, and a finite translation splitting of its connected–étale extension. These are exactly the permitted ascents in Definition 12. They give an equivariant constant frame for \(\mathcal G[p]\) on that stratum. Functoriality and base change for the truncated exterior construction then give an equivariant constant frame for \(\mathcal H_{\det}[p]\). In characteristic \(p\), the invariant differential module of a \(p\)-divisible group is the invariant differential module of its first truncated level: the differential of multiplication by \(p\) is zero. Thus this is already a finite-stage frame of \(\omega_{\mathcal H_{\det}}\), and hence of \(\mathcal N_s\). Only this finite coefficient datum is needed; no entire infinite marking is claimed to occur at one finite level.

The construction commutes with restrictions to successive label boundaries and with their power identifications. Alternatively, pull back the determinant group from \(\mathcal R_s\) first and repeat the same finite first-level marking and splitting on that stratum. Both give the same line by base-change naturality. Frobenius transports preserve \(\nu_s\), whose values lie in \(\mathbb F_p^\times\). Tensor powers and duals of a finite-stage frame remain finite-stage frames. This proves all the admissibility assertions. ◻

Proof. Each quotient of the filtration \(E\otimes_k Q_j\) is \(E\otimes_k k(\nu_s^{e_j})\). By Lemma 14, tensoring with this constant character is tensoring with a power of the admissible universal determinant line. On every required stratum take one common finite frame and splitting level for the filtration of \(E\) and the finitely many determinant-line powers. Refining the tensor filtration then gives equivariantly trivial graded frames. The original filtration is split as a module filtration, since it is obtained by tensoring finite-dimensional \(k\)-vector spaces; the refined filtration has the module splittings required in the definition. The same construction applies on every successive boundary and after root transport. Thus \(E\otimes_k Q\) is admissible. Dualizing the finite filtration reverses its order and replaces \(e_j\) by \(-e_j\), so it also proves the assertion for \(Q^\vee\). Proposition 35 gives the cohomological conclusion. All acting groups in this argument are the full \(P_s\). ◻

Proof. Begin after the first étale basis level and after the finite marking level defining the volume and its character. The tame connected marking adjoins a \((p^t-1)\)st root of the leading height coefficient, with the line-frame convention understood. Its coefficient module on localization is a sum of powers of the invariant line. Subsequent normal finite levels can be taken to have \(p\)-group deck groups; their regular representations over \(k\) have finite filtrations by trivial representations. Associated-bundle descent gives filtrations of the localized coefficient modules, split as module filtrations. The same construction applies to a finite translation diagram after one frame level on which its extra finite actions are defined. In the multiplicative case one first uses its character decomposition.

Choose lifts of bases of the graded lines. The action matrices in these bases are triangular. Their off-diagonal entries have one common pole bound because the acting group is compact and the actions are bounded-pole continuous. Rescale successive lifted bases by powers of \(x\), starting at one end of the filtration. At each step the finitely many off-diagonal denominators are absorbed by the earlier rescalings. The resulting split free lattice is stable. Its successive quotients are line lattices, and the same is true for its pole strips. Factoring \(x\) into label coordinates as in Corollary 6 gives the asserted boundary filtrations.

Choose a divisible stationary step whose Frobenius power fixes \(k\), and denote its coefficient Frobenius exponent by \(Q>1\). Finite freeness of Frobenius and a finite root basis give a constant \(c_0\) with \[S(x^{-c}L)\subset x^{-\lceil c/Q\rceil-c_0}L.\] This is obtained by applying \(S\) to the finitely many root-basis vectors and clearing their denominators; inverse semilinearity then divides the remaining pole order by \(Q\). If \(b\) is larger than the fixed point of this affine bound, \(S(P)\subset P\), and iterating the bound sends every fixed larger lattice into \(P\).

In dual invariant-volume frames, Lemma 11 identifies the transpose with positive Frobenius. Replacing the dual lattice by \(x^b\) times it increases its Frobenius matrix valuations by \((Q-1)b\), up to the fixed denominators of its original matrix. A further increase of \(b\) therefore makes every entry divisible by \(x\). Finally a finite basis of the finite étale algebra over \(C_1[1/x]\) becomes integral over \(C_1\) after multiplying by a large power of \(x\): clear the denominators in a monic equation for each basis element. Increase \(b\) to achieve this simultaneously in the volume/trace frames. The preceding stability and positivity inequalities continue to hold. ◻

Proof. The units in the height-\(n\) division algebra have analytic dimension \(n^2\). They have no \(p\)-torsion in the present range: an element of order \(p\) would embed the degree-\(p-1\) extension \(\mathbb Q_p(\zeta_p)\) in a division algebra of degree \(n\), which is impossible for \(n<p-1\). The compact analytic group results recalled in (Venjakob 2002, secs. 1.1–1.2) give a resolution of the trivial compact \(\mathbb Z_p[[P_n]]\)-module by finitely generated projectives, of length \(n^2\). Reducing it modulo \(p\) and extending scalars to \(k\) gives the corresponding \(k[[P_n]]\)-resolution: the underlying \(\mathbb Z_p\)-modules are flat, so this base change is exact. The residue-field completed group ring is also Noetherian of finite global dimension (Ardakov and Wadsley 2006, Proposition 3.3). The dualizing row and its adjoint action are described in (Beaudry, Goerss, Hopkins, et al. 2022, Propositions 4.16 and 4.40, and Remark 4.23). Its orientation character is trivial here. Indeed, after a splitting field the adjoint representation is conjugation on a full matrix algebra, whose determinant is one. This is the orientable form of analytic Poincaré duality; see (Lazard 1965; Serre 1997).

The finite resolution computes continuous cohomology for compact coefficients, by reduction to finite discrete quotients and inverse limit. The coefficient maps in those inverse systems are surjective on resolution terms. It agrees with the continuous bar model, and also with that model on lattice unions using the specified pole bounds. For a compact lattice all differentials have closed image, so cohomology is compact and dualization is exact. Duality for the finite projective resolution gives the second identification and the cup-action compatibility.

For the first identification use formal coordinates \(w_1,\ldots,w_r\). The residue pairing sends a negative principal monomial times \(dw_1\wedge\cdots\wedge dw_r\) and a power series to the coefficient of \(w_1^{-1}\cdots w_r^{-1}\). Every continuous functional on \(C\) depends on a finite-dimensional power-series quotient, so it is a finite sum of such monomial functionals. Applying this pairing to a free basis proves the assertion for \(L\). The residue transformation law makes the pairing independent of coordinates and accounts for the canonical line. It therefore respects all the stated actions. ◻

Proof. The existence of the chosen bounded resolution follows from Noetherianity and finite global dimension of \(\mathbb F_p[[K]]\) when \(K\) has no element of order \(p\) (Ardakov and Wadsley 2006, Proposition 3.3); finite generation of its terms follows by successively resolving kernels. The condensation functor from profinite \(R_K\)-modules is exact and preserves projectives by (Tang 2026, Theorems 3.2 and 3.14(i)). Moreover, solid \(\mathbb F_p\)-modules with commuting condensed \(K\)-action are precisely solid \(\underline{\mathbb F_p[[K]]}\)-modules by (Tang 2026, Proposition 3.21). In this assertion the solidification of the condensed group ring is the completed group ring, including its multiplication. Thus condensation of \(Q_\bullet\) is a projective resolution in the required coefficient category, proving (29).

Write \(Q_i\) as a summand of \(R_K^{N_i}\). Internal Hom from this finite free module is \(\mathcal V^{N_i}\), and the idempotent defining \(Q_i\) cuts out the corresponding summand. Evaluation at \(*\) preserves finite sums and these idempotents, and is exact: the point is an extremally disconnected projective object of the condensed site. A scalar on \(\mathcal V(*)\) therefore stays that scalar on every term and on its cohomology. No assertion that point evaluation is conservative on solid modules is involved. Filtering \(\mathcal A\) by its cohomology objects gives (30); the finite length of \(Q_\bullet\) controls its group-cohomological direction. ◻

Proof. Integral continuity and compact parameters. On an affinoid perfectoid, integral sections modulo \(\varpi\) compute \(\mathscr A\) up to almost isomorphism. This is affinoid almost acyclicity and its v-descent form (Scholze 2026, Proposition 8.8). Completing a union of affinoid perfectoid rings does not change its reduced integral sections almost: for every \(\epsilon>0\), quasicompactness makes a bound on the limit an eventual bound with norm loss at most \(|\varpi|^{-\epsilon}\). The same estimate applies to the \(\varpi\)-multiples. Finite Cech diagrams preserve these estimates. Alternatively one may pull a tube system to a strictly totally disconnected cover; its terms and nerve are then computed by rational subsets and their buffered limits. This verifies that the coefficient is pulled back in the continuity assertion just stated.

For a compact profinite parameter, reduction modulo a fixed positive-valuation cutoff makes continuous sections locally constant. Uniform approximation on finitely many clopen pieces therefore reduces the calculation to the one without parameters. The same argument applies on the nerve of a compact profinite torsor. Such a torsor is proper, and inverse images of compact bounds and their compact-group translates admit common compact bounds.

Compact groups and the support colimit. We justify passage through the support colimit in the stable almost localization of derived solid \(A_0\)-modules, where \(A_0=\mathcal O_E/\varpi\) and \(A_0^a=R\). For a buffered support \(K\), let \(\mathscr C_K\) be its supported parameter object. The cutoff continuous-function modules above are solid \(\mathbb F_p\)-modules; their derived Cech limits and supported fibers give \(\mathscr C_K\) in this category. The compact-support object is \(\mathop{\mathrm{colim}}_K\mathscr C_K\) there. Every compact frame group \(G\) used here is \(p\)-adic analytic without \(p\)-torsion, since \(p>n+1\). Choose a bounded resolution of \(\mathbb F_p\) by finitely generated projective profinite \(\mathbb F_p[[G]]\)-modules (Ardakov and Wadsley 2006, Proposition 3.3). The continuous solid bar calculation agrees with this resolution (Tang 2026, Theorems 3.2 and 3.14(i), Proposition 3.21); the completed-bar comparison is detailed following Lemma 20. Derived solid scalar extension to \(R\) preserves its augmentation and makes its terms retracts of finite free modules over the \(R\)-linear action algebra. Thus invariants are computed by a bounded complex of retracts of finite sums of the coefficient object. Forming both computations inside the almost category gives \[\mathop{\mathrm{colim}}_KR\Gamma(G,\mathscr C_K) \xrightarrow{\ \sim\ } R\Gamma\bigl(G,\mathop{\mathrm{colim}}_K\mathscr C_K\bigr).\] Only the group resolution is bounded; no amplitude bound on \(\mathscr C_K\) is required.

For the two torsors in (26), choose common cofinal buffered supports upstairs invariant under both groups, using properness. At a fixed support, the supported fiber commutes with Cech descent, and the two orders of descent give the common double nerve \(\mathcal X\times P_n^a\times U^b\). Replacing its two group directions by their bounded resolutions gives a finite double complex whose terms are retracts of finite sums. The support colimit therefore commutes with both directions. This proves that the torsor nerves compute compact supports and that the two presentations agree. The comparison remains the canonical double-nerve map; its support, cup, and trace maps are unchanged by this verification.

Rational-rank incidence. On bounded affine lifts of \(N_t^m\), cover the compact rational Grassmannian by finitely many clopen charts admitting matrix frames. In such a chart, incidence with a plane of dimension \(j\) has equations \[A(L)z=b,\qquad b\in F^{m-j}.\] The entries of \(A(L)\) have a common bound, say \(C_0\), and \(z\) has some bound \(M_0\). Therefore every possible \(b\) lies in the compact ball \(|b|\leq C_0M_0\) in \(F^{m-j}\). Thus all relevant relations are included in a closed analytic incidence over compact parameters. Projection along these parameters is proper. Its image is closed and generalizing: the same equation and the same witness survive generization. Equivalently one may use the generalizing property of maps of locally spatial diamonds (Scholze 2026, Proposition 11.19(iv), Definition 18.1, and Remark 18.2). Its open complement is partially proper by the valuative criterion (Scholze 2026, Definition 18.4).

Choose a finite mesh in the compact parameter chart, bound the values of its relation functions, and let those bounds tend to zero. The resulting rational tubes have this incidence image as their closed limit. Strict radius gaps exclude additional valuations at an outer boundary. The construction is compatible with every quasicompact test and with refinements of the mesh. On the stratum of actual rank \(j\), the smallest plane is unique. The proper incidence projection with this unique fiber has a continuous inverse there. This is precisely the continuity used when contracting the flag choices on an exact-rank stratum. ◻

Proof. Take \(\mathbb P_C^j\) with hyperplane complement \(\mathbb A_C^j\). Proper primitive comparison (Scholze 2013, Theorem 5.1) and (28) identify the compact-support complex with the fiber of the projective-space restriction map. Restriction preserves the powers of the hyperplane class, so cancels the even degrees from \(0\) through \(2j-2\) and leaves \(R(-j)\) in degree \(2j\). The same conclusion follows directly from compact-support primitive comparison for Zariski opens of smooth proper spaces (Pignon-Ywanne 2025, Corollary 5.3.2 and its proof), with algebraic–analytic proper-support compatibility (Bhatt and Hansen 2022, Proposition 3.16). The top affine class is invariant under affine linear changes, and the comparison and localization maps are natural. Lemma 21 proves the parameter assertion.

For clarity, the boundary germ involved here is indeed the coefficient at a point. Shrinking quasicompact discs to that point and applying spatial continuity gives \(R\) in degree zero. One can also see the vanishing directly on a Kummer annulus: a nonconstant fractional monomial of exponent \(u=bp^v\), \(p\nmid b\), has difference operator of norm \(|\epsilon-1|^{p^v}\). Shrinking both Laurent bounds by a ratio at most \(|\epsilon-1|\) makes division by this operator integral, since \(p^v\leq|u|_{\mathbb R}\). This contracts the nonconstant complex. The remaining degree-one Kummer class vanishes on a smaller disc by the binomial-series root. This argument also explains why the same assertion is not being made about compact supports of a perfected open unit disc. ◻

Proof. Let \(\Lambda_s=p^{-s}\mathcal O_F^j\). Choose radii \(|p|^{-r}<\rho_r<|p|^{-(r+1)}\) and let \(D_r\) be the open polydisc of radius \(\rho_r\), with buffered compact supports inside. Then \(F^j\cap D_r=\Lambda_r\), and translates by elements outside \(\Lambda_r\) are disjoint from \(D_r\). For a fixed lattice, compact-group descent commutes with exhaustion by invariant supports. Lemma 22 consequently gives \[\mathop{\mathrm{colim}}_rR\Gamma_c(D_r/\Lambda_s,\mathscr A) \simeq R\Gamma(\Lambda_s,R(-j))[-2j].\] The quotients \(D_r/\Lambda_r\), with radii chosen to contain exactly the prescribed translation lattice, embed openly into \(N_t^j\) and exhaust it. More explicitly, use pairs \((s,r)\) with \(r\geq s\). Increasing \(r\) gives extension on the disc, while increasing \(s\) gives corestriction for the finite cover between lattice quotients. These are the maps induced by the open embedding into the next quotient chart: on the cover, they sum the disjoint translates. The diagonal is cofinal, because a bounded compact support and any finite compact family of translates fit in a later disc. Thus one may first exhaust the affine space at fixed \(s\) and then take the lattice colimit.

Put \(h=tj\). The continuous Koszul complex for the abelian lattice gives \[H^a(\Lambda_s,R)= \bigwedge^a\mathop{\mathrm{Hom}}_{\mathrm{cts}}(\Lambda_s,\mathbb F_p)\otimes_{\mathbb F_p}R.\] In homothety frames the corestriction from \(\Lambda_s\) to \(\Lambda_{s+1}\) multiplies degree \(a\) by \(p^{h-a}\). This follows either from the exterior Koszul comparison or by duality with restriction in complementary degree. Hence the colimit kills every degree except \(h\), where it preserves the top orientation. Together with the affine degree \(2j\) this proves (31). Under inverse substitution an automorphism acts on the top lattice class by the inverse of its \(\mathbb Q_p\)-determinant. A matrix on \(j\) rows acts on \(F^j\) with determinant \((\det)^t\), proving the stated character. The finite Koszul and compact-support maps are functorial, so the argument retains all commuting actions.

For the commuting \(P_t\)-action this functoriality can be expressed without choosing a \(P_t\)-invariant affine chart. The constant-\(\mathbb F_p\) primitive comparison on the affine pieces, followed by the same lattice corestriction colimit, gives the canonical equivalence \[R\Gamma_c(N_t^j,\mathbb F_p)\otimes_{\mathbb F_p}R \xrightarrow{\ \sim\ }R\Gamma_c(N_t^j,\mathscr A).\] The source has a single one-dimensional \(\mathbb F_p\)-cohomology group, in degree \((t+2)j\). The comparison is natural for automorphisms of \(N_t^j\) and for compact parameter families. Its \(P_t\)-action is consequently a smooth finite character, as required when taking compact-group invariants below. ◻

Proof. Choose a basis and an upper triangular Borel. For each partial flag type, filter by the decreasing Bruhat opens consisting of cells of length at least \(b\). Restriction gives the associated graded as compactly supported locally constant functions on the cells of length \(b\). These restriction maps are surjective: compactly supported locally constant functions on a closed stratum extend after a finite refinement by compact open neighborhoods.

A flag projection sends a refined cell indexed by \(v\) to that indexed by the shortest representative \(w\) of its parabolic coset. Its length can only decrease. If the lengths agree, then \(v=w\) and the projection is an isomorphism on the common unipotent cell coordinates. Hence, on the associated graded, only these isomorphisms contribute.

For a permutation \(w\), the resulting summand is the signed Boolean complex on the subsets \(I\) containing its right descent set \(D_R(w)\), with the same cell coefficient in degree \(|I|+1\). If \(D_R(w)\ne\Delta_s\), toggling any element of \(\Delta_s\setminus D_R(w)\), with the incidence sign, contracts this complex. Only the longest permutation survives, in degree \(s\). The filtration is finite, so this proves vanishing in all other degrees. Its top cokernel is the displayed Steinberg quotient.

Over a Grassmannian choose a finite disjoint clopen refinement with quotient frames. Locally constant sections commute with this calculation: they are filtered colimits over finite clopen partitions of finite products. The constructions are natural, so the local identifications glue independently of the frames. ◻

Proof. For \(a\geq1\), let \(\mathcal I_a\) parametrize \[(z,W_0<\cdots<W_{a-1}<\mathbb Q_p^m),\qquad z\in N_t(W_0).\] The maps \(\pi_a:\mathcal I_a\to N_t^m\) are proper by Lemma 21. With the ambient term in degree zero, form the alternating incidence complex \[ \mathscr A_{N_t^m}\longrightarrow \pi_{1*}\mathscr A_{\mathcal I_1}\longrightarrow\cdots \longrightarrow\pi_{m*}\mathscr A_{\mathcal I_m}. \tag{33}\] Forgetting the smallest plane uses restriction from a larger plane fiber to a smaller one; the other face maps forget a flag step.

This complex represents extension by zero from \(Z\). An independent point has no proper containing plane. At a dependent point, the containing-plane poset has an initial object, its smallest plane, so adjoining this plane contracts the augmented nerve. This is a contraction of the supported calculation, not only a calculation of geometric-point ranks: filter by actual rank and use the continuous smallest-plane map of Lemma 21. On its finite clopen matrix charts the contraction acts on the same tube and parameter complexes. Coefficient localization then glues the contraction.

Apply compactly supported cohomology to (33). First compute the plane-fiber direction by Lemma 23. For a fixed smallest plane of dimension \(j\), the remaining complex is (32) for the quotient of dimension \(m-j\), tensored with \(\mathfrak o_t(W)(-j)\). Its term with no further plane lies in incidence degree one. The full incidence sign is the negative of the flag sign; multiplication in flag degree \(r+1\) by \((-1)^r\) identifies the two conventions. Lemma 24 therefore gives the entries \[E_2^{m-j,(t+2)j}=\mathcal V_j\quad(0\leq j<m), \qquad E_2^{0,(t+2)m}=\mathcal V_m.\] There are no further differentials. A differential to an entry with smaller plane dimension \(j'<j\) would require \[r=j-j',\qquad r-1=(t+2)(j-j'),\] which is impossible. The total degrees are distinct, proving the formula. Projection onto the ambient term is the map induced by extension of supports along \(Z\subset N_t^m\), so its top edge has the asserted identification.

A scalar fixes all rational flags. On a plane of dimension \(j\) it acts by \(\overline a^{-tj}\) on the orientation, and trivially on the affine Tate line. This is one for \(a\in1+p\mathbb Z_p\). The assertion follows on every displayed cohomology group. All maps used in the construction are pullbacks, restrictions, extensions of supports, or the lattice corestrictions described above. Thus the comparison retains the commuting actions. ◻

Proof. Use the finite flag resolutions rather than replace them by an infinite representation without its resolution. Shapiro reduces the \(\mathop{\mathrm{GL}}_m\)-direction to compact parabolic groups with a line coefficient. Their intersection with an open subgroup has finitely many orbits on each compact flag space. The commuting \(P_t\)-direction is compact as well. All these groups have finite \(p\)-cohomological dimension and a bounded resolution with finite projective terms: they are compact \(p\)-adic analytic groups without \(p\)-torsion, since \(p>n+1\). Their line coefficients and the finite twists are smooth; after shrinking an open subgroup they are trivial. Thus each computing complex has finitely many finite almost free coefficient terms. The same is true of the finite flag and group filtrations.

For the last assertion normalize the length of \(R\) to one. Subquotients and finite extensions of \(R^b\) have length bounded by the sum of the corresponding integers \(b\); the assertion also holds after almostification, by taking arbitrarily small valuation losses. A direct sum of \(r\) copies of \(R\) has length \(r\). Every finite-dimensional subspace of \(V\) therefore has bounded dimension. Faithfulness of the scalar extension implies that \(V\) is finite dimensional. ◻

Proof. First fix a coefficient stage. Write \(K_a\) for the closed polydisc of radius \(a\), \(B_s\) for the outer closed polydisc of radius \(s<1\), and \(E_{b,s}=\bigcup_i\{b\leq|w_i|\leq s\}\subset B_s\) for its Laurent exterior. For \(0<a<b<c<s<1\), put \[\mathscr C_a=R\Gamma_{K_a}(\mathbb D^{m,\mathrm{perf}},\mathscr M_r), \qquad \mathscr L_{b,s}=\mathop{\mathrm{fib}}\bigl(R\Gamma(B_s,\mathscr M_r) \to R\Gamma(E_{b,s},\mathscr M_r)\bigr).\] The inclusions \(B_s\setminus K_c\subset E_{b,s}\subset B_s\setminus K_a\) give, by restriction and excision at the two ends, maps \[ \mathscr C_a\longrightarrow\mathscr L_{b,s}\longrightarrow\mathscr C_c \tag{37}\] whose composite is the canonical enlargement of supports.

Here is the direction of the maps in the buffered system. Index it by triples \((a,b,s)\) with \(a<b<s\), and let a later triple \((a',b',s')\) satisfy \(s<a'\). Choose \(b<c<s\) and use \(\mathscr L_{b,s}\to\mathscr C_c\to\mathscr C_{a'} \to\mathscr L_{b',s'}\) as its transition. Comparing two choices with a larger intermediate \(c\) shows independence of that choice; (37) also shows that these transitions compose. The genuine support system and this buffered system interleave cofinally, so their colimits agree. Outer-domain comparisons enter through excision; the forward maps enlarge supports.

The intersections in the finite cover of \(E_{b,s}\) are \[U_I=\{b\leq|w_i|\leq s\ (i\in I),\quad |w_j|\leq s\ (j\notin I)\}, \qquad \varnothing\ne I\subseteq\{1,\ldots,m\}.\] Affinoid almost acyclicity computes \(\mathscr L_{b,s}\) by the augmented Cech complex of integral Laurent coefficients modulo \(\varpi\).

For an exponent vector \(\alpha\), a monomial \(w^\alpha\) occurs on \(U_I\) exactly when its negative exponents have indices in \(I\). Wherever it occurs its Gauss weight is \[b^{\sum_{\alpha_i<0}\alpha_i} s^{\sum_{\alpha_i\geq0}\alpha_i}.\] Thus the monomial Cech contraction preserves the norm: it removes every exponent vector except those negative in all coordinates, which remain in degree \(m\) of the restriction fiber. On a fixed buffered chart the weighted coefficients of a convergent Laurent series tend to zero. Modulo any fixed positive-valuation cutoff, only finitely many monomials remain, so this norm-preserving contraction also applies to the completed series.

The support transitions just constructed carry a negative Laurent cocycle to the same principal part, with its coefficients unchanged. For a fixed all-negative \(\alpha\), the integral coefficient bound is \(|c_\alpha|\leq b^{\sum_i(-\alpha_i)}\); the denominator sublattice has the same bound multiplied by \(|\varpi|\). Along the cofinal system \(b\) tends to one, so these bounds tend to \(1\) and \(|\varpi|\). Their colimit quotient is almost \(R\). We obtain the discrete finite-sum principal part module tensored with \(R\), shifted by \(m\).

At a fixed module stage \(r\), the perfected analytic scalars can be computed by allowing a further scalar-root stage \(h\geq r\). This gives two indices: \(r\) for the module transitions (35), and \(h\) for the fractional exponents in its scalar coefficients. After the fixed cutoff calculation every principal part is finite, so its exponents have one common root denominator. It appears at some finite \(h\), and increasing \(r\) to at least \(h\) places it on the diagonal. That diagonal is cofinal. The identity \(\theta_{r+1}\iota_r=F\theta_r\) identifies its algebraic Cech transitions with the direct system on the left of (36). The integral matrices preserve both the coefficient lattices and their \(\varpi\)-multiple submodules.

The comparison map itself is the map from algebraic Cech fractions to the supported analytic Cech complex, permitting arbitrarily small scalar losses before almostification. It is therefore independent of the chosen expression in monomials. The residue change-of-variables formula proves coordinate naturality. On a compact continuous group parameter, uniform polynomial approximation reduces the calculation modulo the cutoff to locally constant finite principal parts. Hence the same map and contractions apply to the parameter complexes. ◻

Proof. Trivialize the finite volume frame, retaining its transformation line. By Lemma 11, the transpose is the actual positive Frobenius on finite etale cover functions in this frame. In particular, the matrices here are not arbitrary semilinear matrices. Their root-stage bases are the roots of these finite-cover functions.

First work on a bounded affinoid chart with \(|x|\) bounded away from zero. A finite free basis norm and the intrinsic integral norm on cover functions are equivalent, with a fixed two-sided factor \(C_0\geq1\). At the \(r\)th root stage this factor is at most \(C_0^{1/Q^r}\). The chosen basis is integral over \(C_1\), so its root basis is power bounded; thus there is an actual map from these integral lattices into the intrinsic integral coefficient. For every \(\epsilon>0\), at sufficiently large \(r\) the inverse comparison loses norm at most \(|\varpi|^{-\epsilon}\). It follows that the cokernel is almost zero. Applying the same two-sided estimates to the \(\varpi\)-multiples proves almost injectivity after reduction. Finally finite fractional polynomials are dense in the completed perfection; modulo a positive-valuation cutoff they are available after a common finite increase in root stage. This proves the desired equivalence on \(\mathbb D^\times\), including the completion comparison.

On the closed locus \(x=0\), every transition of the free extended lattices is zero, since its entries are divisible by the relevant root of \(x\). Hence the direct limit has zero restriction to that locus. In terms of tubes, an element from a fixed stage acquires a positive fractional power of \(x\) at its next stage; the latter basis is power bounded. It therefore vanishes modulo \(\varpi\) on a sufficiently small tube. The tube may depend on the fixed stage. This is enough for the direct-limit restriction, and no common tube for all stages is needed.

The zero locus is generalizing closed, being the inverse limit of its shrinking rational tubes. Coefficient continuity and (28) now identify the direct limit with extension by zero of the coefficient already computed on its open complement. This proves (38).

The construction used the intrinsic finite-cover functions, their actual Frobenius, and the residue/trace frame. All these maps commute with the finite-level operations of Lemma 11. Norm bounds serve only to prove that these same maps are almost equivalences; they do not choose a replacement action. The estimates hold uniformly on compact \(P_n\)-families by a common finite basis and pole bound. This proves the asserted naturalities. ◻

Proof. The comparison has four terms. The two shifts come from group duality and principal parts, respectively: \[ \begin{tikzcd}[row sep=large] \left(\mathop{\mathrm{colim}}_{S^\vee}R\Gamma(P_n,P)^\vee\right)\otimes_kR \arrow[d,"\text{group and residue duality}","\simeq"']\\ R\Gamma\!\left(P_n, (\mathop{\mathrm{colim}}_F H^m_{\mathfrak m}(M))\otimes_kR\right)[n^2] \arrow[d,"\text{principal parts and positive extension}","\simeq"']\\ R\Gamma\!\left(P_n,R\Gamma_c(\mathcal X/U,\mathscr A) \otimes\langle\eta\rangle\right)[m+n^2] \arrow[d,"\text{the common torsor nerve}","\simeq"']\\ R\Gamma\!\left(U,R\Gamma_c(Z,\mathscr A) \otimes\langle\eta\rangle\right)[m+n^2]. \end{tikzcd} \tag{41}\] We justify these arrows with their cochain actions.

The group \(P_n\) has dimension \(n^2\) and no \(p\)-torsion, since \(p-1>n\). Its dualizing orientation is trivial: after a splitting field, the adjoint action is conjugation on matrices and has determinant one. Continuous compact-group duality therefore gives \[R\Gamma(P_n,P)^\vee \simeq R\Gamma(P_n,P^\vee)[n^2].\] One may compute this with a bounded finite projective resolution over \(k[[P_n]]\). The compact and discrete models are compatible by reduction to finite coefficients and inverse limit; the transition maps on the terms are surjective. This is also the compact/discrete duality of Lemma 17. The completed-module interpretation for the corresponding continuous and solid group cochains is compatible with these finite projective resolutions (Tang 2026, Theorems 3.2 and 3.14(i), Proposition 3.21).

Apply (34) and Lemma 27. The bounded resolution lets us commute its finite sums and retracts with the filtered root colimit. Lemma 28 then identifies the root system with the compact-support coefficient in the third term of (41). There is no \(P_n\)-character on the constant volume frame; its remaining character is the auxiliary one displayed here.

Finally use Theorem 19. The two compact torsor presentations of \([Z/U]\) have compatible cofinal compact bounds. At each fixed bound their common double nerve gives the comparison; the bounded \(P_n\) and \(U\) resolutions in Lemma 21 allow the support colimit to pass through both group directions. This is the last arrow of (41). Lemma 21 gives this comparison with every compact parameter and every support restriction needed in the nerves. This proves (39) and its cohomological form.

We check the extra module assertion. On the original presentation, the constant cochain algebra acts through the structural map to \(BP_n\). Compact-group duality sends its cup action to the cap/cup transpose with the canonical graded signs. The residue pairing, the Cech support maps, and the finite-cover Frobenius comparison are all maps over that same presentation. They therefore commute with these operations. The common double nerve carries the identical structural map to \(BP_n\), so the action transported to \([Z/U]\) is the actual one, not an action chosen after comparing dimensions. Finite etale trace and its transpose are compatible with the same residue pairing. Enlarging a frame level hence gives geometric pullback, as asserted. ◻

Proof. Corollary 26 and faithful scalar extension applied to (40) imply that \(\mathop{\mathrm{colim}}_{S^\vee}H^i(P_n,P)^\vee\) is finite dimensional. Its compact dual is the inverse limit with transition \(S\). Projection from that inverse limit has image exactly the intersection displayed above: compactness supplies compatible preimages of every point in the intersection. The intersection is therefore a quotient of a finite-dimensional vector space. ◻

Proof. For \(y\in I\), the nonempty compact sets \(S^{-1}(y)\cap S^aM\) are nested. Their intersection gives a preimage in \(I\), proving \(S(I)=I\); finiteness makes the restriction invertible. The images \(S^a(M/I)\) shrink uniformly to zero. Otherwise a compact closed set outside a neighborhood of zero would meet every one of these nested images and hence their intersection, which is zero.

Choose finitely many lifts of a basis of \((M/I)/S(M/I)\). Repeatedly subtract their linear combinations, and divide the remainder by \(S\). The resulting series converges uniformly in the compact topology and defines a continuous surjection \(k[[z]]^r\to M/I\). The kernel and cokernel of \(z\) on a finite module over this discrete valuation ring are finite-dimensional. The same conclusion for \(S\) on \(M\) follows using its finite invariant subspace \(I\).

The image of \(\ker f\) in \(M/I\) is contained in the \(z\)-power torsion submodule, which is finite-dimensional. Since \(I\) is finite, this proves that \(\ker f\) is finite, without assuming it closed in advance. For \(x\) on the left of (43), its fibre is now a finite closed coset of \(\ker f\). It meets every nested compact set \(S^aM\), and hence meets \(I\). This proves the equality. ◻

Proof. Constant objects in a pro abelian category are closed under kernels and cokernels of maps between constants and under extensions. For the last assertion, one can pass to the opposite ind category: in an extension of two constants, the identity of the constant quotient factors through one presenting term. Add the constant kernel to that term; the resulting relation object is constant, giving a constant presentation of the extension.

We induct downwards in \(i\), starting above the uniform cohomological bound. Suppose the assertion of constancy is known for larger indices. Resolve \(\mathfrak m_\Lambda=(D_1,\ldots,D_d)\) by the truncated completed Koszul resolution. Successive short exact sequences \(0\to K\to F\to Q\to0\) in this finite resolution give \[0\longrightarrow \mathop{\mathrm{coker}}\bigl(\overline T^j(K)\to\overline T^j(F)\bigr) \longrightarrow\overline T^j(Q) \longrightarrow \ker\bigl(\overline T^{j+1}(K)\to\overline T^{j+1}(F)\bigr) \longrightarrow0.\] Starting from its free end, closure of constants under these operations shows that \(\overline T^j(\mathfrak m_\Lambda)\) is constant for \(j>i\). The augmentation sequence then identifies the image of \(\overline T^i(\Lambda)\to T^i(k)\) with the constant kernel of \(T^i(k)\to\overline T^{i+1}(\mathfrak m_\Lambda)\).

This image is represented by the decreasing system \(S^aX^i\subset X^i\). It stabilizes at a single finite index \(h\) in \(\mathscr G\). More explicitly, after removing its constant subobject the decreasing system is pro zero. A sufficiently late transition to any fixed term is therefore zero; since that transition is monic, its source is zero. This gives stabilization, and hence \[S^hX^i/S^{h+1}X^i\] is countable-dimensional at every sufficiently deep frame level.

At each such level put \(M=S^hM^i\). The map from \(M\) to \(S^hX^i\) has countable kernel and cokernel by the strip argument. It follows that \(M/SM\) is countable, and it is a compact quotient because \(SM\) is compact. It is therefore finite. Its stable part is finite by (42). Lemma 31 now shows that \(S\) has finite kernel and cokernel on \(M\), and hence countable kernel and cokernel on \(S^hX^i\). Thus \(S\) is an isomorphism on the stabilized image in \(\mathscr G\). The stationary inverse system is consequently constant, and its augmentation is the inclusion of that stabilized image in \(T^i(k)\), which is monic. This completes the induction. Finally the augmentation is \(\Lambda\)-linear and \(\mathfrak m_\Lambda\) acts trivially on \(k\); its monicity implies the same assertion on \(\overline T^i(\Lambda)\). ◻

Proof. Use the completed free Koszul complex \(K_s=K(D_1^s,\ldots,D_d^s)\) in columns \([-d,0]\). Here \(s\) is an allowed distribution exponent from Lemma 9, not a height. Its cohomology is \(N_s=\Lambda/(D_1^s,\ldots,D_d^s)\) in column zero. After applying the cochain functor the vertical page consists of copies of \(\overline T^j(\Lambda)\). The first differential is zero by Lemma 32. For \(s'>s\), the map lifting the upward inclusion of regular modules is given on the Koszul basis by \[e_I\longmapsto \left(\prod_{j\notin I}D_j^{s'-s}\right)e_I, \qquad |I|=-\text{column}.\] It is the identity in column \(-d\) and is \(\mathfrak m_\Lambda\)-valued in every other column. On vertical cohomology the comparison is therefore the identity in the leftmost column and zero elsewhere.

There are no incoming differentials to the entry \((-d,b_0)\). Increase \(s\) once for each possible outgoing differential. Naturality and the zero map on its target show successively that it vanishes at the increased stage. Only finitely many increases are needed, since there are \(d+1\) columns. The whole leftmost entry therefore survives and is the quotient edge of \(T^{b_0-d}(N_s)\). Projection to the leftmost column followed by augmentation represents a nonzero generator of \(\mathop{\mathrm{Ext}}^d_\Lambda(N_s,k)\), proving the assertion.

The operation just described is also the operation of a finite Yoneda extension. Here are the category comparisons needed for this use. Pro finite-length \(\Lambda\)-modules are the corresponding profinite modules; stable images identify their finite quotients. The completed free modules in \(K_s\) are projective in these resolutions. Moreover Ext between finite modules over the Noetherian local ring \(\Lambda\) agrees with Ext in its \(\mathfrak m_\Lambda\)-torsion category, since injective envelopes of torsion modules remain torsion. A Yoneda extension in that category can be replaced by a finite one: lift a finite set of generators successively through its finite number of arrows and include the resulting finite-length submodules and their relations. The same construction contains any prescribed finite comparison diagram. Hence completed resolutions, torsion-module Ext, and finite Yoneda diagrams induce the same connecting operation under our exact cochain functor. ◻

Proof. Forget the auxiliary group first. Finite Hopf duality identifies coinduction with base change \(D_i\mapsto D_i^q\) of the fundamental Koszul class. Its top coefficient is a unit. The comparison from \(K_s\) to \(K_q\) is the identity on its leftmost column, as in Lemma 33. Hence the composite is a nonzero multiple \(c_qe_s\) in ordinary \(\Lambda\)-Ext.

We must make all these equalities equivariant at a single level. Their differences have the fixed endpoints \(N_s,k\), so they lie in the kernel of the one forgetful map \[\mathop{\mathrm{Ext}}^d_{T\rtimes U}(N_s,k) \longrightarrow\mathop{\mathrm{Ext}}^d_\Lambda(N_s,k).\] We will show that this kernel is finite and that each of its classes vanishes on an open subgroup. A finite intersection will then work for every \(q\). Shrink \(U\) to a pro-\(p\) compact analytic subgroup defining the two chosen finite diagrams. There is a continuous spectral sequence \[H^a\bigl(U,\mathop{\mathrm{Ext}}^j_\Lambda(N_s,k)\bigr) \Longrightarrow \mathop{\mathrm{Ext}}^{a+j}_{T\rtimes U}(N_s,k).\] It can be formed in rational translation representations with smooth auxiliary action. To verify the needed acyclicity, use cofree comodules for the semidirect product: their restriction to \(T\) is cofree, and their \(T\)-invariants are cofree for \(U\). This gives the usual composition-of-invariants spectral sequence. The Koszul resolution shows \[\dim_k\mathop{\mathrm{Ext}}^j_\Lambda(N_s,k)=\binom dj.\] Its terms have continuous finite-dimensional \(U\)-action. Analytic group cohomology is finite-dimensional on such coefficients, so \(\mathop{\mathrm{Ext}}^d_{T\rtimes U}(N_s,k)\) is finite. Since \(k\) is finite, it is a finite set as well.

The finite Yoneda construction also gives finite equivariant representatives here. A finite set of vectors has finite \(U\)-orbits, since the action is smooth and \(U\) is compact. Their \(\Lambda\)-span is finite-dimensional: one power of the augmentation ideal kills the finite orbit set, and that ideal is \(U\)-stable. Using these spans successively in the construction retains equivariance.

Every class in the forgetful kernel vanishes after restriction to some open subgroup. Indeed choose a finite Yoneda zero-comparison for its underlying extension, using the finite comparison property in the preceding proof. All its objects are killed by one power of \(\mathfrak m_\Lambda\). Shrink the auxiliary group so that it fixes that finite quotient of \(\Lambda\) and the finite objects in the original diagram. Give the additional comparison objects the trivial action of this subgroup. This is compatible with their \(\Lambda\)-action, and all the comparison maps are now equivariant. Intersect these open subgroups over the finite forgetful kernel. The resulting \(U'\) kills that entire kernel at once, and hence kills every \(\gamma_q-c_qe_s\).

All the composites in (45) are defined before this common restriction by natural coinduction and transfer. In particular the transfer \(N_q\to k\) is equivariant; its translation-Hom space is one-dimensional, and a pro-\(p\) group has no nontrivial character into \(k^\times\). The argument uses the fixed endpoints \(N_s,k\). It requires no common trivialization of the growing representations \(N_q\). ◻

Proof. We have proved countability for functions at sufficiently deep finite frame levels. It descends to any smaller level by finite flat Čech descent. On cochain terms augmentation is exact: finite étale descent has module sections, or a trace-one contraction, and these preserve bounded poles. Pass first to a normal finite frame cover, with deck group \(D\). Its Čech coefficient in degree \(q\) is a finite product of the deeper-level coefficient indexed by \(D^q\). The acting stabilizer \(P_n\) fixes this indexing set because its action commutes with the frame action. Thus each such term is a finite direct sum of the already treated \(P_n\)-coefficient; no new acting-stabilizer representation is introduced here. In each total degree only finitely many terms of its bounded-below cohomological spectral sequence contribute. Countability is therefore preserved in the descent.

For an admissible extra coefficient, take the finite frame and filtration in Definition 12. Trivial graded pieces reduce to the function calculation. If a finite translation splitting is required, filter its representation over the marked unsplit base. At height greater than one the graded pieces are trivial. At height one a character is a summand of the regular representation of the finite diagonalizable quotient; its associated coefficient is a summand of a finite root ring. Powering identifies this ring, with all actions transported, with the function ring at an adequately deep marking level. The established countability applies to it and its summands. Enlarge any prescribed root ascent to one of the cofinal divisible lift levels; the split filtration pulls back to this level. After the finite marking, its descent uses the translation torsor of Proposition 7. Each degree of its bar construction adds a finite tensor power of the Honda torsion-coordinate algebra. Its \(P_n\)-action is trivial, so the resulting coefficients are finite direct sums of the treated coefficient. The underlying module splittings preserve bounded poles, and only finitely many bar terms contribute in a fixed total degree. Finite extensions and these two descents prove countability for the localized extra.

Now let \(E\) be a compact admissible coefficient on \(C_1\). Its map to \(E[1/x]\) has countable-dimensional cohomological kernel by the strip argument: the boundary restrictions and their conormal twists are admissible and have finite cohomology by induction. Its localized cohomology is countable-dimensional by the preceding paragraph. Hence \(H^i(P_n,E)\) is countable-dimensional. It is also profinite by Lemma 17, and is therefore finite. Only finitely many degrees occur.

For \(E\) on \(A\) first localize at \(x_t\) and pass to the first labeled level. The exact-height countability just proved descends to the unlabeled localization. The strips for powers of \(x_t\) have a finite filtration by coefficients on the next starting-height disc, with the specified conormal twists. These satisfy the induction hypothesis after pullback to its labeled strata. The identical countable-kernel argument and compactness prove finiteness on \(A\).

The argument is unaffected by a fixed finite root change: transport the group, its marking and splitting torsors, the filtrations, and all twists by that same power identification. In particular a universal finite torsion construction restricted to a dead-label boundary may be a Frobenius pullback of the smaller universal construction; its frames and splittings are transported from those of that construction. This checks exactly the extra coefficients and their linear duals used in the smaller-height local duality of Proposition 36. ◻

Proof. We first verify the parameter assertion, including flatness. Write \(h\) for the universal isomorphism, oriented by \[h\circ[p]_{\mathrm{original}}=[p]_{\mathcal G}\circ h, \qquad c=h'(0)\in E_N^\times.\] Successive height-coordinate congruences show that the parameters below \(t\) vanish. Inductively compare the coefficient of \(X^{p^i}\) modulo the earlier height parameters, for \(t\leq i<s\). On that quotient the two \(p\)-series start with \(x_iX^{p^i}\) and \(v_iX^{p^i}\), respectively. The isomorphism identity therefore gives \(cx_i=c^{p^i}v_i\) modulo the earlier parameters. Induction also identifies the ideals generated by those earlier parameters on the two sides. Passing to the conormal module gives \[x_i\equiv c^{p^i-1}v_i+\sum_{j<i}b_{ij}v_j\pmod{J_s^2}.\] Changing the orientation of \(h\) inverts this matrix. In particular the diagonal entries are units. The \(v_i\) and \(x_i\) consequently generate the same transverse ideal and the same ideal-power filtration on every nilpotent neighborhood.

This conormal calculation is accompanied by a calculation of every parameter graded piece. Before marking, regularity of the transverse parameters gives \[\operatorname{gr}_{J_s}(A[1/x_s]/J_s^N) \simeq (A/J_s)[1/x_s][X_t,\ldots,X_{s-1}]/(X)^N.\] Finite étale marking covers are flat, and this equality therefore base changes to their special-fiber algebras. The triangular change just computed identifies it with the associated graded for \(R_N\). Choose a \(k\)-linear lift of the special-fiber algebra into the marked algebra. Multiplication extends that lift to an \(R_N\)-linear map from its tensor product with \(R_N\). The map is an isomorphism on every associated graded piece, hence is an isomorphism of modules because the filtration is finite. This proves flatness over \(R_N\); it is valid regardless of the dimension of the special-fiber algebra as a \(k\)-vector space. Adjoining a finite set of basis lifts is a finite locally free torsor, so it preserves this flatness. Its filtered union does so as well.

The group of compatible lifts is \[ T'=(T_p\mathcal G)^{e_s}. \tag{49}\] On the closed parameter fiber, Proposition 7 identifies its finite lift levels with the successive root rings of the full height-\(s\) marked coefficient. Their union is \(S_s\), which is perfect.

We recall the elementary splitting fact that now applies. Let \(R\) be a finite local Artin \(k\)-algebra, let \(E\) be a flat \(R\)-algebra, and suppose \(S=E/\mathfrak m_RE\) is perfect. Choose \(q=p^a\) large enough to kill the nilpotent ideal by \(q\)th powers and to act identically on \(k\). For \(z\in S\) define \[\sigma(z)=\widetilde{z^{1/q}}^{\,q}\in E,\] where the tilde denotes any lift. The result is independent of the lift because the difference of two lifts has zero \(q\)th power. Using a further root proves independence of sufficiently large \(q\). The characteristic-\(p\) power identities show that \(\sigma\) is additive, multiplicative, and \(k\)-linear. It is functorial in \(E\). Flatness gives \[\operatorname{gr}_{\mathfrak m_R} E =\operatorname{gr}_{\mathfrak m_R}R\otimes_k S.\] Thus \(r\otimes z\mapsto r\sigma(z)\) is an isomorphism on associated graded modules and therefore an isomorphism of algebras. Applying this fact to \(E_N/R_N\) proves (48). Functoriality proves all asserted equivariances and compatibility in \(N\). In particular \(P_n\) acts only on \(S_s\) in this description, whereas \(P_s\) acts on both factors in the universal deformation convention.

We check the topological assertion at the same finite stages. Expansion in the transverse \(x_i\) gives continuous additive sections of their finite truncations. At a finite étale marking stage the coefficient module is finite projective. More explicitly, if its presentation is \(e_N(A[1/x_s]/J_s^N)^d\) and \(e_0\) is the reduced idempotent, let \(\tau\) lift each coordinate by transverse truncation. Then \(z\mapsto e_N\tau(z)\) is a continuous additive section on \(\operatorname{im}(e_0)\). Monomial division in the coordinates, followed by \(e_N\), extracts the parameter layers. A finite lift-torsor algebra has the finite free torsion-coordinate bases of Proposition 4, and the same construction applies in that basis. The finitely many structure constants and the inverse triangular conormal matrix have a common denominator after localizing at \(x_s\). It follows that these additive lifts and successive parameter-coefficient extractions have bounded pole loss and are continuous on a cleared-denominator power-series lattice. For a compact cochain source all of these choices use one finite stage and one bound. In the formula for \(\sigma\), extracting the \(q\)th root means moving to one further finite root stage; taking powers and products preserves the stated continuity and merely changes the finite pole bound. The inverse of (48) extracts the finitely many parameter layers by the same maps. This proves the assertion for cochains. All tensor products used here have the finite-dimensional algebra \(R_N\) as one factor; no completion or infinite parameter tensor product has been introduced. ◻

Proof. Recall first the integral full-section presentation. If \(H\) is a finite monogenic scheme of rank \(p^m\), a label homomorphism \(\ell:\mathbb F_p^m\to H\) is full when the characteristic polynomial of its coordinate equals the product over the label sections, with multiplicities. Equivalently, for every function \(f\) on \(H\) after arbitrary base change, \[ \det(m_f:\mathcal O(H)\longrightarrow\mathcal O(H)) =\prod_{v\in\mathbb F_p^m} f(\ell(v)). \tag{51}\] The equivalence follows by polynomial substitution in the characteristic-polynomial identity, or by the resultant formula for the norm. Thus this condition is intrinsic and is preserved by every automorphism of the finite scheme.

For the Frobenius-divided \([p]\)-kernel, Lemma 5 identifies this full-section algebra with \(C_1\), including after arbitrary nilpotent base change.

Over the split lift tower the divided kernel is \[H=H_c\times E_{\mathrm{et}}, \qquad E_{\mathrm{et}}\simeq\mathbb F_p^{e_s}.\] The projection of a full tuple gives a surjection \[b:\mathbb F_p^{n-t}\twoheadrightarrow\mathbb F_p^{e_s}.\] Indeed, if a component of \(E_{\mathrm{et}}\) were omitted, take the function which is zero on that component and one on every other component. Its norm is zero, but its product over the proposed sections is one, contradicting (51). Projection to the constant étale group is locally constant on the test scheme. We can therefore decompose the full-section scheme into the finitely many open and closed components indexed by these surjections.

Fix \(b\), put \(K=\ker b\), and choose a complement \(W\) to \(K\) in the label space. Its dimensions are \(s-t\) and \(e_s\), respectively. A label homomorphism with this projection is exactly the data of a homomorphism \(K\to H_c\) and \(e_s\) arbitrary points of \(H_c\), the images of a basis of \(W\). On each étale component, the proposed tuple is the translate of the kernel tuple by the associated point of \(H_c\). To test fullness on one component, take an arbitrary function on that copy of \(H_c\) and extend it by \(1\) on every other component. Equation (51) then reduces exactly to the norm identity for that one component. Conversely, the product of the componentwise identities gives the identity for every function on the whole disjoint union. Translation preserves these norm identities. Thus the original tuple is full if and only if its kernel tuple is full for \(H_c\), over every test algebra. The two constructions are inverse without reducing that algebra. The component is therefore \[\operatorname{Full}(H_c)\times H_c^{e_s},\] whose ring is (50).

The argument uses algebraic translations on the finite group scheme. It does not evaluate a formal power series on an arbitrary localized coordinate; the norm identity supplies the required substitution invariance within the finite algebra. Finite freeness follows from the smaller \(C_1\) calculation and finite local freeness of \(H_c\). All operations are functorial for changes of deformation framing. Changes of surviving Tate basis act on these particular data through a finite level, since the tuple and the divided kernel are finite-level objects. A sufficiently deep congruence subgroup fixes the chosen labels, complement, and connected factors. The same finite-level observation applies to translations and to each fixed invariant-line or torsion twist. ◻

Proof. Since \(s\geq2\), the special Cartier dual of \(\mathcal G\) is connected, of dimension \(s-1\). Its formal completion is smooth over \(R'\). The completed distribution algebra of \(T'\) is therefore a power-series algebra over \(R'\) on \(\delta\) parameters. The representations under consideration are locally killed by a parameter power and locally factor through a finite torsion level. These are the continuous torsion modules for this distribution algebra. To check the topology, modulo \(\mathfrak m'\) the finite dual-kernel quotients are cofinal with powers of their augmentation ideal, because the special dual is connected. At a fixed Artin thickness this cofinality lifts by multiplying exponents by a nilpotence bound. Conversely the coordinates of \([p]\) have no linear term in characteristic \(p\), so their iterates tend to zero in the augmentation topology. The inverse limit of these finite distribution quotients is precisely \(\Lambda_N\).

Suppose first that \(a\) conjugates translations by \([a]\), and let \(\sigma_a=[a]^*\) on the distribution algebra of \(T'\). As \(a\) acts identically on the row, the ideal \[J_a=(\sigma_a(f)-f:f\in\Lambda_N)\] annihilates it. Write \(F\) for the formal group law on the Cartier dual. The identity \[F([a](D),[-1](D))=[a-1](D)\] shows that the coordinates of \([a-1](D)\) belong to \(J_a\). Conversely, \(F(D,[a-1](D))=[a](D)\) shows that \([a](D)-D\) belongs to the ideal of those coordinates. Thus \[J_a=([a-1]_1(D),\ldots,[a-1]_\delta(D)).\] If \(a-1=p^hu\) with \(u\in\mathbb Z_p^\times\), the automorphism \([u]\) gives \[J_a=[p^h]^*(I).\] This is the image of the augmentation ideal for restriction to \([p]^hT'\), so that subgroup acts trivially. With \(a^{-1}\) in place of \(a\), its difference from \(1\) has the same \(p\)-adic valuation and the argument is identical. The chosen scalar is uniform on the special cohomology and the coefficient \(M\) is one fixed finite construction, so \(h\) is uniform over its local-cohomology union.

It remains to justify the relative cohomology calculation. The sequence \(D_1,\ldots,D_\delta\) is regular in \(R_N[[D]]\), even though \(R_N\) itself can have nilpotents: successive quotients are the power-series rings in the remaining variables over \(R_N\), and multiplication by the next variable is injective. Its Koszul complex is a finite free resolution of \(R_N\) over \(\Lambda_N\). Applying continuous \(\mathop{\mathrm{Hom}}\) to a row annihilated by \(I\) gives zero differentials, and hence (55). No flatness of the row over \(R_N\) is required.

This Ext calculation is also the torsor-descent calculation being used here. At finite levels the lift algebras are finite locally free, with the unit a local basis vector, and their augmented descent diagrams are split as base-module diagrams on affine covers. Their flat union supplies the relative continuous bar construction. Over \(R_N\), its completed distribution resolution and the Koszul resolution are resolutions split over the base: unit insertion and monomial division give continuous contractions of their augmented complexes. An induced completed bar term \(\Lambda_N\mathbin{\widehat\otimes}_{R_N}W\) lifts along a continuously \(R_N\)-split surjection by its \(R_N\)-linear splitting; finite free Koszul terms have the same lifting property. Induction along the two resolutions therefore gives continuous comparison maps in both directions and comparison homotopies. Alternatively continuous \(\mathop{\mathrm{Hom}}\) from each finite free Koszul term commutes with the union of torsion coefficients. Consequently both resolutions compute the same relative Ext. Its exterior description is canonically the exterior algebra on the dual augmentation conormal, so nonlinear changes of parameters give the asserted action. All the calculations are made at a common Artin thickness before passing to the local-cohomology union. ◻

Proof. Take the full \(k\)-linear dual, with its product topology, and put \(\Omega=k[[V]]\). This duality is exact from discrete vector spaces to pseudocompact vector spaces and exchanges smooth \(V\)-modules with pseudocompact \(\Omega\)-modules. The ring \(\Omega\) is local and Noetherian. The dual bounded complex \(D=C^\vee\) has degreewise finite derived completed reduction \[k\mathbin{\widehat\otimes}^{\mathbb L}_{\Omega}D \simeq R\Gamma(V,C)^\vee.\] We recall why this implies finite generation of each \(H^j(D)\). All differentials in \(D\) have closed image, so its cohomology is pseudocompact. At the highest nonzero cohomological degree \(b\), right t-exactness of derived completed tensor gives \[H^b\!\left(k\mathbin{\widehat\otimes}^{\mathbb L}_\Omega D\right) =k\mathbin{\widehat\otimes}_\Omega H^b(D).\] Thus the ordinary completed reduction of \(H^b(D)\) is finite. Lift a finite set of generators of that reduction to cycles. Their \(\Omega\)-span is closed, because it is the image of a finite free pseudocompact module. Its quotient has zero completed reduction, and is zero by compact Nakayama: a nonzero pseudocompact quotient has a nonzero finite simple quotient, which for the local algebra \(\Omega\) is \(k\). Thus these finitely many cycles generate \(H^b(D)\).

Map the finite free module on these cycles into \(D\), in degree \(b\), and take its cone. The cone has no cohomology in degree \(b\) and still has degreewise finite derived reduction. Repeat downwards. In recovering the cohomology of the preceding complexes, all kernels inside finite free modules are finitely generated by Noetherianity. The exact sequences of these cones therefore prove finite generation of every original \(H^j(D)\); only finitely many cone steps are needed for each degree. For a finitely generated pseudocompact \(\Omega\)-module \(M\), choose a resolution of \(M\) whose terms are finite-rank free pseudocompact \(\Omega\)-modules. Such a resolution is constructed successively: the kernel at each step is closed and finitely generated because \(\Omega\) is Noetherian. Completed tensoring this resolution with \(k\) gives finite-dimensional terms in every degree. Consequently every \(\operatorname{Tor}^{\Omega}_a(k,M)\) is finite-dimensional. The finite projective dimension of the trivial module \(k\) supplies the vanishing range \(a>\dim(V)\). Applying these facts to the finitely generated cohomology modules of \(D\) and using duality proves the assertion for \(H^a(V,H^i(C))\). The bounded range for \(i\) gives the asserted finite total range. ◻

Proof of Proposition 36. For an intermediate support row \(t<s<n\), perform the marked and split ascent of Lemmas 37 and 38. The resulting coefficient is (53). First take \(P_n\)-cohomology, then forget the subgroup \(T'_h\) supplied by Lemma 39. The cohomology rows are the tensors in (54) with the finite exterior bundles in (55).

Take next the invariants of a sufficiently small surviving matrix group \(V\), chosen also to fix all finite label and torsion factors being considered. The finite resolution for \(V\) commutes with the discrete union (52). Its individual cohomology terms consequently have the form \[ Q\otimes_k H^r_{\mathfrak m'}(M\otimes_{R'}F). \tag{58}\] Here \(Q\) is one of the actual rows (57), and \(F\) is a finite free tensor construction from the Lie bundle of \(\mathcal G^\vee\), invariant lines, and the fixed additional finite torsion-coordinate bundles. In particular \(Q\) is finite and its \(P_s\)-composition factors are norm powers. Every fixed bar degree adds only such universal torsion-coordinate factors to \(F\); the special coefficient producing \(Q\) remains \(B_{\mathrm{full},s}^{\mathrm{perf}}\).

We finally take \(P_s\)-cohomology. The residue pairing on the \(r\)-dimensional disc identifies the continuous dual of the local term in (58) with \[L_{M,F}=\mathop{\mathrm{Hom}}_{R'}(M\otimes_{R'}F,\omega_{R'}), \qquad \omega_{R'}=\bigwedge^r\Omega^1_{R'/k,\mathrm{cts}}.\] The group \(P_s\) is orientable of dimension \(s^2\), and continuous group duality therefore gives \[ H^b\!\left(P_s, Q\otimes_kH^r_{\mathfrak m'}(M\otimes F)\right)^\vee \simeq H^{s^2-b}(P_s,Q^\vee\otimes_kL_{M,F}). \tag{59}\] First we verify that \(L_{M,F}\) is an admissible compact coefficient at the smaller pair \((s,t)\), whose total height is strictly less than \(n\). The algebra \(M\) is free over \(C_1^{(s,t)}\) and consists of finite torsion-coordinate factors. On each successive labeled boundary stratum of that smaller disc, finite root transport identifies the construction with the corresponding height-stratum construction. After finite connected and étale markings and a finite splitting level, its divided connected kernels and finite étale sections have the required constant frames. The Lie and invariant-line factors have the same property. Finite tensor constructions and linear duals preserve these filtrations. If a multiplicative connected kernel occurs, its character factors are summands of the finite regular translation representation; its associated coefficient is the finite root ring allowed in the compact assertion. Finally finite duality gives \[\mathop{\mathrm{Hom}}_{R'}(M\otimes F,\omega_{R'}) \simeq \mathop{\mathrm{Hom}}_{C_1^{(s,t)}}(M\otimes F,\omega_{C_1^{(s,t)}}),\] and the volume construction and conormal adjunction supply the required filtration for this canonical-line twist. This proves admissibility of \(L_{M,F}\).

Now use the finite \(P_s\)-stable filtration of \(Q^\vee\) from (57). Tensoring it with \(L_{M,F}\) gives a finite filtration with successive coefficients \[L_{M,F}\otimes_k k(\nu_s^{-j}).\] Lemma 14 identifies the constant norm line through the determinant of the universal height-\(s\) \(p\)-divisible group and proves its admissibility on every required stratum. Thus these successive coefficients are admissible by Corollary 15. Their compact \(P_s\)-cohomology is finite by the earlier compact assertion at \((s,t)\). The finite-dimensional filtration of \(Q^\vee\) splits over \(k\) as a filtration of vector spaces; after tensoring with \(L_{M,F}\), its coefficient sequences therefore have continuous underlying module splittings. Their long exact cohomology sequences prove finiteness for \(Q^\vee\otimes_kL_{M,F}\). Consequently (59) is finite-dimensional. The acting group in this application is \(P_s\) throughout.

These successive computations are legitimate descent spectral sequences for the coefficient under consideration. At an Artin stage, adjoining finite free translation coordinates commutes with \(P_n\)-cochains because those coordinates come from \(\mathcal G/R'\) and \(P_n\) fixes \(R'\). Flat torsor descent and the finite-module splittings described above give the bar construction before taking the local row. The compatible finite-thickness calculations then give it on that row. Thus one may filter first by \(P_n\)-, then by translation-, and then by matrix-group cohomology before taking \(P_s\)-cohomology. Each final contribution is a group of the form (59). All group degrees are nonnegative, and the one local-cohomology shift is the fixed integer \(r\).

There is still the finite translation quotient \(T'/T'_h\) to forget. Use its augmented bar construction. In bar degree \(q\) it adds \(q\) finite torsion-coordinate factors of \(\mathcal G\), of one fixed finite torsion order. They are finite free over \(R'\), and \(T'_h\) acts trivially on them. The calculation just given applies in every bar degree, with these factors included in \(M\otimes F\). A further matrix congruence restriction fixes their finite-level label actions. For every resulting open matrix group, the representation assertion of Proposition 50 still applies to its special cohomology rows. The factors added in the finite quotient bars remain in the universal bundle \(M\otimes F\), so the same norm-filtration argument applies in each bar degree. There are only finitely many bar degrees in a fixed total degree, so this descent preserves finite-dimensionality in each degree. The same reasoning forgets the finite quotients used to fix label components or to replace a matrix group by a small open subgroup. Forgetting the full connected marking is exactly the \(P_s\) step above. We have proved that every support row in (46) has finite-dimensional \(P_n\)-cohomology. Its intrinsic \(P_n\) cohomological bound is \(n^2\), so its total cohomology is finite.

The endpoint \(s=n\) was treated by compact duality at the beginning of the section. The finite height-support filtration therefore has finite total \(P_n\)-cohomology. Compact finiteness on \(C_1\), together with the localization triangle \[R\Gamma_{(x)}(C_1\otimes\omega^e) \longrightarrow C_1\otimes\omega^e \longrightarrow C_1[1/x]\otimes\omega^e,\] proves the first assertion of the proposition.

For completeness, this yields every finite frame level as follows. After the first étale basis level, the leading connected marking is tame and decomposes into powers of the invariant line. All subsequent sufficiently deep normal finite joint levels have finite \(p\)-group deck transformations. Their regular representations over \(k\) have finite filtrations by trivial representations. The associated coefficient bundles consequently have finite filtrations by the invariant-line coefficients already treated. The filtrations split as modules on localization, so they give exact sequences of bounded-pole continuous cochains. This proves finite total cohomology at all such sufficiently deep levels. Any prescribed finite joint level is below one of these. Finite torsor descent and its cohomological spectral sequence prove finite-dimensionality in each degree there as well. The fixed cohomological bound for \(P_n\) again gives finite total dimension. Every step admits arbitrary compact profinite-source parameters by the common-thickness, common-stage, and bounded-pole constructions above. The asserted finiteness is finiteness of the resulting cohomology groups over \(k\). ◻

Proof. Put \(M=H^i(P_n,P)\) and \(X=H^i(P_n,B_U)\). Both are finite-dimensional: this is Proposition 35 for \(M\) and Proposition 36 for \(X\). Their comparison kernel is locally transfer-nilpotent. Indeed a primitive in the bounded-pole complex has some finite pole bound, and sufficiently many transfers carry that primitive into \(P\).

Choose cocycle representatives for a basis of \(X\). They have a common pole bound, which is carried into \(P\) by an iterate of transfer. Hence \(S^aX\) is contained in the image of \(M\) for some \(a\). On a finite vector space, the stable image \(X_{\mathrm{inv}}=\bigcap_a S^aX\) is the largest subspace on which \(S\) is invertible; the complementary generalized zero-eigenspace is nilpotent. The same holds for \(M\). The two preceding observations show that \(M_{\mathrm{inv}}\to X_{\mathrm{inv}}\) is surjective and has zero kernel.

The direct limit of the dual transfer system is the dual of the stable image. This proves the comparison on cohomology in every degree. For the assertion on complexes, use the natural zigzag \[C^\bullet(P_n,B_U)^* \longrightarrow C^\bullet(P_n,P)^* \longleftarrow C^\bullet(P_n,P)^\vee\] and take the direct limits of transpose transfer. The second arrow is the inclusion of continuous functionals. The compact cochain terms and their closed images have exact continuous duality; algebraic vector space duality is exact as well. Since the compact cohomology groups are finite, the second arrow is a cohomology isomorphism. The stable-image argument gives the same assertion for the first arrow after the direct limit. Filtered colimits are exact and the complexes are bounded, so the zigzag consists of quasi-isomorphisms after that limit.

All its arrows are restriction, inclusion of functionals, or transpose transfer. Their compatibility with cup products and with finite-frame trace is the compatibility in Lemma 11 and Proposition 29. No choice of a basis of \(M\) or \(X\) is used to define these arrows. ◻

Proof. First keep the translation level finite. If \(U'\triangleleft U\) is a smaller frame level, its coefficient algebra is a finite torsor for \(H=U/U'\). Over the quotient algebra \(A\), faithfully flat trivialization makes it a regular \(A[H]\)-module. Projectivity over \(A[H]\) descends, and \(A[H]\) is free as a \(k[H]\)-module; hence the additive coefficient module is projective for the finite deck group. The same argument applies to the descended coefficient bundles. Coinvariants identify with the lower-level coefficient by trace: on a trivialized torsor this is the augmentation of the regular module. The fixed finite projective \(P_n\)-resolution takes equivariant finite sums and retracts, so both assertions hold on its cochain terms.

Dual terms are consequently acyclic for deck-group invariants, with invariants equal to the required dual coefficient at the lower level. Passing to the smooth union preserves this calculation. Indeed a continuous cochain with discrete smooth values has finite image on a compact source; that image and all its stabilizers are visible at one finite quotient. Thus the continuous bar calculation is the filtered union of these finite calculations.

For translations, test a term of (60) against any finite-dimensional rational representation \(W\). Its equivariant Hom is the dual of the associated-bundle functor evaluated on \(W^*\). That functor is exact. Hence the term is injective in the category of rational \(T\)-representations. Here finite tests suffice: in extending a map from a subcomodule, any new vector lies in a finite-dimensional subcomodule \(W\). Its intersection with the domain is finite dimensional, so exactness extends the map across \(W\) and then across their sum. The maximal-extension argument proves injectivity. The restriction to \(T_h\) remains injective: induction across the finite quotient \(T/T_h\) is exact, as is seen on its finite locally free Hopf algebra.

Translation invariants of these terms are therefore computed without higher termwise cohomology. They recover the dual coefficient of the trivial translation representation; \(T_h\)-invariants recover the indicated finite lift level. The bounded complex permits taking this calculation before the compact frame-group calculation. This proves (61).

At a finite deck level, dualizing pullback gives trace. The induced map on the invariant calculation is precisely corestriction, by the regular-module norm formula. This proves the assertion about frame transitions, including its direction. The remaining assertions follow from the equivariant powering identities of Lemma 11. In particular, powering preserves constant \(\mathbb F_p\)-cochains even before a scalar extension has been chosen to make every stationary operator \(k\)-linear. ◻

Proof. Apply Proposition 29 at a sufficiently deep fixed frame level. Lemma 41 replaces its compact lattice by the bounded-pole coefficient without changing the dual transfer colimit. We obtain, naturally in the frame level, the comparison with the compact support calculation on \([Z/U]\), with the shift \(m+n^2\) and the volume character shown in (62).

Now pass up all finite frame covers using transpose trace. On the geometric presentation this removes compact invariants. More explicitly, finite cover descent computes the supported complexes with locally constant parameters on the compact deck groups. A finite diagram of those parameters is defined at one finite quotient. Passing to all covers therefore gives the supported complex on \(Z\). This is the same smooth-union descent used in Lemma 42. The buffered support and coefficient comparisons of Proposition 29 are made on those finite diagrams before taking the union, so their arrows pass to this limit as well.

Continuous compact-group duality supplies the transpose of the cup action, the residue pairing supplies the shift by \(m\), and the geometric torsor comparison supplies the structural map to \(BP_n\). These are comparisons of complexes with their actions. Thus the result retains the asserted constant-cochain action, rather than only the dimensions of its cohomology groups. ◻

Proof. Suppose first that \(t>1\). In the distribution algebra \(\Lambda\), the ideal generated by \(f\circ[a]-f\) annihilates \(M\); replacing \(a\) by \(a^{-1}\) does not change the argument. The formal group identity \[[a-1](z)=F([a](z),[-1](z))\] shows that the coordinates of \([a-1]\) belong to this ideal. This follows also by setting the two inputs of \(F(X,[-1](Y))\) equal: its coordinates vanish when \(X=Y\), and hence lie in the ideal generated by \(X_i-Y_i\). Writing \(a-1=p^h u\) with \(u\in\mathbb Z_p^\times\) shows that the ideal contains the coordinates of \([p]^h\). The image of the augmentation ideal under the distribution-algebra map for \(T_h\hookrightarrow T\) therefore annihilates \(M\). This is exactly triviality of the \(T_h\)-action.

At \(t=1\), decompose \(M\) into characters of the diagonalizable translation group. Conjugation by \(a\) carries the character \(\chi\) to \(a\chi\). If \(a\) acts identically on \(M\), every character occurring in \(M\) satisfies \((a-1)\chi=0\). It is consequently trivial on \(T_h\) for \(h\geq v_p(a-1)\). Both arguments depend only on \(a\), not on a choice of vectors in \(M\). ◻

Proof. Shrink \(K\) so that it acts trivially on \(M\). Compact group duality identifies the dual of corestriction in degree \(c\) with restriction in degree \(t^2-c\), with the dual coefficient and the orientation character. The latter character is trivial here. In a restriction colimit over open subgroups, positive-degree continuous cohomology with finite discrete coefficients vanishes: a normalized cocycle and its finite diagram factor through a finite quotient, and restriction to the kernel makes its positive-degree class zero. Degree zero retains the underlying coefficient. This proves the assertion by duality.

For the complex assertion, use the bounded spectral sequence \[E_2^{c,j}(K)=H^c(K,H^j(C)) \Longrightarrow H^{c+j}\bigl(R\Gamma(K,C)\bigr).\] All its groups at a fixed level are finite over the finite field \(k\), so their inverse systems are Mittag–Leffler. Taking the inverse limit is exact on these systems and retains only the column \(c=t^2\). There are finitely many groups and pages in each total degree; no derived inverse-limit term remains. This proves (63). Naturality of the spectral sequence and of compact duality proves naturality also for a map \(C\to C'[r]\). Conjugation preserves the chosen orientation because its adjoint determinant is one, so normal levels preserve the stated \(P_t\)-action as well. ◻

Proof. By Proposition 25, every scalar in \(1+p\mathbb Z_p\) acts identically on the compact support cohomology of \(Z\): it fixes the plane and flag parameters and has trivial reduced orientation character. Its character on the volume line is also trivial. Proposition 43 and faithful almost scalar extension therefore give the same scalar identity on \(H^*(\mathcal D)\).

Choose such an \(a\ne1\). By Lemma 44, a single \(T_h\) acts trivially on all these cohomology groups. Increase \(h\) into the prescribed divisible progression. The translation hypercohomology spectral sequence then has rows \[ H^q(\mathcal D)\otimes_k \bigwedge^b(\mathfrak m_{\Lambda_h}/\mathfrak m_{\Lambda_h}^2)^*, \qquad 0\leq b\leq d. \tag{64}\] This is the augmentation Koszul calculation for the regular distribution algebra. The induced auxiliary action on Ext is the linear action on its cotangent space, even though an auxiliary automorphism of the full distribution algebra may be nonlinear. At \(t=1\) only the column \(b=0\) is present, by exactness of invariants.

Apply \(V\)-cohomology to the rows in (64). After tensoring with \(R\), the flag resolutions of Proposition 25 and Shapiro’s lemma reduce their cohomology to that of compact parabolic subgroups with finite-dimensional coefficient representations. Those groups are compact analytic groups without \(p\)-torsion; their finite projective resolutions give finite-dimensional cohomology in a bounded range. There are only finitely many flag types and finite exterior factors. The resulting bounds descend before almost scalar extension by the bounded almost-length argument of Corollary 26: every finite-dimensional subspace of a \(k\)-cohomology group contributes its dimension to the length after tensoring with \(R\), whereas the finite flag and compact group resolutions give one upper bound for that length. Smooth cohomology commutes with this flat tensor, by the fixed finite projective group resolution. Thus the \(V\)-cohomology groups just obtained are finite-dimensional smooth \(P_t\)-modules.

Take \(K\)-cohomology next and then inverse limit with corestrictions in \(K\). Lemma 45 retains only the top connected group degree \(t^2\), with the underlying finite-dimensional coefficients. All cohomological ranges are bounded by the dimensions of the compact analytic groups, the support range, and \(d\). Mittag–Leffler therefore permits using the inverse limit on these spectral sequences. By Lemma 42, the result is the algebraic dual of \[\mathop{\mathrm{colim}}_K H^*(P_n,B_{K,V}^{(h)}).\] It is finite-dimensional, so the displayed colimit is finite-dimensional as well. Power transport identifies it with the asserted group for \(B_{\mathrm{conn},V}\). Rectangular subgroups form a cofinal family, so this proves the finiteness statement at every compact open \(V\).

It remains to check that a scalar subgroup can be chosen uniformly when both frame levels vary. The scalar \(a\) above is central in \(G\), so it acts trivially by conjugation on every \(V\). On (64) it is the identity: it is the identity on \(H^q(\mathcal D)\), and its action on the cotangent factor is given by reduction of scalar multiplication, hence is also the identity. It therefore acts identically on the successive spectral-sequence terms used above. The filtration lengths are bounded independently of \(K,V\): the dimensions are always \(t^2\) and \(m^2\), the translation range is \([0,d]\), and the complex \(\mathcal D\) is bounded by \(n^2\). A further fixed \(p\)-power of \(a\) consequently acts identically on the cohomology of every sufficiently deep rectangular-level calculation: an operator acting identically on a filtration of length \(l\) satisfies \((a-1)^l=0\), and \(a^{p^j}-1=(a-1)^{p^j}\) in characteristic \(p\). These deep rectangular levels are cofinal in the full-frame union.

Dualizing and passing to the algebraic frame union proves scalar identity on \(H^*(P_n,B_{\mathrm{full}})\). Its action is smooth, so identity for a topological generator implies identity for the closed open scalar subgroup it generates. Finally, every root stage is identified by powering with the same finite-level calculation, with the same scalar action. Cohomology commutes with the algebraic root union under the common-stage convention, so the same subgroup works there. This proves the scalar assertion used on strata of higher starting height in the subsequent induction steps. ◻

Proof. For \(s=1\), the finite diagonalizable translation representations split into character summands. The transpose of inclusion is the projection dual to inclusion of those summands, which is normalized finite-Hopf corestriction; it preserves the trivial character. This proves the first assertion in that case.

Now suppose \(s>1\). At a finite translation level use the invariant trace pairing of Lemma 11 to identify the regular representation with its distribution-algebra model \[N_q=k[D_1,\ldots,D_d]/(D_1^q,\ldots,D_d^q).\] The constant function corresponds to the normalized top socle element. Inclusion of functions at the next finite quotient is therefore, in this model, \[N_q\longrightarrow N_{q'},\qquad f\longmapsto \left(\prod_iD_i^{q'-q}\right)f .\] Here compatible invariant pairings give the displayed formula; for other Frobenius-pairing coordinates it is multiplied by a unit of the target algebra. Its image is the submodule annihilated by all \(D_i^q\). The opposite-direction normalized Hopf transfer is the quotient \(N_{q'}\twoheadrightarrow N_q\). These descriptions follow either from the invariant pairing or by pairing the two maps with the monomial basis; the normalization fixes the top functional. The root-ring presentation in Proposition 7 identifies the first map with the actual root inclusion. Taking its dual in the regular-module construction of \(\mathcal D\) gives corestriction from \(T_{h'}\) to \(T_h\). Thus the transition here is distinct from the transpose of Hopf transfer used to define \(\mathcal D\).

Here is the cochain calculation, with directions explicit. Write \[A=\Lambda_h=k[[x_1,\ldots,x_d]],\qquad A'=\Lambda_{h'}=k[[y_1,\ldots,y_d]],\qquad y_i\longmapsto x_i^q ,\] where \(q>1\) is the relative exponent in the divisible progression. The Koszul complex \(K_A(x_1,\ldots,x_d)\) resolves \(k\), and \[A\otimes_{A'}K_{A'}(y_1,\ldots,y_d) =K_A(x_1^q,\ldots,x_d^q)\] resolves \(A/(x_1^q,\ldots,x_d^q)\). The socle inclusion \(k\to A/(x_i^q)\) is lifted by the chain map \[ e_I\longmapsto \left(\prod_{j\notin I}x_j^{q-1}\right)f_I . \tag{65}\] Indeed both sides of its differential identity have, for an index \(i\in I\), the factor \(x_i^q\prod_{j\notin I}x_j^{q-1}\). Applying \(\mathop{\mathrm{Hom}}_A(-,\mathcal D)\) gives \[\mathop{\mathrm{RHom}}_{A'}(k,\mathcal D)\longrightarrow\mathop{\mathrm{RHom}}_A(k,\mathcal D),\] the asserted corestriction. On a row \(M=H^j(\mathcal D)\) annihilated by \((x_1,\ldots,x_d)\), its degree-\(b\) map is multiplication by the complementary product in (65). It is zero if \(b<d\) and the identity if \(b=d\).

Multiplying the socle inclusion by a unit multiplies these components by that unit; on \(M\) its effect is its nonzero augmentation. Thus the vanishing and top isomorphism do not require a choice of compatible unit normalizations. The maps themselves are the finite Hopf inclusion and its transpose, so they are natural under all frame actions, including nonlinear augmented automorphisms of the distribution algebra. The cohomology-row calculation is therefore equivariant. We do not need to identify the full complexes at a fixed root level with their top rows. All finite coefficient diagrams commute with frame changes and their traces, giving the remaining compatibilities. ◻

Proof. The finite flag calculation and the translation-row spectral sequence in the proof of Proposition 46 show that \(H^*(A_h)\) has finite total dimension. This uses only \(V\)-cohomology of the support flag representations; no cohomology of a coefficient twisted by an arbitrary \(P_s\)-representation is used. Finiteness is obtained before almost scalar extension by Corollary 26. The dimensions of the cohomological ranges are bounded by \(n^2\), \(m^2\), and \(d\), independently of \(h\).

First take the connected-frame limit at a fixed \(h\). Choose a cofinal sequence of normal principal open subgroups \(K\triangleleft P_s\). Lemma 42 identifies \(R\Gamma(K,A_h)\) with the finite-frame dual. Compact duality and inverse limit with corestriction therefore give \[ \mathop{\mathrm{Hom}}_k\!\left( H^i(P_n,B_{\mathrm{conn},V}^{(h)}),k\right) \simeq H^{-i-s^2}(A_h). \tag{68}\] This is the calculation of Lemma 45, applied to the bounded finite cohomology rows of \(A_h\). It retains the full \(P_s\)-action: an element of \(P_s\) acts on \(K\) by conjugation as well as on its coefficient. In compact duality its additional character is the determinant of the adjoint action on \(\mathop{\mathrm{Lie}}(P_s)\), which is one. For example, after a splitting field extension the Lie algebra is \(M_s\) and conjugation has determinant one. Thus the top-degree survivor in (68) carries exactly the original \(P_s\) coefficient action. Normality of \(K\) makes these actions compatible throughout the inverse system. The finite cohomology groups are Mittag–Leffler, so no derived inverse-limit term occurs.

All these operations precede scalar extension. The actual power-transport isomorphism at a fixed \(V\) identifies the dimensions of \(H^*(P_n,B_{\mathrm{conn},V}^{(h)})\) with the same fixed finite dimensions for every \(h\) in the progression. These isomorphisms give a uniform bound on dimensions; they do not identify the root inclusions with isomorphisms. Consequently the root direct limit is finite dimensional, and duality identifies it with the inverse limit of the finite spaces in (68). Every cohomological inverse system here is Mittag–Leffler.

This proves (66). It remains to prove the character test. For \(s>1\), fix a row \(M=H^q(\mathcal D)\) and consider the bounded spectral sequence \[H^a(V,H^b(T_h,M)) \ \Longrightarrow\ H^{a+b}\bigl(V,R\Gamma(T_h,M)\bigr).\] Here \(H^b(T_h,M)=M\otimes E_h^b\). By Lemma 47, root corestriction acts as zero on every row with \(b<d\). On the \(\chi\)-part the remaining \(b=d\) row is zero by (67). Thus every transition is zero on this row page after projecting to \(\chi\). The ranges \(0\leq a\leq m^2\) and \(0\leq b\leq d\) give a uniform finite filtration on its abutment. A transition zero on the associated graded lowers that filtration, so a bounded number of successive transitions is zero on the \(\chi\)-part of the abutment.

Next use the bounded cohomology filtration of \(\mathcal D\) to compute \(A_h\) from these row calculations. Its number of nonzero cohomology rows is at most \(n^2+1\). Repeating the same filtration argument therefore gives one integer \(N\), independent of \(h\), such that the composite of \(N\) successive root transitions is zero on \(H^j(A_h)_\chi\) in every degree. For example, one may enlarge the bound to \((n^2+1)(m^2+d+1)\). This two-stage argument retains all derived translation and \(V\)-extensions; it assumes no equivariant formality. The \(\chi\)-part of the inverse system is uniformly pro-zero. Its inverse limit is consequently zero, as is its derived inverse-limit contribution.

For \(s=1\), translation invariants are exact character projection. Once \(T_{h_0}\) acts trivially on \(H^*(\mathcal D)\), only \(b=0=d\) occurs. Condition (67) kills the corresponding \(\chi\)-part directly by the bounded cohomology filtration of \(\mathcal D\). Throughout the proof, inverse limits were evaluated on the finite \(k\)-cohomology systems before tensoring with \(R\); no interchange of an infinite inverse limit with almost scalar extension is required. ◻

Proof. We record the connected-frame characters in the support calculation. The orientation line for a rational plane of dimension \(r\) comes by scalar extension from a one-dimensional \(\mathbb F_p\)-space, naturally for automorphisms of \(N_s\). Indeed the natural compact primitive comparison \[R\Gamma_c(N_s^r,\mathbb F_p)\otimes_{\mathbb F_p}R \longrightarrow R\Gamma_c(N_s^r,\mathscr A)\] is computed by the same buffered affine exhaustion and lattice corestriction calculation as Lemma 23. The affine finite-coefficient complex is \(\mathbb F_p(-r)[-2r]\); the lattice corestriction limit retains its top \(sr\)-dimensional orientation. Thus the source has one \(\mathbb F_p\)-cohomology line in degree \((s+2)r\), and the comparison is an equivalence. The calculation is natural, so it applies also to \(P_s\) automorphisms which do not preserve a chosen affine/lattice presentation. The compact-parameter construction gives a continuous action on that finite line. Its \(P_s\) character consequently takes values in \(\mathbb F_p^\times\).

Every continuous character \(P_s\to\mathbb F_p^\times\) is a power of \(\nu_s\). Its restriction to the pro-\(p\) group \(P_s^1\) is trivial; the quotient is the cyclic group \(\mathbb F_{p^s}^\times\), and every homomorphism from that group to \(\mathbb F_p^\times\) is a power of the finite-field norm. Denote the resulting orientation character for an \(r\)-plane by \(\nu_s^{c_r}\); its exact exponent is not needed. The rational plane and flag parameters are fixed by \(P_s\), since \(P_s\) commutes with the action on the \(m\) row indices. Hence each finite flag term for the support row is a representation with this scalar \(P_s\) character.

There are also the volume and top translation Ext lines. Let \(a\in\mathbb F_{p^s}^\times\) be a Teichmüller element, acting on the tangent line of the fixed Honda group by \(a\). With the inverse-substitution convention of (15), the characters are \[\langle\eta\rangle(a)=a^{-m},\qquad E_{h_0}^d(a)=a^{-m(p+p^2+\cdots+p^{s-1})}.\] For the second identity, let \(h=[a]\) act on the Honda group. Its action on functions is \(h\cdot f=(h^{-1})^*f\), so the dual distribution action is \(\lambda\mapsto(f\mapsto\lambda(h^*f))\). Under Cartier duality this is pullback by \(h^D(\rho)=\rho\circ h\). To compute these characters, let \(O=W(\mathbb F_{p^s})\subset\operatorname{End}(\Gamma_s)\) act on the covariant Dieudonne module \(M\) over \(W(k)\). The Honda endomorphism calculation identifies a Teichmuller element \(a\) with the endomorphism \(aX\) (Ravenel 2004, Lemma A2.2.16 and Theorem A2.2.18). The \(s\) idempotents of \(W(k)\otimes_{\mathbb Z_p}O\) split \(M\) into its eigensummands. Its semilinear Frobenius commutes with \(O\) and is invertible after inverting \(p\), so it permutes the rationalized eigensummands cyclically and forces their ranks to be equal (Zink 2002, Example 14 and Proposition 15). The covariant Hodge sequence for \(M/pM\) has total rank \(s\), the sum of the dual dimension \(s-1\) and the dimension one of \(\Gamma_s\) (Lau 2010, sec. 5). Each eigensummand therefore has rank one, and the residue eigencharacters are \(a,a^p,\ldots,a^{p^{s-1}}\). In this functorial Hodge sequence the quotient \(\mathop{\mathrm{Lie}}(\Gamma_s)\) has character \(a\), and the subspace \(\omega_{\Gamma_s^D}\) is the cotangent space of the distribution formal group, with characters \(a^p,\ldots,a^{p^{s-1}}\). Degree-one translation Ext is the linear dual of this cotangent space; top Ext is its top exterior power. This gives the stated negative exponents. In particular the action used here is cotangent pullback by \(h^D\), not the tangent action of \(\rho\mapsto\rho\circ h^{-1}\) on dual points. The divisible progression fixes these characters when transported to \(T_{h_0}\). Therefore \[ (\langle\eta\rangle\otimes E_{h_0}^d)(a) =a^{-m(1+p+\cdots+p^{s-1})} =\nu_s(a)^{-m}. \tag{69}\] For \(s=1\) the second line is trivial and the same formula holds. This is a calculation on the fixed Honda group and its actual translation representations. It does not identify a parameter-dependent universal Lie line with the constant extension of its special-fibre character.

To apply (67), use Proposition 43 on each \(H^q(\mathcal D)\otimes E_h^d\) after tensoring with \(R\). By the finite flag resolution, the resulting support rows have only norm characters of \(\Delta_s\), namely the products of \(\nu_s^{c_r}\) with (69). \(V\) commutes with \(\Delta_s\). Its finite projective resolution therefore preserves this condition, and commutes with the flat scalar extension. Faithfulness detects (67) before scalar extension for every nonnorm \(\chi\). Lemma 48 now kills all these isotypic pieces of the perfected dual. The set of norm characters is closed under inversion, so dualizing proves the assertion on the primal coefficient. ◻

Proof. First take the algebraic union over the matrix levels in Lemma 49. The fixed finite \(P_n\)-resolution and the common-stage cochain convention give \[H^i(P_n,B_{\mathrm{full}}^{\mathrm{perf}}) =\mathop{\mathrm{colim}}_{V'} H^i(P_n,B_{\mathrm{conn},V'}^{\mathrm{perf}}).\] The maps here are actual coefficient pullbacks and are \(P_s\)-equivariant. Since \(\Delta_s^0\) acts identically on every term, it acts identically on this individual \(P_n\)-cohomology group. This step uses an algebraic coefficient colimit; it does not deduce an individual row from a total-cohomology spectral-sequence abutment.

The group \(V\) commutes with \(P_s\), in particular with \(\Delta_s^0\). A bounded finite projective resolution for \(V\) computes its cohomology by retracts of finite sums of the displayed coefficient module. Thus \(\Delta_s^0\) also acts identically on \(Q\). Its finite-dimensionality is the individual-row finiteness assertion of Lemma 40. The hypotheses of that lemma hold for the perfected coefficient as well: finite-frame descent identifies its total \(V'\)-cohomology with the connected-frame calculation, and the uniform finite root-stage dimensions used in Lemma 48 give finite dimensions after the root union, for every open \(V'\). The cohomological ranges stay bounded by the fixed group dimensions.

Finally the image \(\Pi\) of \(P_s^1\) in \(\mathop{\mathrm{GL}}_k(Q)\) is a finite normal \(p\)-group. Its augmentation ideal \(J\subset k[\Pi]\) is nilpotent. The finite filtration \(J^bQ\) is \(P_s\)-stable, and its quotients have trivial \(P_s^1\) action. They therefore split over \(k\) into characters of \(P_s/P_s^1=\mathbb F_{p^s}^\times\). Exactness of restriction to \(\Delta_s\) and of its character projections shows that each such character is trivial on \(\Delta_s^0\). Each is consequently a power of \(\nu_s\). Refine the filtration by these one-dimensional summands. Duality reverses this finite filtration and inverts its characters, proving the final assertion. ◻

Proof. We specify the finite-coefficient input. Theorem 1.1 of (Dotto and Le Hung 2025), comprising Propositions 5.5 and 5.7 there, computes the cohomology of an unramified split Chevalley group and proves restriction to its Iwahori to be an isomorphism when \(p>h+1\), where \(h\) is the Coxeter number. For \(\mathop{\mathrm{GL}}_a/\mathbb Q_p\), \(a\geq2\), this is \(h=a\); the degree of the ground field and its ramification index are both one. Thus its hypothesis is exactly \(a<p-1\), and its exterior degrees are \(1,3,\ldots,2a-1\). For the division algebra, (Dotto and Le Hung 2025, sec. 5.2, Proposition 5.12) applies to an unramified ground field and a division algebra of invariant \(1/a\) with \(a<p-1\). It gives the same exterior algebra over the residue field of \(D_a\), which here is \(\mathbb F_{p^a}\). Extension of the finite constant field commutes with continuous cochains. In particular, that calculation gives the stated degreewise dimensions over \(\mathbb F_p\). For \(a=1\) all three groups are \(\mathbb Z_p^{\times}=\mu_{p-1}\times(1+p\mathbb Z_p)\), so the assertion follows directly from the two-term resolution for \(\mathbb Z_p\) and exactness of \(\mu_{p-1}\)-invariants.

We include the integral deduction. None of these groups has an element of order \(p\). Such an element in \(\mathop{\mathrm{GL}}_a(\mathbb Q_p)\) would require the cyclotomic polynomial \(\Phi_p\), of degree \(p-1\), in its rational representation; in \(D_a^{\times}\) it would embed \(\mathbb Q_p(\zeta_p)\) as a subfield of a division algebra of degree \(a\). Both are impossible for \(a<p-1\). The compact \(p\)-adic analytic group theorem therefore gives \(p\)-cohomological dimension \(a^2\). Its completed group algebra is Noetherian, and the trivial module has a bounded resolution by finitely generated projectives. One can obtain finite generation of the resolution terms from Noetherianity and truncate at the cohomological dimension. The compact-module form of these analytic-group results is recalled in (Venjakob 2002, secs. 1.1–1.2); see also (Lazard 1965; Serre 1997). In particular, \(H^q(G,\mathbb Z_p)\) is finitely generated over \(\mathbb Z_p\), and rational base change computes continuous rational cohomology.

Write \(r_q\) for its rank and \(t_q\) for the number of cyclic summands in its finite \(p\)-primary torsion subgroup. The coefficient sequence gives \[0\longrightarrow H^q(G,\mathbb Z_p)/p \longrightarrow H^q(G,\mathbb F_p) \longrightarrow H^{q+1}(G,\mathbb Z_p)[p]\longrightarrow0,\] and consequently \[ \dim_{\mathbb F_p}H^q(G,\mathbb F_p)=r_q+t_q+t_{q+1}. \tag{72}\] Rational Lazard comparison identifies the rational cohomology with the invariant Lie-algebra calculation. The matrix Lie algebra is \(\mathfrak{gl}_a\); after a splitting field extension the division Lie algebra is the same Lie algebra. Its cohomology is exterior in degrees \(1,3,\ldots,2a-1\), and conjugation acts trivially (Barthel et al. 2025, Proposition 3.8.1 and Lemma 3.8.2). The rational and mod-\(p\) dimensions therefore agree in every degree. Equation (72) forces \(t_q=t_{q+1}=0\) for every \(q\). The coefficient sequence for \(p^r\) now proves (71). Restriction to \(I_a\) is an isomorphism integrally as well, since its reduction modulo \(p\) is an isomorphism between finite free modules of equal rank.

For completeness, the finite-field extension in the division calculation does not conceal an algebra-descent assertion. Choose over \(\mathbb F_p\) a homogeneous basis of the indecomposable quotient of the positive-degree cohomology algebra. After extension to \(\mathbb F_{p^a}\) there is precisely one basis element in each of the indicated odd degrees. Their lifts generate the original graded algebra by induction on degree; their squares vanish because \(p\) is odd. The resulting surjection from the exterior algebra is an isomorphism by the degreewise dimension calculation. Lift these generators to integral cohomology using (71). Graded commutativity and torsionfreeness give a map from the integral exterior algebra, whose reduction modulo \(p\) is an isomorphism. Nakayama’s lemma in each degree proves the integral presentation. ◻

Proof. For \(a=1\) the groups are equal. Otherwise let \(\mathcal B_a\) be the reduced Bruhat–Tits building. The group \(J_a\) preserves vertex types, acts without inversions, and has a closed chamber as fundamental domain. The stabilizer \(K_v\) of a vertex is a conjugate of \(K_a\). Indeed a matrix stabilizing the homothety class of a lattice is a scalar times a lattice automorphism, and determinant valuation zero forces the scalar valuation to be zero. Every face stabilizer \(K_\sigma\) is compact and lies between the Iwahori of the chamber and one of its vertex stabilizers.

The index \([K_\sigma:I_a]\) is prime to \(p\): after reduction it is a flag count for the corresponding Levi blocks, and each such flag count is congruent to one modulo \(p\). Transfer therefore makes \(H^*(K_\sigma,\mathbb F_p)\to H^*(I_a,\mathbb F_p)\) injective. The restriction \(H^*(K_v,\mathbb F_p)\to H^*(I_a,\mathbb F_p)\) is an isomorphism by Theorem 51, and factors through this injection. It follows that the face restriction is an isomorphism too. Identifying every face group with \(H^*(I_a,\mathbb F_p)\) by its actual restriction makes all incidence maps identities.

The equivariant cellular resolution of the contractible building gives \[ E_1^{r,s}=\bigoplus_{\substack{\sigma\text{ in the chamber}\\ \dim\sigma=r}} H^s(K_\sigma,\mathbb F_p) \ \Longrightarrow\ H^{r+s}(J_a,\mathbb F_p). \tag{73}\] The sum is finite, the chamber has dimension \(a-1\), and the compact face groups have cohomological dimension \(a^2\). Thus this is a bounded spectral sequence. Its rows are the ordinary cochain complex of a simplex with constant coefficient \(H^s(I_a,\mathbb F_p)\). They have only degree zero cohomology. The resulting edge map is restriction to a vertex; this proves the assertion for \(H\mathbb F_p\).

A finite abelian \(p\)-group has a finite filtration with quotients \(\mathbb F_p\). Coefficient exact sequences and then Postnikov fibre sequences prove the assertion for every stated \(E\). This proves a comparison of the actual restriction maps, natural in \(E\). The integral assertion follows by taking the inverse limit of the finite-coefficient comparisons. The groups on the compact side are finite at each finite coefficient level, so the cohomological inverse systems are Mittag–Leffler. The same is then true on the \(J_a\) side. ◻

Proof. Use the convention that \(V_n\) has slope \(1/n\). A point of projective space specifies a length-one quotient at the untilt divisor, \[0\longrightarrow W\longrightarrow V_n' \longrightarrow i_*\mathcal L\longrightarrow0,\] where \(V_n'\) is a form of \(V_n\). The kernel has rank \(n\) and degree zero. If a proper subbundle of rank \(r\) had positive degree, its inclusion in the stable bundle \(V_n'\) would give degree strictly less than \(r/n<1\), a contradiction. Thus \(W\) is semistable of slope zero. The relative slope-zero equivalence identifies it with a rank-\(n\) \(\mathbb Q_p\)-local system.

Conversely, an upper modification of a trivial rank-\(n\) bundle has nonnegative slopes and total degree one. Nonnegativity follows because a negative-slope quotient would receive no map from the trivial bundle, whereas that bundle has only a torsion cokernel. The positive part is therefore a single stable bundle of degree one; the rest, if present, has slope zero. A slope-zero quotient is precisely a rational linear functional annihilating the modification direction. Its absence is exactly the condition that the direction lie in \(\Omega^{n-1}_C\). On that locus the middle bundle is of type \(V_n\).

The relative classification of these bundles and the slope-zero/local-system equivalence make the frames torsors in the \(v\)-topology, so the two descriptions are inverse on families as well as on geometric points. This is the basic EL modification description underlying the duality of (Scholze and Weinstein 2013, Theorem 7.2.3); the slope argument above specifies its basic locus in the present case.

We explain the determinant reduction. Fix a lattice in the rank-one slope-zero local system \(\det(V_n)(-\infty)\). Require the determinant of the frame of \(W\) to carry \(\mathbb Z_p\) to that lattice. Its permitted rational frame changes are exactly \(J_n\). On the positive-bundle side the corresponding condition is \(v_p(\operatorname{Nrd}(g))=0\) on \(D_n^{\times}\), whose subgroup is \(P_n\). Hence the same determinant condition cuts out the two groups in (74). The determinant-matching construction is the one appearing, for the positive-height extension towers, in (Barthel et al. 2026, Theorem 2.0.1 and Theorem 2.6.3).

Choose the positive Honda isocrystal over the unramified field containing its endomorphism field. Arithmetic then acts on the period factor and commutes with its division-algebra endomorphisms. Subtracting the untilt divisor in its determinant produces a rank-one \(\mathbb Q_p\)-local system with compact arithmetic image; equivalently, its integral Tate lattice is preserved. More intrinsically, the valuation of an arithmetic determinant action is a continuous homomorphism from a profinite group to \(\mathbb Z\) and hence is zero. The chosen determinant component can therefore be retained arithmetically. On the split-frame directions this is the usual arithmetic action on Drinfeld space: changing the trivialization of the determinant only changes a line scale, which disappears in projective directions. Thus both torsor presentations and both structural maps commute with arithmetic. ◻

Proof. Arithmetic separation and the specified degree-zero edges. We first prove (75). The projective-space calculation is \[H^{2j}(\mathbb P^{n-1}_C,\mathbb F_p)=\mathbb F_p(-j),\quad 0\leq j<n, \qquad H^{2j+1}(\mathbb P^{n-1}_C,\mathbb F_p)=0.\] Its geometric automorphisms act trivially on these lines, and \(\gamma\) acts by \(u^{-j}\). For Drinfeld space, apply the integral and finite-coefficient theorem (Colmez et al. 2021, Theorems 1.1 and 5.1) with ground field \(\mathbb Q_p\), matching the rational \(\mathbb Q_p\)-hyperplanes defining \(\Omega_C^{n-1}\), and then restrict its arithmetic action to \(\mathop{\mathrm{Gal}}(\overline K/K)\). It gives compatible topological isomorphisms \[ \begin{split} H^j_{\mathrm{\acute et}}(\Omega^{n-1}_C,\mathbb Z_p(j)) &\simeq\operatorname{Sp}_j(\mathbb Z_p)^*,\\ H^j_{\mathrm{\acute et}}(\Omega^{n-1}_C,\mathbb F_p(j)) &\simeq\operatorname{Sp}_j(\mathbb F_p)^*, \qquad 0\leq j<n. \end{split} \tag{78}\] Here \(\operatorname{Sp}_j\) is the integral generalized Steinberg representation, the star is its continuous integral or finite-field dual, and arithmetic acts trivially on that factor. Cohomology vanishes above \(n-1\). Thus the untwisted \(j\)th group has exactly the arithmetic scalar \(u^{-j}\); degree zero is precisely the constants. In particular, (78) is an integral and finite-coefficient statement. We neither reduce a rational pro-étale calculation nor discard analytic terms by tensoring it with \(\mathbb F_p\).

Apply geometric descent to the two presentations of \(\mathcal M_n\). Lemma 20 shows that every group-cohomology row arising from geometric degree \(j\) has the same arithmetic scalar \(u^{-j}\). For the projective presentation its geometric degree is \(2j\); for the Drinfeld presentation it is \(j\). Both filtrations are bounded, using the compact group resolution on the first side and the finite chamber together with compact face resolutions on the second. Since \[1\leq j\leq n-1<p-1,\] all the higher geometric rows have eigenvalue different from one. An equivariant differential between generalized summands with relatively prime eigenvalue polynomials is zero. Exactness of primary decomposition therefore shows that the generalized-one part of the abutment is exactly the geometric degree-zero row.

We identify its edge using the unit \(\underline{\mathbb F_p}\to\mathscr C_{Y,\mathbb F_p}\). For each of \(Y=\mathbb P_C^{n-1}\) and \(Y=\Omega_C^{n-1}\), its induced point map \(\mathbb F_p\to H^0_{\mathrm{\acute et}}(Y,\mathbb F_p)\) is an isomorphism: the displayed row calculations make the target one-dimensional, and the constant section \(1\) is nonzero. For each compact group in the descent, apply the finite projective resolution \(Q_\bullet\) of Lemma 20. Each evaluated internal-Hom term is an idempotent summand of finitely many copies of the point value, as in (29). The unit commutes with these idempotents and the differentials, so it is an isomorphism on the bottom-row computing complexes. The bar comparison identifies the map so obtained with \(\eta_{Y,G,H\mathbb F_p}\) from continuous cochains. On the Drinfeld side make the same check on each compact face and then take the finite chamber: naturality of the unit with respect to face restrictions holds before the projective resolutions are chosen. Thus both degree-zero edges are the specified finite constant-section maps. This proves (75), with its specified maps. The generalized-one part is closed under products, and these unit maps are multiplicative, proving the assertion about cup products.

Finite Postnikov coefficients and the sphere limit. The finite-coefficient comparison must now retain actual lifts. Put \[Q(T)=\prod_{j=1}^{n-1}(T-\widetilde{u^{-j}})\in\mathbb Z[T],\] where the tildes are integer lifts; the empty product at \(n=1\) is one. Then \(Q(1)\) is a \(p\)-adic unit. The preceding bounded filtrations imply that, on the finite-coefficient cohomology, some powers of \(T-1\) and \(Q(T)\) give the two coprime primary factors for \(T=\gamma\). Coefficient extensions and Postnikov fibre sequences retain this property, with possibly larger exponents. They also retain a finite bound on the cohomological amplitude of \(C^*(\mathcal M_n;E)\).

For clarity, the primary part can be taken on spectra and not merely on their homotopy groups. A bounded Postnikov spectrum with bounded \(p\)-primary exponent is killed, as an object, by a sufficiently large power of \(p\). A polynomial in an endomorphism that vanishes on every homotopy group becomes zero as a map after a bounded power, by the Postnikov filtration. Increase the two primary exponents accordingly. The Bezout identity for the coprime factors \((T-1)^N\) and \(Q(T)^N\) modulo that power of \(p\) then gives complementary idempotents. Splitting them defines \(C^*(\mathcal M_n;E)_{(1)}\) functorially. Equivalently, localization of the \(\gamma\)-action by \(Q(\gamma)\) selects this summand. The construction is exact on fibre sequences and is independent of the larger exponents used to express the two primary factors.

The finite units \(\eta_{X,G,E}\) are \(\gamma\)-equivariant by their base-change naturality, and their continuous sources have trivial \(\gamma\)-action. They therefore land in this summand. Applying coefficient exact sequences to these same units extends (75) from \(H\mathbb F_p\) to finite \(p\)-primary Eilenberg–Mac Lane spectra. Induction through the finite Postnikov tower of \(E\) proves the first two equivalences in (76). The last is Lemma 53. This proof uses the maps of cochain spectra throughout, and their coefficient naturality gives Hurewicz naturality. Product compatibility means compatibility with a specified coefficient pairing \(E\wedge F\to E'\). The termwise cup maps for \(\eta\) respect this pairing, and a tensor product of two unipotent arithmetic actions is again unipotent, so they preserve the generalized-one parts.

In particular, write the continuous sphere cochains as \[ C^*_{\mathrm{cts}}(G;S) =\varprojlim_{r,N} C^*_{\mathrm{cts}} (G;\tau_{\leq N}(S/p^r)). \tag{79}\] Every displayed coefficient has finite \(p\)-primary homotopy groups. Take the inverse limit of the already constructed finite units and equivalences, defining the middle generalized-one object by this same limit. Every geometric pullback of a sphere class means this inverse limit of the finite constant-section pullbacks. We do not interchange an eigenspace localization with an unrestricted inverse limit. On the classifying groups, finite cohomological dimension controls the Postnikov limit in each degree; finite coefficient cohomology and the finite building comparison control the coefficient limit. This agrees with continuous totalization of termwise \(p\)-completed locally constant sphere cochains. Consequently a sphere-cochain representative on \(K_n\) transports to one on \(P_n\) with its specified mod-\(p\) Hurewicz image.

The action on a fixed connected quotient. It remains to prove (77). Fix a length-one quotient \(V_t\to i_*\mathcal L\) and a rational frame, with determinant lattice, of its slope-zero kernel \(W_t\). On the full connected frame space the universal sequence is \[0\longrightarrow L_m\otimes\mathcal O \longrightarrow V_n'\longrightarrow V_t\longrightarrow0.\] Modify it by the fixed quotient of \(V_t\). On \([Z/K_m]\) this produces \[ \begin{gathered} 0\longrightarrow W_n\longrightarrow V_n' \longrightarrow i_*\mathcal L\longrightarrow0,\\ 0\longrightarrow L_m\otimes\mathcal O \longrightarrow W_n\longrightarrow W_t\longrightarrow0. \end{gathered} \tag{80}\] The first line defines a map \(f:[Z/K_m]\to\mathcal M_n\) whose positive-bundle classifying map is \(b\). All bundles in the second line have slope zero. Exactness of the slope-zero/local-system equivalence therefore identifies its kernel with the actual rank-\(m\) Tate local system \(L_m\otimes\mathbb Q_p\), and its quotient with the fixed framed \(\mathbb Q_p^t\). Locally on the pro-étale, hence on the \(v\)-site, this sequence admits rational frames and splittings.

Its permitted frame changes form \[ H_{m,t}= \left\{\begin{pmatrix}g&v\\0&1_t\end{pmatrix}: g\in K_m,\ v\in M_{m,t}(\mathbb Q_p)\right\}\subset J_n. \tag{81}\] The determinant condition is the product of the determinant lattice of \(L_m\) and the fixed one of \(W_t\), using the matching convention in Lemma 54. The quotient \(H_{m,t}\to K_m\) has its displayed block-diagonal section. The composition \([Z/K_m]\to BH_{m,t}\to BK_m\) is exactly \(q\).

The additive group \(M_{m,t}(\mathbb Q_p)\) has no positive continuous cohomology with constant finite \(p\)-primary coefficients. Indeed, exhaust it by \(p^{-r}\mathbb Z_p^{mt}\). The cohomology of these compact lattices with \(\mathbb F_p\)-coefficients is the exterior algebra on their continuous duals. Restriction to the preceding lattice multiplies each degree-one generator by \(p\), and is zero in every positive degree. In degree zero it is the identity. The countable inverse-limit sequence has no derived-limit error: these cohomology systems are Mittag–Leffler, and restriction of continuous cochain terms is surjective because a compact open subset admits extension of locally constant functions. Thus only degree zero survives. Coefficient exact sequences and Postnikov induction give the same assertion for every finite Postnikov \(E\).

For the extension descent, the same lattice argument applies with an extra compact profinite parameter \(T\). At \(\mathbb F_p\) coefficients, finite clopen rectangle refinements give \[\operatorname{LC}(T\times(p^{-r}\mathbb Z_p^{mt})^\bullet;\mathbb F_p) =\operatorname*{colim}_{T\twoheadrightarrow T_i} \prod_{t\in T_i}C^\bullet_{\mathrm{cts}}(p^{-r}\mathbb Z_p^{mt};\mathbb F_p).\] Filtered colimits and finite products are exact, so its vertical cohomology is the locally constant \(T\)-family of the compact-lattice cohomology just computed. The lattice restrictions are still the identity in degree zero and zero in positive degrees, and the cochain term restrictions are surjective by extension by zero from the clopen subset \(T\times(p^{-r}\mathbb Z_p^{mt})^a\). The same Mittag–Leffler argument therefore applies. The lattices are \(K_m\)-stable; take \(T=K_m^b\) in the outer bar for \(1\to M_{m,t}(\mathbb Q_p)\to H_{m,t}\to K_m\to1\). The resulting vertical equivalences commute with the outer faces. Totalizing this external double bar, and then using coefficient and Postnikov induction, shows that inflation from \(K_m\) is an equivalence on the finite constant cochains of \(H_{m,t}\). Its inverse is restriction along the displayed section.

For a \(J_n\)-class \(c_J\) whose restriction to \(K_n\) is \(c_K\), that section followed by \(H_{m,t}\to J_n\) is the upper-left block inclusion through \(K_n\). Hence \[\operatorname{res}^{J_n}_{H_{m,t}}c_J =\operatorname{inf}^{H_{m,t}}_{K_m} \operatorname{res}^{K_n}_{K_m}c_K.\] The \(H_{m,t}\)-torsor above induces the split-frame torsor of \(f\) along \(H_{m,t}\to J_n\) and the torsor of \(q\) along \(H_{m,t}\to K_m\). The finite torsor naturality of \(\eta\) therefore identifies the pullback of the left side with the \(J_n\) structural image pulled along \(f\), and the pullback of the right side with the \(K_m\) structural image pulled along \(q\).

The structural images of corresponding \(c_P\) and \(c_J\) in \(\mathcal M_n\) agree under (76) inside the generalized-one summand itself. Pull that equality along \(f\). Its positive-bundle torsor is \(b\) by (80), and the preceding torsor calculation identifies its split-frame side. Over the base \(\mathcal B\) of this diagram the resulting equality is \[b^*\bigl(\eta_{\mathcal B,P_n,E}(c_P)\bigr) =q^*\bigl(\eta_{\mathcal B,K_m,E} (\operatorname{res}^{K_n}_{K_m}c_K)\bigr),\] which is (77) under the classifying-pullback convention. The same equality persists under (79). For ordinary cohomology it is an equality of the classes that act by cup product on the supported complex on \(Z\). Once this action is identified with the \(K_m\)-action, the \(K_m\)-equivariant map to the ambient affine support calculation preserves it. There is no need to extend the \(BP_n\)-classifying map away from the independent locus. ◻

Proof. First fix a coefficient exponent \(p^l\) and a bounded Postnikov interval. The \(p\)-adic continuity theorem applies to \(\mathbb Z_p\), since it is local, \(p\)-torsion-free, and henselian along \((p)\). It identifies \[K(\mathbb Z_p)/p^l\longrightarrow \{K(\mathbb Z/p^r)/p^l\}_r\] on the pro systems of homotopy groups in every fixed degree; see (Geisser and Hesselholt 2006, Theorem C). The mod-\(p\) formulation also appears in (Clausen et al. 2021, sec. 5.2, Theorem 5.10); the assertion for \(p^l\) follows from it by the finite coefficient exact sequences. The relevant homotopy groups of the finite rings are finite. In positive degrees this is the usual finiteness of the \(K\)-groups of a finite ring (Weibel 2013, IV, Proposition 1.16); it can also be obtained from stability for its finite linear groups and rational acyclicity. Degree zero becomes finite after reduction modulo \(p^l\).

Consequently the displayed map is a pro-Postnikov equivalence on the chosen interval. Explicitly, the inverse systems of finite groups are Mittag–Leffler. An isomorphism from a finite constant system to their limit has pro-zero kernel and cokernel: after passing to stable images, the system is an inverse system of surjections, and the finite limit detects its stable image at every fixed stage. Induction through the finitely many Postnikov fibers gives the corresponding statement for spectra on the interval. Alternatively, the pro homotopy-group form of the cited continuity theorem gives this directly.

For these bounded targets the natural maps out of the constant system can therefore be computed after passage far enough down the finite-ring tower. The representative \(\kappa_{a,r}\) then factors through the finite congruence quotient of \(G_a\) and is a continuous cochain representative. The pro equivalence retains the homotopies between these representatives, so the construction is compatible as both the Postnikov interval and \(l\) increase. Taking these complete limits defines \(\kappa_a\) and its composites with \(f\). The finite-stage construction commutes with block sum, and therefore so do the limits. Finally, every comparison was induced by reduction of the usual standard representation, which proves the assertion about discrete evaluation.

Only bounded Postnikov intervals are needed to form any fixed ordinary cohomology component. Thus this construction of the ordinary components is available at arbitrary ranks, even when we will use a finite cohomological dimension bound only in the range \(a\leq p-2\). ◻

Proof. Use the connective-cover comparison between local algebraic \(K\)-theory and topological cyclic homology (Hesselholt and Madsen 1997, Theorem D and Addendum 5.2), together with the odd-prime decomposition of \(\operatorname{TC}(\mathbb Z)^\wedge_p\); see (Blumberg and Mandell 2023, sec. 2 and 7). The summands indexed by twists \(i=2,\ldots,p-2\) are the connective suspensions \(\Sigma^{2i-1}\ell_p\). Projection to one of them gives \(f_i\). Only these nonexceptional twist residues are used. In particular the argument uses the local calculation and imposes no regular-prime condition on global algebraic \(K\)-theory.

Put \(d=2i-1\) and \(X=K(\mathbb Z_p)^\wedge_p\). The same decomposition gives \[\pi_dX=\mathbb Z_p,\qquad \pi_{d-1}X=0,\qquad g_i:=(f_i)_*:\pi_dX\xrightarrow{\sim}\pi_0\ell_p=\mathbb Z_p.\] To check the adjacent degree, write \(L\) for the periodic \(p\)-complete Adams summand and \(J=L_{K(1)}S\). The decomposition has summands \(j\), \(\Sigma j'\), and \(z_0,\ldots,z_{p-2}\). Here \(j\) and \(j'\) are the connective covers of \(J\) and \(J'\), with \(J'\) noncanonically equivalent to \(J\); \(z_k\) is the \(1\)-connected cover of \(\Sigma^{2k-1}L\) for \(k\ne0\), and \(z_0\) is the \((-2)\)-connected cover of \(\Sigma^{-1}L\). Passing to \(X\) takes the connective cover of this wedge. Now \(\pi_*L\) is concentrated in degrees divisible by \(2(p-1)\), while \(\pi_*J\) is concentrated in degree zero and in degrees congruent to \(-1\) modulo \(2(p-1)\). Thus, in the range \(3\leq d\leq2p-5\), only the bottom group of the summand \(\Sigma^d\ell_p\) contributes in degree \(d\), and no summand contributes in degree \(d-1\). These descriptions are in (Blumberg and Mandell 2023, sec. 2).

Write \(K_j(A;\mathbb Z/p^r)=\pi_j(K(A)/p^r)\). For a connective spectrum, the completion map is an equivalence modulo \(p^r\): it is an equivalence modulo \(p\), and the Moore spectrum for \(p^r\) is built by finitely many extensions from the Moore spectrum for \(p\). The coefficient exact sequence and \(\pi_{d-1}X=0\) therefore identify \[(\pi_dX)/p^r\xrightarrow{\sim}K_d(\mathbb Z_p;\mathbb Z/p^r).\] The localization sequence for \(\mathbb Z_p\subset\mathbb Q_p\) has residue term \(K(\mathbb F_p)\). By (Hesselholt and Madsen 1997, Theorem B and the following paragraph), \(K(\mathbb F_p)^\wedge_p\simeq H\mathbb Z_p\), so \(K(\mathbb F_p)/p^r\simeq H(\mathbb Z/p^r)\). Since \(d\geq3\), localization and (Hesselholt and Madsen 2003, Theorem A, the descent spectral sequence (6.1.9), and Theorem 6.1.10) consequently give compatible isomorphisms \[(\pi_dX)/p^r \xrightarrow{\sim}K_d(\mathbb Q_p;\mathbb Z/p^r) \xrightarrow{\sim}H^1(\mathbb Q_p,\mu_{p^r}^{\otimes i}).\] The last theorem applies to the complete characteristic-zero discrete valuation field \(\mathbb Q_p\), whose residue field is perfect of characteristic \(p>2\). Its odd-degree isomorphism is the unique \(H^1\) edge of the canonical etale comparison and is natural in the coefficient maps. Continuous integral cochains are the inverse limit of the finite-coefficient cochains. Their transitions are surjective by choosing set sections of the finite coefficient maps, and the finite groups in degree zero have vanishing \(\varprojlim^1\). Taking the inverse limit gives an isomorphism \[\operatorname{edge}_i:\pi_dX\xrightarrow{\sim} H^1(\mathbb Q_p,\mathbb Z_p(i)).\] Consequently \(g_i\circ\operatorname{edge}_i^{-1}\) is an integral isomorphism from this etale group to \(\mathbb Z_p\). The universal Chern regulator used below is compared with these coordinates in Lemma 61. ◻

Proof. The standard representation is additive for block sum and each \(f_i\) is a map of spectra. Thus every rational ordinary component of \(z_{a,i}\) is primitive under block sum. To interpret this statement stably, work first in a fixed bounded Postnikov interval of the coefficient spectrum. Lemma 57 constructs its components at all matrix ranks, compatibly. Clear the finitely many denominators on this interval and then rationalize the resulting continuous integral cohomology classes. This gives the usual stable primitive condition without any exchange of an unbounded Postnikov limit with rationalization.

The primitive space in stable rational matrix Lie cohomology is one-dimensional in each degree \(2j-1\), generated by \(b_j\), and its restriction to rank \(a\) vanishes for \(j>a\). The possible ordinary components of (83) have degrees \[2i-1+2r(p-1)=2\bigl(i+r(p-1)\bigr)-1, \qquad r\geq0.\] If \(r>0\), the corresponding primitive index exceeds \(a\), since \(a\leq p-2\). These components therefore vanish. The component for \(r=0\) is a multiple of \(b_i\), as asserted. ◻

Proof. Put \(q_0=2p-2\), and choose \(q\in\mathbb Z_p^\times\) whose Adams operation generates the pro-Adams action and whose finite residue is primitive. The operation \(\psi^q-1\) on \(\ell_p\) is zero on \(\pi_0\), so it factors through the positive connective cover, identified with \(\Sigma^{q_0}\ell_p\). Denote the resulting map by \[\theta:\ell_p\longrightarrow\Sigma^{q_0}\ell_p, \qquad j_p=\mathop{\mathrm{fib}}(\theta).\] This is the connective image-of-\(J\) fiber, and the sphere unit lifts to a map \(S\longrightarrow j_p\).

For any fixed suspension of \(\ell_p\), spectral sequence (82) has finitely many contributing groups on each total degree, since \(s\leq a^2\). All of them are finite free over \(\mathbb Z_p\) by Theorem 51. The rational spectral sequence has zero differentials, because the rational coefficient spectrum is a direct sum of ordinary Eilenberg–Mac Lane spectra in this finite range. Induction on the pages shows that the integral differentials also vanish: a map between the free groups on a page which vanishes rationally is zero. The resulting finite extensions of free \(\mathbb Z_p\)-modules are free. In particular the abutment in each fixed total degree is torsion-free and injects into its rationalization.

By Lemma 59, the rational class \(z_{a,i}\) has only its degree-zero coefficient component. The map \(\theta\) kills this component. Thus \(\theta z_{a,i}\) is rationally zero, and the torsion-freeness just proved makes it zero integrally. A choice of its nullhomotopy lifts \(z_{a,i}\) to cochains with coefficients in \(j_p\).

The classical odd-primary image-of-\(J\) range identifies \(S\to j_p\) on homotopy below the first beta stem \[b_p=2p(p-1)-2.\] We use only degrees at most \(a^2\), and \[ a^2\leq(p-2)^2<2p(p-1)-2. \tag{84}\] The finite continuous cohomological dimension bound shows that coefficients above degree \(a^2\) cannot affect a negative-degree cochain class. Indeed a coefficient in degree \(t>a^2\) contributes only total degrees \(t-s>0\). Therefore \(S\to j_p\) induces the needed isomorphism in degree \(1-2i\) on continuous cochains, and the lifted \(j_p\) class has a sphere representative. We use here the usual first image-of-\(J\) range, as recorded in the classical calculation of the odd-primary stable stems; see (Ravenel 2004, Theorems 1.1.14 and 1.5.19(a)).

For \(i=1\), normalize the logarithm by \[\lambda:\mathbb Z_p^\times\longrightarrow\mathbb Z_p, \qquad \lambda(u)=\frac{\log u}{\log(1+p)}.\] It kills the Teichmuller factor and sends \(1+p\) to \(1\). The group \(\mathbb Z_p^\times\) has \(p\)-cohomological dimension one. In total degree \(-1\), its connective sphere-cochain spectral sequence therefore has just the entry \(H^1(\mathbb Z_p^\times,\mathbb Z_p)\). Thus \(\lambda\) lifts to a sphere cochain, whose pullback along \(\det:G_a\to\mathbb Z_p^\times\) defines \(c_{a,1}\) and \(\alpha_{a,1}\). ◻

Proof. Write \(M_i=H^1(\mathbb Q_p,\mathbb Z_p(i))\). First match the coefficient conventions. For a connective \(K\)-theory spectrum \(Y\), the mod-\(q\) Moore space \(P^d(q)\) used in (Weibel 1993, sec. 1, Example 1.4) has suspension spectrum \(\Sigma^{d-1}S/q\). Adjunction and the fiber/cofiber triangle give \[[P^d(q),\Omega^\infty_0Y] =\pi_{d-1}F(S/q,Y) =\pi_{d-1}\Sigma^{-1}(Y/q) =\pi_d(Y/q).\] Here \(d\geq3\), so the connected Moore space maps to the identity component. With the common cofiber orientation these identifications commute with ordinary reduction and coefficient maps.

Compose the finite-coefficient identifications in Lemma 58 with the universal etale Chern maps \[c_{i,r}^{\mathrm S}:(\pi_dX)/p^r \xrightarrow{\sim}K_d(\mathbb Q_p;\mathbb Z/p^r) \xrightarrow{c_{i,1}}H^1(\mathbb Q_p,\mu_{p^r}^{\otimes i}).\] The last map is defined in (Weibel 1993, sec. 2, equation (2.1)) by the standard representation’s universal Chern class, its Kunneth component, and the mod-\(p^r\) Hurewicz map. Its coefficient prime is invertible in \(\mathbb Q_p\). Proposition 2.4 there gives additivity for odd coefficients, and Proposition 2.8 gives compatibility with coefficient reductions for \(d\geq2\). Their inverse limit is therefore a continuous additive, hence \(\mathbb Z_p\)-linear, map \(c_i^{\mathrm S}:\pi_dX\to M_i\). The compatible etale edges identify the projection to coefficients \(\mathbb Z/p\) with the quotient \(M_i\to M_i/pM_i\), and thus \(M_i/pM_i\cong H^1(\mathbb Q_p,\mu_p^{\otimes i})\).

We prove that the mod-\(p\) Chern map over \(\mathbb Q_p\) is onto. Put \(L=\mathbb Q_p(\zeta_p)\). The polynomial \(\Phi_p(1+T)\) is Eisenstein, so \([L:\mathbb Q_p]=p-1\). By (Weibel 1993, Proposition 5.2), the cokernel of \[c_{i,1}:K_{2i-1}(L;\mathbb Z/p)\longrightarrow H^1(L,\mu_p^{\otimes i})\] is annihilated by \((i-1)!\). Its hypotheses are \(i\geq2\) and \(q\not\equiv2\pmod4\) for a field containing \(1/q\) and a primitive \(q\)th root. They hold for \(q=p\) and \(L\) above. Since \((i-1)!\) is a unit modulo \(p\), this map is onto.

For the finite separable extension \(\mathbb Q_p\subset L\), (Weibel 1993, Proposition 4.4) gives \[i\bigl(c_{i,1}(N_{L/\mathbb Q_p}y) -\operatorname{cor}_{L/\mathbb Q_p}c_{i,1}(y)\bigr)=0 \qquad (y\in K_d(L;\mathbb Z/p)).\] Because \(i\) is a unit modulo \(p\), the two maps agree. On \(H^1(\mathbb Q_p,\mu_p^{\otimes i})\), restriction followed by corestriction is multiplication by \(p-1\). Given \(h\) in this group, choose \(y\) with \(c_{i,1}(y)=(p-1)^{-1}\operatorname{res}_{L/\mathbb Q_p}h\). Then \(c_{i,1}(N_{L/\mathbb Q_p}y)=h\), proving surjectivity over \(\mathbb Q_p\). The compatible edges identify both the source and target of this mod-\(p\) map with \(\mathbb F_p\). It is the reduction of \(c_i^{\mathrm S}\), so that reduction is an isomorphism. Both integral modules are free of rank one; consequently \(c_i^{\mathrm S}\) is an isomorphism.

It remains to identify this integral map on discrete classes. All the coefficient identifications above are induced by completion and localization, so the reduction of \(\widehat x\) corresponds to the ordinary reduction of the localized class \(x\). By (Weibel 1993, Lemma 2.3), the finite Chern map on this reduction is evaluation of the same universal etale Chern class on the integral Hurewicz image of \(x\). The proof of Proposition 2.8 there makes these evaluation maps compatible already in the coefficient triangles. For a fixed \(x\), its Hurewicz image is represented by a finite bar cycle at a finite matrix rank. Evaluating the compatible integral universal class on that cycle and reducing therefore gives \(c_{i,r}^{\mathrm S}(\widehat x\bmod p^r)\) at every \(r\). The isomorphism \(M_i=\varprojlim_r H^1(\mathbb Q_p,\mu_{p^r}^{\otimes i})\) makes the integral evaluation uniquely equal to \(c_i^{\mathrm S}(\widehat x)\).

The regulator in (Huber and Kings 2011, sec. 1.3) evaluates the universal standard-representation etale Chern class with \(\mathbb Q_p(i)\) coefficients on the same Hurewicz image. It is the coefficient image of the usual integral class just used: these Chern classes are normalized on line bundles and characterized by the splitting principle. Hence, with the common suspension convention, \[r_p(x)=c_i^{\mathrm S}(\widehat x)\otimes1 \quad\text{in }H^1(\mathbb Q_p,\mathbb Q_p(i)) \qquad (x\in K_d(\mathbb Z_p)).\] Here the target is \(M_i[1/p]\). Indeed a continuous map from a compact power of \(G_{\mathbb Q_p}\) to \(\mathbb Q_p(i)\) has bounded image, so it lies in \(p^{-N}\mathbb Z_p(i)\) for some \(N\). Thus rational continuous cochains are the localization of integral continuous cochains, and localization is exact. This comparison evaluates a fixed finite bar cycle and uses the inverse limit in the target; it makes no exchange of an unbounded group-cohomology limit with rationalization. ◻

Proof. For \(i\geq2\), the rational group \(H^1(\mathbb Q_p,\mathbb Q_p(i))\) is crystalline and is identified with \(\mathbb Q_p\) by the Bloch–Kato exponential. The integral group \(H^1(\mathbb Q_p,\mathbb Z_p(i))\) is free of rank one. Indeed local Galois cohomology is finitely generated, its rational rank is one, and its torsion vanishes because \(H^0(\mathbb Q_p,(\mathbb Q_p/\mathbb Z_p)(i))=0\): the mod-\(p\) cyclotomic character to the \(i\)th power is nontrivial in the stated range.

We shall construct one primitive integral extension using Fontaine–Laffaille theory over the absolutely unramified field \(\mathbb Q_p\). We use covariant filtered Frobenius conventions with weights \(-i\) and \(0\). Taking the filtered dual puts these weights in \([0,i]\subset[0,p-2]\), where the original contravariant Fontaine–Laffaille functor applies. Its exact lattice construction is (Fontaine and Laffaille 1982, sec. 7.14 and Proposition 7.15(i)); its full faithfulness on the reductions modulo \(p\) is (Fontaine and Laffaille 1982, Theorem 6.1). The latter applies since the interval avoids filtration degree \(p-1\). The rational comparison in (Fontaine and Laffaille 1982, Theorem 8.4(ii)) identifies the functor on the filtered dual with the original covariant crystalline representation. Thus strongly divisible extensions in our conventions give integral crystalline extensions of \(\mathbb Z_p\) by \(\mathbb Z_p(i)\), compatibly with reduction and rationalization.

The rank-one endpoint has the standard period scale. Let \(C=\widehat{\overline{\mathbb Q_p}}\), \(R=\mathcal O_{C^\flat}\), and \(\theta:W(R)\to\mathcal O_C\). We also write \(\theta\) for the residue map \(B_{\mathrm{dR}}^+\to C\). Choose \(\xi=[x_0]+p\) with \(x_0^\sharp=-p\), so \(\ker\theta=(\xi)\). At \(q=p\), the period ring and its filtered pieces are \[\mathscr S=W(R)[\xi^p/p],\qquad \mathscr S^j=(\xi^j,\xi^p/p)\quad(0<j<p)\] by (Fontaine and Laffaille 1982, sec. 2.5 and Lemma 5.4); write \(\widehat{\mathscr S}^{\,j}\) for their separated \(p\)-adic completions as in Section 7.14 there. Choose a generator \(e_{\mathrm{cyc}}\) of \(\mathbb Z_p(1)\), represented by a primitive compatible root system \(\epsilon=(1,\zeta_p,\zeta_{p^2},\ldots)\), and put \(\mu=[\epsilon]-1=\xi a\) with \(a\in W(R)\). The logarithm \(t_{\mathrm{cyc}}=\log[\epsilon]\) belongs to \(\widehat{\mathscr S}^{\,1}\). Indeed, if \(k=pm+r\) with \(0\leq r<p\), \[\frac{\mu^k}{k} =\frac{p^m}{k}\,\xi^r a^k\left(\frac{\xi^p}{p}\right)^m.\] Here \(m-v_p(k)\geq0\) and tends to infinity with \(k\). Each term lies in \(\mathscr S^1\) (when \(r=0\), \(m\geq1\)), and the logarithm series converges in its \(p\)-adic completion. The displayed ideals give \((\mathscr S^1)^i\subseteq\mathscr S^i\), while \(\phi(t_{\mathrm{cyc}})=p t_{\mathrm{cyc}}\) (Fontaine 1982, sec. 4.11). Here the completed divided Frobenius is \(\phi_i=p^{-i}\phi:\widehat{\mathscr S}^{\,i}\to\widehat{\mathscr S}\). Thus \(t_{\mathrm{cyc}}^i\in\widehat{\mathscr S}^{\,i}\) and \(\phi_i(t_{\mathrm{cyc}}^i)=t_{\mathrm{cyc}}^i\).

The same ideal formula shows that \(x\mapsto\theta(x/\xi^i)\) sends \(\mathscr S^i\) to \(\mathcal O_C\): the generator \(\xi^p/p\) contributes zero because \(i<p\). This map extends \(p\)-adically to \(\widehat{\mathscr S}^{\,i}\) and, under the filtered period embedding in the proof of (Fontaine and Laffaille 1982, Theorem 8.4(ii)), is the same residue map. For the primitive period just chosen, the proof of (Fontaine 1982, Proposition 2.17, pp. 543–544) computes \(v_p\theta(t_{\mathrm{cyc}}/\xi)=1/(p-1)\). Consequently \[v_p\theta\left(\frac{t_{\mathrm{cyc}}^i/p}{\xi^i}\right) =\frac{i}{p-1}-1<0,\] so \(t_{\mathrm{cyc}}^i/p\notin\widehat{\mathscr S}^{\,i}\). For the positive rank-one object of weight \(i\) with \(\phi_i(f_i)=f_i\), its realization is \(L_i=\{s\in\widehat{\mathscr S}^{\,i}:\phi_i(s)=s\}\). Rank preservation and rational comparison make \(L_i\) a rank-one lattice in \(\mathbb Q_pt_{\mathrm{cyc}}^i\). It contains \(t_{\mathrm{cyc}}^i\) but not \(t_{\mathrm{cyc}}^i/p\), hence \(L_i=\mathbb Z_pt_{\mathrm{cyc}}^i\). The unit \(1\) is primitive at weight zero since \(\theta(\widehat{\mathscr S})\subseteq\mathcal O_C\). The corresponding covariant basis is therefore the standard period basis \(e_i=t_{\mathrm{cyc}}^{-i}\otimes e_{\mathrm{cyc}}^{\otimes i}\).

Here is the lattice calculation in those conventions. Let \(e_i\) be the standard basis of the rank-one filtered Frobenius object for \(\mathbb Q_p(i)\), with \(\phi(e_i)=p^{-i}e_i\), and choose an integral lift \(e_0\) of the quotient basis. Write \[\operatorname{Fil}^0=\mathbb Z_p(e_0+x e_i), \qquad \phi(e_0)=e_0+y e_i.\] Strong divisibility is exactly \[ x\in\mathbb Z_p,\qquad z:=y+p^{-i}x\in\mathbb Z_p. \tag{85}\] Indeed the second condition says that \(\phi(\operatorname{Fil}^0)\) is integral. The condition in filtration \(-i\) follows because \(p^iy=p^iz-x\) is integral, and generation holds because \(p^i\phi(e_i)=e_i\) while the degree-zero filtered generator maps to a lift of \(e_0\). These are all the conditions for the two-step filtered object. Subtracting the rational Frobenius splitting gives the exponential parameter \[ x-\frac{y}{1-p^{-i}} =\frac{x-z}{1-p^{-i}}. \tag{86}\] As \(x,z\) range independently over \(\mathbb Z_p\), this ranges exactly over \(p^i\mathbb Z_p\), since \(p^i-1\) is a unit. Replacing \(e_0\) by \(e_0+b e_i\), \(b\in\mathbb Z_p\), replaces both \(x\) and \(z\) by their differences with \(b\), so (86) is independent of this change of integral lift.

To see that the resulting lattice is the whole integral Galois cohomology lattice, take \(x=1\) and \(z=0\). Its extension modulo \(p\) does not split in the filtered category. This can be checked directly in the standard positive interval: if \(f_i,f_0\) are dual to \(e_i,e_0\), the dual lattice has \[\phi(f_0)=f_0,\qquad \phi(f_i)=p^if_i+f_0,\qquad \operatorname{Fil}^{j}=\mathbb Z_p(f_i-f_0)\quad(1\leq j\leq i).\] Its divided Frobenius satisfies \(\phi_i(f_i-f_0)=f_i\). A filtered section of the weight-\(i\) quotient modulo \(p\) must lift its basis to \(f_i-f_0\); commuting with \(\phi_i\) would require its image to be \(f_i-f_0\) instead of \(f_i\). Thus no such section exists. In the original coordinates this is the invariant \(x-z=1\) under all changes of lift, even after reduction. Full faithfulness on finite objects implies that the corresponding Galois extension modulo \(p\) does not split either: a Galois splitting would lift to a filtered splitting. Its class in \(H^1(\mathbb Q_p,\mathbb Z_p(i))\) is consequently not divisible by \(p\), and hence generates that rank-one free module. Its exponential parameter is \(p^i/(p^i-1)\) by (86). This proves \(\exp_{\mathrm{BK}}^{-1}H^1(\mathbb Q_p,\mathbb Z_p(i))=p^i\mathbb Z_p\) using the exact construction, finite full faithfulness, and rational comparison just cited. ◻

Proof. Choose a nonidentity Teichmuller root of unity \(\xi\in\mathbb Z_p^\times\) and put \(F=\mathbb Q(\xi)\). The symbol \([\xi]_i\) lies in degree one of de Jeu’s rational weight-\(i\) polylogarithmic complex over \(F\) (Jeu 1995), in the presentation of (Besser and Jeu 2003, 868). Its differential is \([\xi]_{i-1}\otimes\xi\) for \(i>2\) and \((1-\xi)\wedge\xi\) for \(i=2\); both vanish because \(\xi\) is torsion in \(F^\times\), hence zero in \(F^\times\otimes_\mathbb Z\mathbb Q\). The first map in (Besser and Jeu 2003, Theorem 1.12) sends the resulting degree-one cohomology class to \(K_{2i-1}(F)\otimes_\mathbb Z\mathbb Q\). For the chosen embedding \(F\to\mathbb Q_p\), let \(\mathcal O\) be the localization of the integers of \(F\) at the corresponding prime above \(p\). Both \(\xi\) and \(1-\xi\) are units there. The same theorem sends the rational \(K\)-class through the localization isomorphism and the map to local algebraic \(K\)-theory before applying the syntomic regulator. This number-field case requires no conjectural weight assumption; see (Besser and Jeu 2003, 871, the paragraph preceding Theorem 1.10, and Remark 1.13). The regulator here is the syntomic Chern character in the normalized coordinate of (Besser and Jeu 2003, Definition 4.6 and Appendix A). Its value is \(\pm(i-1)!\operatorname{Li}_i(\xi)\), since the logarithmic correction terms vanish at \(\xi\).

There is some such \(\xi\) for which this value is nonzero. To see this directly, use the Coleman measure identity \[ (1-p^{-i})\operatorname{Li}_i(\xi) =\lim_{r\to\infty} \frac{\displaystyle \sum_{\substack{1\leq b<p^r\\p\nmid b}}\xi^b b^{-i}} {1-\xi^{p^r}}, \qquad \xi^p=\xi\neq1; \tag{87}\] see (Besser and Jeu 2008, Proposition 4.8 and Corollary 4.9), where the proof of Proposition 4.8 attributes the measure formula to (Coleman 1982, Lemma 7.2). This is integration of \(x^{-i}\) on \(\mathbb Z_p^\times\) against the bounded measure whose mass on \(b+p^r\mathbb Z_p\) is \(\xi^b/(1-\xi^{p^r})\). The masses are compatible under refinement by the geometric-series identity, so their integral modulo \(p\) is computed at level one. This finite-polylogarithm reduction is the one developed in (Besser 2002, Proposition 2.1 and Corollary 2.2). Its numerator is the finite polylogarithm \[P_i(X)=\sum_{b=1}^{p-1} b^{-i}X^b\in\mathbb F_p[X].\] This polynomial is nonzero, has degree \(p-1\), and vanishes at both \(0\) and \(1\); the latter uses \(1\leq i\leq p-2\). It cannot vanish at every element of \(\mathbb F_p\), since that would give at least \(p\) distinct roots. Some \(\overline\xi\in\mathbb F_p^\times\setminus\{1\}\) therefore has \(P_i(\overline\xi)\neq0\), and its Teichmuller lift makes the right side of (87) nonzero. The denominator is a unit. Since \(1-p^{-i}\ne0\), this proves \(\operatorname{Li}_i(\xi)\ne0\), and hence the normalized Chern-character regulator above is nonzero.

On Hurewicz images from positive algebraic \(K\)-groups, evaluation of the usual Chern-character component of degree \(i\) is \((-1)^{i-1}/(i-1)!\) times evaluation of \(c_i\); see (Tamme 2014, the remark after Definition 4.23 and Section 5.9). The comparison from the syntomic theory used by Huber–Kings to the modified theory above preserves Chern classes, by (Besser and Jeu 2003, sec. 4) and (Huber and Kings 2011, Propositions 2.2.7 and 2.2.9). Because the normalized coordinate is an isomorphism, its nonzero character value implies a nonzero syntomic Chern value in the Huber–Kings theory. Their isomorphism \(\eta:H^1_{\mathrm{syn}}(\mathbb Z_p,i)\xrightarrow{\sim}\mathbb Q_p\) is defined in (Huber and Kings 2011, Definition 2.2.5). By Proposition 2.3.4 there, evaluation of this Chern class followed by \(\eta\) and the Bloch–Kato exponential is the etale regulator \(r_p\). The exponential is an isomorphism for \(i>1\) by Section 1.3 there. Thus the preceding construction supplies a discrete rational algebraic \(K\)-theory class \(x\) with \(r_p(x)\ne0\). If it is initially regarded over \(\mathbb Q_p\), it has a preimage in \(K_{2i-1}(\mathbb Z_p)\otimes\mathbb Q_p\) by (Huber and Kings 2011, Remark 1.3.1), since \(i>1\); the regulator extends \(\mathbb Q_p\)-linearly. ◻

Proof. For \(i\geq2\), put \(g_i=(f_i)_*\) and define \[t_i=\exp_{\mathrm{BK}}^{-1} \bigl(c_i^{\mathrm S}(g_i^{-1}(1))\bigr).\] Lemma 62 and Lemma 61 give \(t_i\in p^i\mathbb Z_p^\times\). Write \(\alpha_i\) for the stable compatible family of leading classes, and let \(\alpha_i^\delta\) and \(b_i^\delta\) denote the stable discrete restrictions of the compatible continuous classes. The bottom Postnikov composite \[K(\mathbb Z_p)^\wedge_p\xrightarrow{f_i}\Sigma^{2i-1}\ell_p \longrightarrow\Sigma^{2i-1}H\mathbb Z_p\] induces \(g_i\) on \(\pi_{2i-1}\). Its fundamental ordinary class is the leading class \(\alpha_i\). Lemma 57 and naturality of Hurewicz evaluation give \[\langle\alpha_i^\delta,\operatorname{hur}(x)\rangle =g_i(\widehat x) \qquad (x\in K_{2i-1}(\mathbb Z_p)),\] and the same identity extends \(\mathbb Q_p\)-linearly to \(K_{2i-1}(\mathbb Z_p)\otimes_{\mathbb Z}\mathbb Q_p\). The fundamental class of the Eilenberg–Mac Lane target evaluates as the identity on its homotopy group, so no additional normalization factor occurs. By (Huber and Kings 2011, Definition 1.2.3 and Theorem 1.3.2) and Lemma 61, \[\langle b_i^\delta,\operatorname{hur}(x)\rangle =\exp_{\mathrm{BK}}^{-1}r_p(x) =t_i g_i(\widehat x) =t_i\langle\alpha_i^\delta,\operatorname{hur}(x)\rangle.\] Choose \(x\) as in Lemma 63. By Lemma 59, write the stable continuous primitive as \(\alpha_i\otimes\mathbb Q_p=\lambda_i b_i\). The evaluation identities above are nonzero on \(x\) and give \(\lambda_i=t_i^{-1}=\varepsilon_i p^{-i}\) for a unit \(\varepsilon_i\in\mathbb Z_p^\times\). Both \(g_i\) and \(c_i^{\mathrm S}\) were fixed independently of matrix rank, so this is the same unit at every rank. This uses one detected discrete evaluation and requires no density assertion about discrete local algebraic \(K\)-theory.

Finally, \(b_1\) corresponds under the Lazard comparison to the ordinary determinant logarithm. Since \(\log(1+p)\) has valuation one, the normalization in Lemma 60 gives \(\alpha_{a,1}=\varepsilon_1p^{-1}b_1\) with \(\varepsilon_1\in\mathbb Z_p^\times\). ◻

Proof. For the first assertion, we use the Chern-class comparison and volume calculation of Huber–Soergel (Huber and Soergel 2010, Propositions 1.3, 3.3, and 4.1). The compatibility with suspension in their Corollary 3.4 identifies precisely the primitives \(b_i\) fixed above, rather than arbitrary rational generators. On the compact real form \(U(a)\) of \(\mathop{\mathrm{GL}}_a(\mathbb C)\), these transgressed Chern classes, divided by their Tate factors \((2\pi\sqrt{-1})^i\), are integral odd generators, and their product integrates to \(1\) up to orientation. Successive last-column projections \(U(j)\to S^{2j-1}\) compute the entry-volume integral as \[\int_{U(a)}\omega_a =\pm\frac{(2\pi\sqrt{-1})^{a(a+1)/2}} {\prod_{j=1}^a(j-1)!}.\] One obtains this formula by using the sphere volume \(2\pi^j/(j-1)!\) at the \(j\)th step; each complex off-diagonal tangent entry contributes its \(2\sqrt{-1}\) factor and the imaginary diagonal entry its \(\sqrt{-1}\) factor. Comparing with the product of the transgressions gives exactly \(u_a=\pm\prod_{j=1}^a(j-1)!\) in the chosen normalization. Since \(a\leq p-2\), this coefficient is a \(p\)-unit. In particular this calculation does not introduce the possibly nonunit factor \((a^2)!\) into the primitive product.

We give a direct integral verification of the second assertion that also handles that latter factorial. Put \(e=v_p(d!)\), and take \(r\) sufficiently large; for example \(r\geq d+1\) is sufficient for the estimates below. In the affine coordinates \(g=1+p^rY\) on \(U_r\), the normalized form is \[p^{-rd}\omega_a =\det(1+p^rY)^{-a}\bigwedge_{r',s'}dY_{r's'}.\] Integrate this form formally, term by term, over affine straight simplices with vertices in \(U_r\). The integral of a monomial of total degree \(h\) on a \(d\)-simplex has denominator dividing \((h+d)!\). The expansion \[\det(1+p^rY)^{-a} =1+\sum_{h\geq1}p^{rh}Q_h(Y)\] has integral homogeneous polynomials \(Q_h\). Hence the resulting series of simplex integrals converges \(p\)-adically. Left multiplication on matrices is affine and preserves \(\omega_a\). Formal Stokes and the faces of a simplex therefore give a homogeneous group cocycle. In inhomogeneous notation call it \(C_r\), evaluated on the vertices \[1,\ g_1,\ g_1g_2,\ \ldots,\ g_1\cdots g_d.\] It is a continuous rational analytic cocycle representing the normalized form under differentiation. Its leading term is \[\frac{1}{d!}\det(X_1,\ldots,X_d), \qquad X_j=(g_j-1)/p^r.\] Differentiation alternates in the \(d\) variables and cancels precisely this \(d!\), so the corresponding Lie cohomology class is \(p^{-rd}\omega_a\) with the stated normalization. This description of the rational Lazard map agrees with the analytic differentiation comparison in (Huber and Kings 2011, sec. 4); an all-degree integral comparison for uniform groups is also supplied by (Huber et al. 2011, Theorem 3.3.3, Proposition 4.2.4, and Remark 4.2.5).

The following reduction proves the integral assertion directly. Multiply \(C_r\) by \(d!\) and work modulo \(p^{e+1}\). Every term with \(h\geq1\) has valuation at least \[e+rh-v_p((h+d)!)\geq e+1\] for the chosen \(r\). For instance \(v_p((h+d)!)\leq(h+d)/(p-1)\) proves this inequality when \(r\geq d+1\). Furthermore successive-product vertices differ from their additive partial sums by multiples of \(p^r\). Since \(r\geq e+1\), these differences do not affect the reduction. Writing \(A_e=\mathbb Z/p^{e+1}\), the entry functions \[\chi_j:U_r\longrightarrow A_e, \qquad \chi_j(g)=((g-1)/p^r)_j\pmod {p^{e+1}}, \quad 1\leq j\leq d,\] are genuine group homomorphisms. Indeed the additional product term is divisible by \(p^r\). We consequently obtain \[ [d!C_r] =d![\chi_1\smile\cdots\smile\chi_d] \quad\hbox{in }H^d_{\mathrm{cts}}(U_r,A_e). \tag{89}\] Here the determinant cocycle is the alternating sum of the ordered cups. Graded commutativity makes each term equal to the ordered cup after its permutation sign is included, giving the factor \(d!\).

The group \(U_r\) is uniform, and the reductions of the \(\chi_j\) are its standard degree-one basis. Their ordered cup is the nonzero top generator in mod-\(p\) cohomology. Orientable compact-group duality gives \[H^d_{\mathrm{cts}}(U_r,\mathbb Z_p)=\mathbb Z_p, \qquad H^{d+1}_{\mathrm{cts}}(U_r,\mathbb Z_p)=0.\] Reduction thus identifies \(H^d(U_r,A_e)\) with \(H^d(U_r,\mathbb Z_p)/p^{e+1}\). The ordered cup in (89) is a unit in this cyclic module. The integral cocycle \(d!C_r\) therefore represents a class of exactly valuation \(e\) in \(H^d(U_r,\mathbb Z_p)\). Dividing its rational class by \(d!\) proves that \([C_r]\) is an integral unit generator. This argument does not divide by \(d!\) inside \(A_e\); the valuation is first determined by reduction and only then divided in rational cohomology. It proves the second assertion even when \(p\mid d!\).

For the last assertion, the determinant of the adjoint action of \(G_a\) is one, so its orientation is trivial. Top restriction from \(G_a\) to \(U_r\) multiplies an integral orientation generator by \([G_a:U_r]\), up to a unit; this follows equally by dualizing degree-zero corestriction or by the restriction–corestriction identity and oriented duality. Its index has valuation \[\begin{align*} v_p[G_a:U_r] &=(r-1)a^2+v_p|\mathop{\mathrm{GL}}_a(\mathbb F_p)|\\ &=(r-1)a^2+\frac{a(a-1)}2 =ra^2-\frac{a(a+1)}2. \end{align*}\] The rational Lie algebra is unchanged under restriction. Combining this index with the lattice \(p^{-ra^2}\omega_a\) gives \(p^{-a(a+1)/2}\omega_a\) for the full group. ◻

Proof of Theorem 56. By Lemmas 64 and 65, the product of the leading integral classes has rational image \[\alpha_{a,1}\cdots\alpha_{a,a} =\left(\prod_{i=1}^a\varepsilon_i\right) p^{-a(a+1)/2}\,b_1\cdots b_a\] and is an integral unit generator in top degree \(a^2\). In particular its mod-\(p\) reduction is nonzero.

All the classes have odd degree, and \(2\) is invertible in \(\mathbb Z_p\), so they define a map from the indicated exterior algebra. If the graded kernel of the mod-\(p\) map contained a nonzero homogeneous element, the perfect top-degree pairing in the source exterior algebra would multiply it to a nonzero top class in the kernel. The nonvanishing top image therefore makes this map injective. Theorem 51 gives the same graded dimensions on both sides, so it is an isomorphism. Integral torsion-freeness, finite generation in each degree, and reduction modulo \(p\) then show that the integral map is also an isomorphism. This proves the generator assertions, with actual leading classes rather than a choice of abstract exterior generators.

The definitions of \(\kappa_a\) and \(f_i\) make their leading classes compatible with every upper-left block restriction. For \(i>b\), the rational primitive restricts to zero at rank \(b\). Integral torsion-freeness at that rank makes the integral restriction zero as well. For \(i\leq b\) it is exactly \(\alpha_{b,i}\), since the projection \(f_i\) and its free-coordinate normalization were fixed independently of rank. To choose compatible sphere representatives, first use Lemma 60 at rank \(N\) and then restrict those chosen representatives to every smaller rank. Their images in \(\ell_p\) and their leading integral classes are the ones already constructed there. No simultaneous canonical choice of nullhomotopies is needed.

Finally apply Proposition 55 to these rank-\(N\) sphere classes. That proposition transports continuous stable lifts through the common modification stack and preserves their mod-\(p\) Hurewicz images. It also identifies their action after choosing the connected quotient frame with the actual upper-left matrix-block restriction. Thus the transported classes \(u_i\) have the leading classes \(\xi_i\) stated in the theorem, and for \(i\leq m\) their action on the extension space is through \(\overline\alpha_{m,i}\). This proves all the asserted compatibilities. ◻

Proof. We give the group-cohomological weight bound used in the argument. Let \(P\subset G\) be the compact stabilizer of a rational flag, written with its successive subspaces first. Its matrix entries below the corresponding block diagonal are zero. Let \(P^+\) consist of its matrices whose reduction is upper unitriangular, and let \[\Delta=\{\operatorname{diag}(a_1,\ldots,a_m): a_i\in\mu_{p-1}\}.\] The subgroup \(P^+\rtimes\Delta\) has index prime to \(p\) in \(P\). Indeed its index is the product of the complete flag counts in the diagonal Levi blocks, each congruent to one modulo \(p\). Restriction to this subgroup is therefore injective on \(k\)-cohomology, by restriction followed by transfer.

The pro-\(p\) group \(P^+\) has an ordered valued basis consisting of elementary upper entries, principal diagonal entries, and the allowed principal lower entries. Give these entries the values \[ \begin{array}{c|c} \text{entry}&\text{value}\\ \hline E_{ij}\ (i<j)&(j-i)/m\\ pE_{ij}\ (i>j)\text{ in one flag block} &1+(j-i)/m\\ pE_{ii}&1 . \end{array} \tag{94}\] Successive \(p\)-powers increase the value by one. When \(m=1\) only the diagonal entry occurs. For \(m>1\) the smallest value is \(1/m>1/(p-1)\). Matrix multiplication respects the resulting filtration: the row-index terms add along a product \(E_{ij}E_{jk}\), and the \(p\)-adic valuations add. The block zero conditions are closed under these products. Logarithm and exponential converge in this filtration because a term of degree \(s\) has valuation at least \(s/m-v_p(s!)\), tending to infinity; the same estimates identify the required \(p\)-roots when their valuation exceeds the saturation threshold. Thus (94) is a saturated \(p\)-valuation with values in \(\frac1m\mathbb Z\).

Apply the valued-group cohomology spectral sequence of (Sorensen 2021, Theorem 1.1). In its trivial-coefficient specialization, the \(E_1\)-page is the cohomology of the finite-dimensional graded Lie algebra obtained from the ordered basis by setting the power-shift parameter to zero. The group-ring construction is functorial for automorphisms preserving the valuation. It is therefore \(\Delta\)-equivariant; this equivariant use is also the construction in (Dotto and Le Hung 2025, sec. 3 and equation (4.2)). The Chevalley cochain complex gives the following sufficient bound: every weight in \(H^*(P^+,k)\) is a sum of distinct allowed cochain-root weights \[ \epsilon_j-\epsilon_i \quad\text{for each allowed tangent entry }E_{ij}; \tag{95}\] diagonal entries contribute zero. This statement only bounds the weights of the abutment; it does not assert degeneration of this spectral sequence. It follows because its Lie cochains are exterior powers of the dual leading-symbol space, and every subsequent page is a subquotient.

The same bound holds with the finite coefficient representation in (91), after adding one of its weights. To see this without assuming a trivial unipotent action, filter that representation by powers of the augmentation ideal of the finite \(p\)-group through which \(P^+\) acts. This is a finite, \(\Delta\)-stable filtration. Its quotients have trivial \(P^+\)-action and decompose into \(\Delta\)-characters, since \(|\Delta|\) is invertible in \(k\). Their character weights occur among the original coefficient weights. Apply the preceding calculation to each quotient and use the coefficient spectral sequence. Tensoring with \(R\) preserves both this filtration and its weight vanishing.

Consider a flag with smallest plane of dimension \(r<m\). Use its finite flag resolution from Proposition 25. At each term, Shapiro reduces to a group \(P\) as above whose first block has size \(r\). For a weight in (91), the sum of the exponents on the last \(m-r\) coordinates is strictly positive. A root internal to either of the two large blocks contributes zero to that sum. The allowed crossing roots are precisely the cochain roots from an upper entry with \(i\leq r<j\); by (95) each contributes \(+1\) to the last-block sum. There are no crossing roots with the opposite sign, since the lower crossing entries are absent from \(P\). The resulting full weight therefore cannot be zero as an integral weight.

This integral test also tests the actual finite torus. Each coordinate of a sum of distinct allowed roots lies between \(-(m-1)\) and \(m-1\). Equation (91) gives \[|\text{each resulting exponent}| \leq t+m-1=n-1<p-1.\] The only multiple of \(p-1\) in this range is zero. Hence a trivial \(\Delta\)-character would require every integral exponent to be zero, contrary to the last-block sum. Exactness of \(\Delta\)-invariants and injectivity of restriction prove vanishing for each of these parabolic terms. Their finite flag resolution proves it for each proper-plane support cohomology group.

For the ambient term put \(r=m\). Its coefficient exponents are \(1-t+e_i\leq0\), with total \[\sum_i(1-t+e_i)=b-m(t-1)=b-d.\] Every root has total zero. If \(b<d\), the same finite-torus bound therefore excludes a trivial weight. If \(b=d\), then every \(e_i=t-1\), and the coefficient is a line with character \[(\det)^{-t}\,(\det)\,(\det)^{t-1}=1.\] This cancellation is an equality of \(G\)-characters, since the three determinant characters are defined on \(G\), not only on \(\Delta\).

These calculations prove (92). The map to ambient support is an isomorphism on the top geometric cohomology group by Proposition 25; its other cohomology groups are the proper-plane groups just killed. The bounded hypercohomology spectral sequence of its cone therefore proves (93). The flag maps, weight filtrations, and torus projections commute with the connected-frame action, which is consequently retained throughout. ◻

Proof. Suppose \(t>1\) and write \(\Lambda=\Lambda_h\). The residue field \(k\) is a perfect \(\Lambda\)-module, of projective dimension \(d\). Finite Koszul duality gives the natural tensor-Hom identification \[\mathop{\mathrm{RHom}}_\Lambda(k,\mathcal D) \simeq \mathop{\mathrm{RHom}}_\Lambda(k,\Lambda)\otimes_\Lambda^{\mathbb L}\mathcal D.\] The first factor on the right has one cohomology group: \[\mathop{\mathrm{Ext}}_\Lambda^i(k,\Lambda)=0\quad(i\ne d), \qquad \mathop{\mathrm{Ext}}_\Lambda^d(k,\Lambda) =\det(\mathfrak m_\Lambda/\mathfrak m_\Lambda^2)^*=E_h^d.\] Truncation therefore gives a canonical identification \(\mathop{\mathrm{RHom}}_\Lambda(k,\Lambda)\simeq E_h^d[-d]\), where \(\Lambda\) acts on the line through its residue field. Consequently \[ \mathop{\mathrm{RHom}}_\Lambda(k,\mathcal D) \simeq (k\otimes_\Lambda^{\mathbb L}\mathcal D)\otimes_k E_h^d[-d]. \tag{97}\] The map \(\Lambda\to k\) gives the natural map \(\mathcal D\to k\otimes_\Lambda^{\mathbb L}\mathcal D\). Tensor it with \(E_h^d[-d]\) and use (97). This is (96), since rational translation cohomology is the displayed derived Hom.

These constructions are natural for every augmented automorphism of \(\Lambda\). In particular, the orientation identification is the canonical top Ext identification, so it is equivariant even when a frame change acts nonlinearly on chosen parameters. One may use coordinates to check the row map: for a module \(M\) annihilated by \(\mathfrak m_\Lambda\), the Koszul differential is zero, and the map sends \(M\otimes E_h^d[-d]\) to the top Ext summand of \(\mathop{\mathrm{RHom}}_\Lambda(k,M)\). This coordinate check defines no choices in the map itself. Tensor-Hom duality and the augmentation are natural for all commuting coefficient maps, including the constant-cochain action.

Filter \(\mathcal D\) by its bounded cohomology filtration. On a row \(H^q(\mathcal D)\), the target has translation cohomology \(H^q(\mathcal D)\otimes E_h^b\) in degrees \(0\leq b\leq d\). The row map is the just-described insertion at \(b=d\). After tensoring with \(R\), Proposition 43 and Lemma 66 kill all the other columns on applying \(R\Gamma(G,-)\). The same lemma identifies the top column on source and target. The spectral sequences are bounded, since \(\mathcal D\) is bounded, \(k\) has \(\Lambda\)-projective dimension \(d\), and \(G\) has \(p\)-cohomological dimension \(m^2\). The comparison is therefore an equivalence after almost scalar extension, hence before it by faithful flatness. All its maps commute with \(P_t\).

For \(t=1\), rational representations of \(T_h\) decompose into characters, and the natural projection onto the trivial character is exact. It defines (96) with \(d=0\). All cohomology rows are already in the trivial character, so the projection is a quasi-isomorphism; the same argument with \(G\) and the support calculation applies. ◻

Proof. Put \[C_G=R\Gamma\bigl(G,R\Gamma(T_h,\mathcal D)\bigr).\] Lemma 42 identifies \(R\Gamma(K,C_G)\) with the dual of the coefficient complex at level \((K,G,h)\). The proof of Proposition 46 gives finite cohomology for \(C_G\) before almost scalar extension, with its smooth \(P_t\)-action. Applying (63) therefore gives \[ \mathop{\mathrm{Hom}}_k\!\left(H^i(P_n,B_{\mathrm{conn},G}^{(h)}),k\right) \simeq H^{-i-t^2}(C_G). \tag{99}\] Here corestriction is the transpose of coefficient pullback, so its inverse limit is exactly the dual of the connected-frame union. The finite Mittag–Leffler calculation takes place before tensoring with \(R\). Power transport removes the root stage while preserving ordinary \(\mathbb F_p\)-cochains.

To compute \(C_G\), apply Lemma 67 and then Proposition 43 after the faithful almost extension. The ambient map (93) and Lemma 23 leave the ambient line in geometric degree \((t+2)m\). The three \(G\)-characters cancel as in Lemma 66.

If \(a\) is the \(G\)-cohomological degree, the dual cochain lies in degree \(a+(t+2)m+d+t^2-(m+n^2)\). It is consequently dual to primal degree \[ i=n^2+m-(t+2)m-d-t^2-a=m^2-a. \tag{100}\] Here \(n=m+t\) and \(d=m(t-1)\), including \(d=0\) when \(t=1\). The remaining Tate and connected orientation factors are one-dimensional. A choice of their bases over the fixed geometric coefficient field gives (98); changing those bases changes the comparison by a nonzero scalar and does not change its module assertion.

To identify the action, return to the support calculation on \(Z\) before the ambient map. Both the support comparison and the top translation insertion preserve the actual pullback of constant \(P_n\)-cochains along the middle-bundle map. A constant class of degree \(r\) thus acts by a \(P_t\)-equivariant map \(C_G\to C_G[r]\) in the derived category. The naturality for degree-shifted coefficient maps in (63) shows that its action in (99) is the underlying map on \(H^*(C_G)\) after forgetting \(P_t\). Equivalently, the connected-group spectral sequence retains only degree \(t^2\); a positive connected-filtration component would raise that degree beyond the cohomological dimension and vanishes. This uses no equivariant splitting of \(C_G\). Compact duality retains the dual cup signs throughout.

After fixing the connected quotient frame, Proposition 55, specifically (77), identifies this ordinary action on \([Z/G]\) with pullback from \(BG\) of the actual restriction along \(g\mapsto\operatorname{diag}(g,1_t)\). That equality comes from the slope-zero extension with its fixed quotient and the vanishing of positive finite-coefficient cohomology of the additive rational upper block; it is an equality of pullback classes, so applies to their cup action on compact supports. The map from \(Z\) to ambient support is \(G\)-equivariant and hence linear for these \(BG\)-classes. Thus it preserves the now-identified action. The middle positive bundle need only be of standard type on \(Z\); no extension of its \(BP_n\)-map to dependent tuples is used. The ambient line is \(G\)-trivial, so the remaining action is ordinary cup product on \(A_G\). This proves the assertion. ◻

Proof. By Theorems 51 and 56, \[A_G=\bigwedge_k (\overline\alpha_{m,1},\ldots,\overline\alpha_{m,m}), \qquad \sum_{i=1}^m(2i-1)=m^2.\] The stable-generator theorem identifies the block restriction of \(\overline\alpha_{n,i}\) with \(\overline\alpha_{m,i}\) for \(i\leq m\), and with zero for \(i>m\). Proposition 68 consequently identifies the dual of the desired module, after tensoring with \(R\), with the reindexed regular module of this specified exterior algebra.

An exterior algebra has its perfect top-degree pairing \[A_G^j\otimes_k A_G^{m^2-j} \longrightarrow A_G^{m^2}\simeq k.\] Indeed the ordered monomial for a subset pairs, up to its exterior sign, with the monomial for its complement; all other pairings in complementary degree are zero. Thus the reindexed dual of the regular \(A_G\)-module is again its regular module, with a generator in degree zero. This proves that the target of (101), after tensoring with \(R\), is free of rank one under the displayed exterior action.

We identify its generator. The preceding comparison shows that \(H^0(P_n,B_{\mathrm{conn},G})\) is one-dimensional over \(k\): it is finite by Proposition 46, and its almost scalar extension has the length of one copy of \(R\). The unit \(1\) is nonzero in it. More explicitly, the algebraic coefficient rings and their frame pullbacks are unital, so \(k\cdot1\) injects into their union; this line is \(P_n\)-invariant, and no negative-degree cochains can give a boundary in degree zero. Hence \(1\) spans that one-dimensional \(k\)-space. After tensoring with \(R\) it is a basis of the degree-zero line, and therefore generates the entire regular module.

The cup map in (101) is consequently an isomorphism after the faithful almost extension. Its kernel and cokernel are \(k\)-vector spaces, and exactness and faithfulness of that extension make both zero. The map is a map of algebras: group cohomology with commutative coefficients has graded-commutative cup product, and each \(\xi_i\) has odd degree, so its square vanishes because \(p\) is odd. This proves the algebra presentation and the asserted product basis. The zero restrictions for \(i>m\) give zero actions after extension by Proposition 68, hence before extension by the same faithfulness. All comparisons preserve the actual classes and powering maps, so the proof applies to the one fixed family at rank \(n\), rather than choosing new generators at each height. ◻

Proof. Morava \(E\)-theory attached to the chosen perfect field \(k\) is descendable in the \(K(n)\)-local category (Mathew 2016, Proposition 10.10). Descendability says that the augmented partial-totalization tower has nilpotent error; equivalently, it belongs to the thick class generated by constant and nilpotent towers (Mathew 2016, Propositions 3.10, 3.20 and 3.27). Here a nilpotent tower is one for which a fixed finite number of consecutive transition maps has null composite. Such a tower has zero inverse limit. This is the quickly convergent form of descent, whose universal totalization property is explained in (Mathew 2018, Definition 2.19 and Propositions 2.20, 2.21 and 2.26).

The inclusion of \(K(n)\)-local spectra into spectra is exact. We verify that each full partial totalization here is a finite limit. For \(s\geq0\), send a nonempty subset of \(\{0,\ldots,s\}\) to its ordered set in \(\Delta_{\leq s}\). This functor is homotopy initial. Indeed, its comma category at \([q]\), \(q\leq s\), is the nonempty face poset of the complex with simplices \[(i_0,j_0),\ldots,(i_b,j_b),\qquad 0\leq i_0<\cdots<i_b\leq s,\quad 0\leq j_0\leq\cdots\leq j_b\leq q.\] Call this complex \(P(s,q)\). It is contractible for \(s\geq q\). Start with the cone with vertex \((s,q)\) on \(P(s-1,q)\), and attach in descending order the cones with vertices \((s,j)\), \(j<q\). The attaching subcomplex is \(P(s-1,j)\), which is contractible by induction since \(j\leq s-1\). The first cone is contractible even when \(q=s\); the induction begins with \(P(0,0)\), a point. This proves the claim.

Consequently the full partial totalization is the limit of the punctured \((s+1)\)-cube with vertex \(L_{K(n)}(E_n^{\wedge|T|})\) at nonempty \(T\subseteq\{0,\ldots,s\}\) and unit-insertion maps. This is the finite cubical comparison in standard descent machinery; compare (Mathew et al. 2017, Proposition 2.14). It applies to the full Amitsur totalization, including its codegeneracies. Thus the inclusion and then \(F\) preserve these finite limits, the augmentation, and the stated nilpotence of the error tower. The inverse limit of the transformed error tower is therefore zero. This proves (103) and carries along the convergence control. Bounded cohomological support then gives the stated finite, complete filtration on the abutment. ◻

Proof. Theorem 56 and Proposition 55 give classes \[u_i\in\pi_{1-2i}C^*_{\mathrm{cts}}(P_n;S)\] with these integral Hurewicz images. The range applies because \(n\leq p-2\).

By Remark 52, the action of \(\Gamma\) on \(H^*_{\mathrm{cts}}(P_n;\mathbb Z_p)\) is trivial.

As \(f\) is a unit in \(\mathbb Z_p\), restriction from \(G_n\) identifies continuous sphere cochains with the \(\Gamma\)-homotopy fixed points of the \(P_n\)-cochains, and the averaging idempotent realizes invariants on homotopy groups. In concrete terms, normalized transfer lifts \[\overline u_i=\frac1f\sum_{\gamma\in\Gamma}\gamma u_i\] to a class on \(G_n\). Its Hurewicz image on \(P_n\) remains \(\alpha^P_{n,i}\). This also follows from the finite-group homotopy-fixed-point spectral sequence: all its positive \(\Gamma\)-cohomology vanishes, since multiplication by \(f\) is invertible on its coefficient groups.

Map these classes through the totalization of (104) and use Proposition 70 to obtain the \(y_i\). Multiplication supplies the stated representatives of all the \(y_I\). The empty product maps to the augmentation unit itself. ◻

Proof. A function into the finite ring \(R/\mathfrak m^a\) is constant on some finite clopen partition of \(Q\); lifting its finitely many values proves surjectivity of reduction. The kernel equality follows by continuous coefficient division among the finitely many monomial generators of \(\mathfrak m^a\). This division is performed in the power-series coordinates, including the \(p\)-adic coefficient expansion, and is continuous for their adic topologies. The same coordinate argument applies to the displayed height ideal.

The locally constant function module modulo \(\mathfrak m^a\) is a filtered colimit of finite free \(R/\mathfrak m^a\)-modules, indexed by finite clopen partitions, and is therefore flat. Furthermore \[N_Q=\varprojlim_a C_{\mathrm{lc}}(Q,R/\mathfrak m^a).\] The adic flatness criterion over the Noetherian ring \(R\) now gives flatness of \(N_Q\). Equivalently, lifting a basis of \(C_{\mathrm{lc}}(Q,k)\) and successively lifting modulo \(\mathfrak m^a\) identifies \(N_Q\) with a completed free \(R\)-module. This also proves its asserted completeness. ◻

Proof. Ordinary cooperations before residue. For \(M=\operatorname{Map}^c(Q,E_n)\), the ordinary module identity is \[E_t\wedge M\simeq (E_t\wedge E_n)\otimes_{E_n}M.\] Lemma 72 gives flatness of \(\pi_*M\) over \(E_{n,*}\), so the module Kunneth spectral sequence has no positive \(\operatorname{Tor}\). Its convergence is bounded by the finite graded global dimension of the regular ring \(E_{n,*}\). Thus \[ \pi_*(E_t\wedge M)= \pi_*(E_t\wedge E_n)\otimes_{E_{n,*}} C_{\mathrm{cts}}(Q,E_{n,*}). \tag{107}\] Ordinary Landweber exact base change identifies the first factor with the two-sided base change of the formal-group-isomorphism Hopf algebroid. This is the ordinary cooperation algebra; the Landweber flatness assertions on both sides apply to it (Hovey and Strickland 2005, Example 0.1(d), Lemma 2.2 and Corollary 2.3). One may compute using \(BP_*BP\): isomorphisms between the typical laws are the formal-group sums of their typical curves, and the ratio of the periodic frames supplies the invertible linear coefficient. Thus this computation includes all isomorphisms of the underlying formal groups (Goerss 2008, Definition 2.40, Remark 2.41(3) and Corollary 2.45).

The height ideal and the algebraic marking ring. Flatness and (107) keep the test-side sequence \(p,u_1,\ldots,u_{t-1}\) regular. Indeed, it is regular on the cooperation algebra, and tensoring its successive injections with the flat function module preserves injectivity. Taking its iterated cofibers therefore produces only even homotopy groups. Under the universal formal-group isomorphism the successive height ideals agree on the two sides; their next generators differ by a unit modulo the preceding ideal. Consequently the residue imposes \[p=x_1=\cdots=x_{t-1}=0,\] and invertibility of the test height-\(t\) coefficient inverts \(x_t\). Lemma 72 then leaves \(C_{\mathrm{cts}}(Q,A)[1/x_t]\) on the source side.

We make the remaining cooperation algebra explicit, including the source constants and periodicity. The two constant fields are initially independent, and \[k\otimes_{\mathbb F_p}k\cong\prod_{\sigma\in\Gamma}k, \qquad a\otimes b\longmapsto(a\sigma(b))_\sigma,\] where the first factor is the test field. Let \(\mathcal B_{n,t}^{[\sigma]}\) be the connected-marking algebra after the source constants are embedded into the test field by \(\sigma\). Its universal marking is an isomorphism from the Honda height-\(t\) group to that source deformation. In particular, \(\mathcal B_{n,t}^{[1]}=\mathcal B_{n,t}\). With the test period \(v\) fixed, the ordinary cooperation calculation gives an isomorphism of graded algebras \[ \frac{\pi_*(E_t\wedge E_n)} {(p,u_1,\ldots,u_{t-1})} \ \cong\ \prod_{\sigma\in\Gamma} \bigl(\kappa_{t,*}\otimes_k\mathcal B_{n,t}^{[\sigma]}\bigr). \tag{108}\] Here the ideal on the left acts through the test factor.

To check the map and its inverse, use degree-zero formal-group coordinates and let \(\alpha\) denote the Honda-to-source isomorphism. If \(v_n\) is the source period, its linear coefficient is \(c=v_n/v\). Comparison of the leading terms in \([p]_{\mathrm{source}}\alpha=\alpha[p]_{\mathrm{Honda}}\) gives \[x_t c^{p^t-1}=1.\] Thus the cooperation data determine the embedding \(\sigma\) and the full marking \(\alpha\), including \(c\). Conversely, a marking and the fixed test period determine \(v_n=cv\); rescaling the coordinates by these periods makes \(\alpha\) strict and recovers the corresponding \(BP_*BP\)-isomorphism. These constructions are inverse and commute with composition. In particular, the source period is already recorded by the linear coefficient of the marking; there is no further frame parameter. No étale Tate basis is chosen.

The full algebraic isomorphism ring is a filtered union of finite étale marking algebras (Lurie 2010, Lecture 14, Theorem 1). These stages retain coefficients of full markings together with their extension equations, as in Proposition 7; they are not schemes of arbitrary finite-bud isomorphisms.

The topology produced by the tensor product. For a finite marking algebra \(J\) over \(B\), choose an idempotent presentation \(J=eB^a\). Then \[ J\otimes_A C_{\mathrm{cts}}(Q,A) =e\bigl(C_{\mathrm{cts}}(Q,A)[1/x_t]\bigr)^a. \tag{109}\] The right side consists exactly of functions admitting a common power of \(x_t\) as denominator and continuous coordinates in a finite \(A\)-lattice. For example, \(J\cap A^a\) is such a lattice: it is finite over the Noetherian ring \(A\), it spans \(J\) after inverting \(x_t\), and it is closed in \(A^a\). Multiplying by a common denominator places any element of the right side in continuous functions into this lattice; the converse follows from its inclusion in \(A^a\). Passing to the algebraic union of finite \(J\)’s requires one finite stage for each element of the tensor product. This proves exactly the bounded-pole, common-stage convention on every profinite parameter \(Q\). No completion is taken after inverting \(x_t\) or forming this algebraic union.

Cofaces and constants. Apply this calculation with \(Q=G_n^q\), retaining every source-embedding factor in (108). For the first coface, composition with the deformation-change isomorphism transports the tautological marking by the actual stabilizer action. Every finite coefficient of this composite uses only finitely many coefficients of the two isomorphisms. The continuous action preserves the height ideal and takes \(x_t\) to a unit times \(x_t\) modulo that ideal. Compactness of the stabilizer parameter therefore gives one finite marking stage and one pole bound for the resulting coefficient. The other cofaces and the codegeneracies are pullbacks along the usual profinite group-nerve maps. They preserve the same convention. The unit is the constant function \(1\); multiplication is ordinary multiplication of the marking coefficients. This proves compatibility with constant cup products as well as with the differential.

It remains to take source-field descent. With the test field fixed, \(\Gamma\) permutes the source-embedding factors simply transitively; the full product of the corresponding \(P_n\)-cochain coefficients is a coinduced module for \(P_n\subset G_n\). Group descent, or Shapiro’s lemma, identifies its \(G_n\)-cochains with the \(P_n\)-cochains of the identity-embedding factor. This is the step that changes the termwise cooperation calculation into the quasi-isomorphism of \(P_n\)-cochain complexes in (106). Equivalently, exact \(\Gamma\)-descent on the full product followed by evaluation at that factor gives this identification. Taking invariants of an isolated factor would not make sense, since that factor is not \(\Gamma\)-stable. The test periodicity is fixed throughout, so \(\kappa_{t,r}\) is an untwisted coefficient line. This gives (106), naturally for constants and their products. ◻

Proof. Use Proposition 70 with the exact functor \(\kappa_t\wedge-\). Lemma 73 identifies the totalization spectral sequence as \[ E_2^{s,r}=\kappa_{t,r}\otimes_k H^s_{\mathrm{cts}}(P_n;\mathcal B_{n,t}) \Longrightarrow\pi_{r-s}(\kappa_t\wedge D). \tag{111}\] By Proposition 69, this page is free over \(\kappa_{t,*}\) with basis \(\xi_I\cdot1\), for \(I\subseteq\{1,\ldots,m\}\) and \(m=n-t\). These are the images of the reductions of the transported integral classes \(\alpha^P_{n,I}\). The basis vector indexed by \(I\) lies in bidegree \((s,r)=(d(I),0)\). In particular, the page is zero for \(s<0\) or \(s>m^2\). This statement uses the constant cup action and the nonzero image of \(1\) in that proposition.

We now identify the actual representative of each basis vector. There are two filtrations to distinguish: the source tower of constant sphere cochains is built from connective spectra, whereas the residue tower computing \(\kappa_t\wedge D\) is periodic. We first locate a representative in the source filtration and then transport it to the residue filtration.

The empty subset is represented by the unit itself. For a nonempty subset put \(b=d(I)>0\). The sphere-cochain representative of \(y_I\) has filtration at least \(b\). Indeed, the \((b-1)\)-st partial totalization of the connective sphere-cochain diagram is \((1-b)\)-connective and hence has zero homotopy in degree \(-b\). The representative therefore lifts to the fiber of the map to that partial totalization. Its component in filtration \(b\), modulo the cochain boundary, is its integral Hurewicz class, whose restriction to \(P_n\) is \(\alpha^P_{n,I}\). The map (104) sends this chosen lift to the completed descent tower. Ordinary residue smash is exact and preserves the partial totalizations and their fibers, so it transports the lift to the residue tower as well. Lemma 73 identifies its leading component in (111) with precisely the specified image of \(\alpha^P_{n,I}\). Connectivity was needed only in the source tower.

These representatives come from the source totalization and hence supply compatible lifts through every subsequent partial totalization. Their leading terms cannot support an outgoing differential. Products give the representatives for all subsets. For a coefficient in \(\kappa_{t,*}\), choose a lift in \(\pi_*E_t\). Its natural action on the residue tower carries these compatible lifts to representatives of the corresponding multiple of each basis vector. Therefore every element of the \(E_2\)-page has such a lift, so \(d_2=0\). Inductively, if the earlier differentials vanish, the same representatives span the next page and force its outgoing differential to vanish. This proves that all differentials vanish; in particular none of the specified vectors can be lost as an incoming boundary.

Strong convergence and the bound \(0\leq s\leq m^2\) now give a finite filtration on \(\pi_*(\kappa_t\wedge D)\). The actual images of the \(y_I\) lift its specified associated-graded basis. The coefficient action preserves every filtration step, so this is a filtration by graded \(\kappa_{t,*}\)-submodules. Finite filtered linear algebra over the graded field \(\kappa_{t,*}\) shows that these images themselves form a basis: independence and spanning follow successively from their lowest nonzero filtration components. It follows that the actual map \[\bigvee_{I\subseteq\{1,\ldots,m\}} \Sigma^{-d(I)}\kappa_t \longrightarrow\kappa_t\wedge D\] is an equivalence. Since \(\kappa_t\) detects \(K(t)\)-equivalences and ordinary residue smash agrees with the relative scalar test, this proves (110) with all its stated components. ◻

Proof. This is (Barthel et al. 2025, Theorem A), with the generator degrees displayed in homotopical grading. Replacing the sphere by its \(p\)-completion does not change its positive-height localization. The dimension formula is the monomial basis of the exterior algebra. Only this additive formula and the natural scalar action are needed below; no identification of the individual \(\zeta_i\) with the classes \(y_i\) is required. ◻

Proof of Theorem 1. Proposition 71 supplies the one family \(y_i\), its ordered products, and \(y_\varnothing=1\). For \(n>1\), take \(z_I=y_I\), for \(I\subseteq\{1,\ldots,n-1\}\), and take the specified map \(u=y_\varnothing:S\to D\) to be the unit. Theorem 74 verifies all the simultaneous local basis hypotheses of Proposition 2; Theorem 75 verifies its rational dimension hypothesis. That proposition constructs the successive quotient blocks and pulls them back to a filtration of \(L_{n-1}D\), with the required individual cofibers. Under the first-stage identification \(F_1=L_{n-1}S\), the composite \(F_1\to L_{n-1}D=X_{n,p}\) is the localized unit. All maps and cofibers are \(S\)-linear, as required.

When \(n=1\), Proposition 71 still supplies \(y_1\); only the positive-height tests are vacuous. The unit calculation in Section 2 and the dimensions of Theorem 75 verify the remaining hypotheses of Proposition 2. Its height-one case gives the two cofibers \(H\mathbb Q_p\) and \(\Sigma^{-1}H\mathbb Q_p\), with the composite from the first stage to \(X_{1,p}\) equal to the canonical localization unit. ◻