Proof. Since \(k\) is perfect, every \(f\in A\) has a unique expansion \[f=\sum_{0\leq i,j<p}a^it^j f_{ij}^{\,p}, \qquad f_{ij}\in A.\] Ordinary derivations therefore have the usual partial-derivative formula and are determined freely by the values of \(a,t\). The continuous partial derivatives show that these values are independent. This proves the assertion for \(A\). The corresponding one-variable expansion proves the assertion for \(K\). A \(p\)-basis is preserved by a separable algebraic extension: Frobenius base change along a separable extension is an isomorphism, so the basis of \(K\) over \(K^p\) remains a basis after that base change. The differential statement follows by the same argument. For a finite étale algebra it holds on each field factor, and it passes to the filtered union defining \(L\). ◻

Proof. Define the closure as the image of the finite \(A_2\)-algebra of \(p^l\)-torsion in the reduced generic algebra of the indicated primitive locus. It is finite and reduced, every component dominates \(\mathop{\mathrm{Spec}}A_2\), and its closed fiber is supported at \(s_l=0\). It is therefore a complete local ring with residue field \(k\), dimension one, and maximal ideal \((t,s_l)\). The element \(s_1\) is nonzero on every generic component.

The series \([p](T)\) factors through \(T^d\), and its reduction at \(t=0\) is \(T^q\). Its coefficient of \(T^d\) is \(t\). Dividing \([p](s_1)=0\) by \(s_1^d\) in the reduced generic algebra gives \[t\,u(s_1^d,t)+s_1^{q-d}=0, \qquad u(0,t)=1.\] Here \(u\) is a unit power series. This identity holds in the closure by generic density. It also holds in the complete finite subalgebra \(\mathcal C_l\): the series for \([p^{l-1}]\) factors through \(T^{d^{l-1}}\), so \(s_1^d\) is a series in \(t\) and \(s_l^{d^l}\) with zero constant term in the latter variable. Since \(q-d\geq d\), the identity puts \(t\) in the ideal \((s_l^{d^l})\) of \(\mathcal C_l\). Its maximal ideal is consequently generated by \(s_l^{d^l}\). Its dimension is one because it is integral over \(A_2\). The surjection \(k[[W]]\twoheadrightarrow\mathcal C_l\), \(W\mapsto s_l^{d^l}\), is an isomorphism: a nonzero ideal in \(k[[W]]\) would lower dimension. The same argument, now with generator \(s_l\), proves the first identity in (9). These are identifications of the inclusion \(\mathcal C_l\subset\mathcal B_l\), not just abstract regularity assertions.

Write \([p^l](T)=g_l(T^{d^l})\) and prepare \(g_l\). The linear coefficient of \(g_l\) is a unit multiple of \(t^{1+d+\cdots+d^{l-1}}\). The invariant-differential identity for this Frobenius-divided homomorphism shows that its derivative, in its prepared kernel algebra, is that coefficient times a unit. After inverting \(t\) the prepared algebra is therefore étale, of rank \(p^l\). Its kernel before dividing by Frobenius has connected rank \(d^l\). The primitive images are distinguished exactly by \(s_l^{d^l}\), so their generator algebra is \(\mathcal C_l[1/t]\).

The scheme of lifts of this generator is finite locally free of rank \(d^l\). Its coordinate algebra surjects onto \(\mathcal B_l[1/t]\) by the same point coordinate. But (9) makes the latter free of rank \(d^l\) over \(\mathcal C_l[1/t]\), with basis \(1,s_l,\ldots,s_l^{d^l-1}\). The surjection is an isomorphism. In particular, reduction in the definition of the closure has not removed any infinitesimal part of the lift scheme on the exact-height locus.

Every primitive étale generator at level \(j\) lifts at level \(l\) after a geometric extension. The generic transition is consequently surjective on spectra and injective on the reduced coordinate algebras; generic density gives injectivity on the closures. The transition is finite because \(\mathcal B_l\) is finite over \(A_2\). Finally, \([p^{l-j}](T)\) factors through \(T^{d^{l-j}}\). Substituting the expression for \(t\) in \(k[[s_l^{d^l}]]\) proves (10). ◻

Proof. Use the spectral norm and choose \(\lambda<1\) bounding \(|t|\). The coefficients of \(P_t(T)-T^q\) and of \(G_t-\Gamma_3\) have norm at most \(\lambda\). Integral power series are \(1\)-Lipschitz on the open unit ball, and factorization through \(T^d\) gives \[ |P_t^{\circ r}(v)-P_t^{\circ r}(w)| \leq |v-w|^{d^r}. \tag{12}\] Successive approximations in the first formula in (11) differ by at most \(\lambda^{d^r}\). They therefore converge; their limits are topologically nilpotent and satisfy \[|\Phi_t(z)_j-z^{1/q^j}|\leq\lambda, \qquad P_t(\Phi_t(z)_{j+1})=\Phi_t(z)_j.\] For a compatible sequence \((b_j)\), consecutive normalized coordinates differ by at most \(\lambda^{q^j}\). Hence its displayed inverse exists and satisfies \[|z-b_j^{q^j}|\leq\lambda^{q^j}.\] After taking the relevant root this error is at most \(\lambda\); applying (12) makes it tend to zero. This proves one inverse identity. Raising the first displayed estimate to \(q^j\) proves the other. No common upper bound strictly less than one for all the individual \(|b_j|\) is required.

The coefficients of \(\Gamma_3\) are fixed by \(q\)-power Frobenius. The discrepancy between the specialized addition law and the Honda addition law is bounded by \(\lambda\); after normalization, or after \(r\) applications of \(P_t\), it tends to zero by the same estimates. Thus the formulas respect addition. A change of deformation frame differs from its Honda reduction by topologically small coefficients. Its Honda reduction commutes with \(q\)-power Frobenius, since its coefficients lie in \(\mathbb F_{p^3}\). The identical estimate proves frame equivariance. The maps commute with integral multiplication and with inverse multiplication by \(p\), so are \(\mathbb Q_p\)-linear. Continuity, including continuous scalar families, follows from uniform convergence on smaller balls.

Over an algebraically closed field, preparation applied to \(P_t(T)-w\) gives a monic polynomial whose roots are small whenever \(w\) is small. Choosing roots successively proves surjectivity of the zeroth projection. Its kernel consists of compatible \(p^j\)-torsion points. Connected finite groups have only the identity as a point over a perfect field, so this kernel is precisely the Tate module of the étale quotient. ◻

Proof. Retain the reduced closures from Lemma 6, including their point at \(t=0\). After perfection, a point of the étale quotient has a unique lift through the connected factor. The resulting inverse tower is thus the tower of the rings \(\mathcal B_l\).

We check its complete algebras after extension to an algebraically closed perfectoid field \(C\) of characteristic \(p\). Fix \(0<\rho<1\) in its value group and put \(y_l=s_l^{q^l}\). At level \(l\) take the closed disc \(|y_l|\leq\rho\). At a point of this disc, write \(\lambda=|t|\). The proof of Lemma 6 at level one, and finite telescoping of normalized coordinates, give \[\lambda\leq |s_1|^d, \qquad |s_1|\leq\max(\rho^{1/q},\lambda).\] If \(\lambda>\rho^{1/q}\) these inequalities would imply \(0<\lambda\leq\lambda^d<\lambda\). Consequently \[ \lambda\leq\rho^{d/q},\qquad |y_{l+1}-y_l|\leq\lambda^{q^l} \leq\rho^{d q^{l-1}}<\rho. \tag{14}\] The finite transition maps preserve these discs. Conversely, every geometric point of a level-\(l\) disc has a lift by finite surjectivity, and the strict inequality in (14) forces its lifts to remain in the next disc. Pullback is therefore isometric for the spectral norm, also after completed perfection.

In the completion of the direct limit of these perfected Gauss algebras, \(y_l\) converges to \(z\). Send the coordinate of the perfected radius-\(\rho\) disc to \(z\). The resulting homomorphism is isometric: for a polynomial in finitely many fractional powers, evaluation at \(y_l\) has exactly its Gauss norm, since \(s_l\) is a free disc coordinate and \(q^l\) is a power of \(p\). These evaluations converge to evaluation at \(z\). For a nonzero polynomial, their difference is eventually smaller than its fixed norm, so the limiting norm is the same. Completion preserves this isometry.

The image is dense. For \(l\geq j\), write \(s_j=F_{j,l}(s_l^{d^{l-j}})\) as in (10), with \(F_{j,l}\in k[[T]]\). Replacing \(s_l\) by \(z^{1/q^l}\) changes this expression by at most \[\lambda^{d^{l-j}}\leq\rho^{(d/q)d^{l-j}},\] which tends to zero with \(l\). The substituted series converges on the \(z\)-disc. The same approximation after taking roots proves that every \(s_j\) and each of its \(p\)-power roots belongs to the closed image. These generate the completed limit algebra. The isometry is therefore onto.

Here is the relative descent explicitly. Let \(E=\widehat{k((u))^{\mathrm{perf}}}\), with \(u\) an auxiliary variable, and choose \(C=\widehat{\overline E}\). For an affinoid perfectoid test \(S=\mathop{\mathrm{Spa}}(A,A^+)\), a pseudouniformizer \(\pi\in A^+\) and its unique compatible roots give a continuous map \(E\to A\), \(u\mapsto\pi\). The field extension \(\mathop{\mathrm{Spa}}(C,C^\circ)\to\mathop{\mathrm{Spa}}(E,E^\circ)\) is a \(v\)-cover, so its base change \[S_C=S\times_{\mathop{\mathrm{Spa}}(E,E^\circ)}\mathop{\mathrm{Spa}}(C,C^\circ)\longrightarrow S\] is a \(v\)-cover. The calculation over \(C\) established an isomorphism of complete algebras, hence of perfectoid spaces, on every smaller disc. It therefore applies to all \(C\)-tests, including \(S_C\), rather than only to geometric points. The coordinate map is defined over \(k\); its inverse over this cover is unique and agrees on the Čech overlap, so it descends to \(S\). This uses base change of an established isomorphism, not preservation of a spectral isometry by an arbitrary completed tensor product. Smaller discs exhaust the comparison: on a quasicompact test \(t\) is uniformly small, and the first torsion coordinate and (14) bound all normalized coordinates uniformly away from radius one.

It remains to identify the puncture. If \(z=0\) on a geometric fiber, Lemma 7 gives \(|s_l|\leq\lambda\) for all \(l\). For fixed \(j\), \[|s_j|=|P_t^{\circ(l-j)}(s_l)| \leq\lambda^{d^{l-j}}\longrightarrow0.\] Thus all \(s_j\) vanish, and the primitive-coordinate equation forces \(t=0\). Conversely, at \(t=0\) the reduced torsion coordinates vanish. Removing these corresponding zero loci gives (13). Equivariance is the frame and scalar equivariance of Lemma 7. Every estimate used a uniform spectral norm; taking the supremum over a compact profinite parameter preserves each estimate and each density argument. ◻

Proof. The Lubin–Tate universal-cover description of sections of \(\mathcal O(1)\) is (Fargues and Scholze 2026, Proposition II.2.2). Applying it with the unramified coefficient field of degree \(i\) and pushing forward gives \(\mathcal O(1/i)=V_i\) and the Honda height-\(i\) cover. The positive-section calculation, including the perfected-ball description, is also (Fargues and Scholze 2026, Proposition II.2.5). The classification on geometric curves and the relative pure-family theorem give the stated local forms and automorphisms; see (Fargues and Scholze 2026, Theorems II.2.14 and II.2.19). Equivalently, the basic stratum is the classifying stack of the corresponding division-algebra group (Fargues and Scholze 2026, Theorem III.4.5). The Honda forms are defined over our \(k\): commuting with \([p](T)=T^{p^i}\) puts every coefficient of a geometric Honda endomorphism in \(\mathbb F_{p^i}\subset k\).

For precision about maps, let \(\tau\) denote the morphism of sites used for relative curve cohomology. Anschütz–Le Bras prove that \[R\tau_*:\mathop{\mathrm{Perf}}(X_S)\longrightarrow D(S_v,\underline{\mathbb Q_p})\] is fully faithful for every small \(v\)-stack \(S\) (Anschütz and Le Bras 2025, Corollary 3.11). The same section calculation gives \(R\tau_*\mathcal O=\underline{\mathbb Q_p}\) and \(R\tau_*V_i=\tau_*V_i[0]\); the latter uses positivity. Taking degree-zero morphisms in this fully faithful functor proves the last assertion. These statements are relative on \(S\), so they apply over the finite-field base site and to forms obtained by descent. In particular, the assertion about \(\mathcal O\) is vanishing of its higher relative sheaves, not an assertion of vanishing on every unrefined test object. ◻

Proof. Work first on a geometric curve. The saturation of the image of a nonzero section \(\mathcal O\to V_3\) is a line bundle of nonnegative integral degree. Stability of \(V_3\) bounds that degree by \(1/3\), so it is zero. The section has no zeros, since a nonzero effective divisor has positive degree. Its quotient has rank two and degree one. A quotient of that quotient of nonpositive slope would also be a quotient of \(V_3\), contrary to semistability. Its slopes are therefore positive. The classification of vector bundles, and integrality of the degree of each stable summand, force the quotient to be \(V_2\).

An extension (15) has no negative-slope quotient, since neither end has a nonzero map to a negative-slope bundle. If it has a slope-zero quotient, that quotient receives a nonzero map from the kernel \(\mathcal O\), because \(\mathop{\mathrm{Hom}}(V_2,\mathcal O)=0\). The quotient is a sum of copies of \(\mathcal O\); compose with a linear functional nonzero on that kernel image and rescale. This gives a retraction onto the kernel, so the extension splits. A nonsplit extension thus has only positive slopes. A positive bundle of rank three and degree one can have only the stable type \(V_3\). The converse is immediate from stability. The asserted vanishing is the negative-slope section vanishing.

These arguments use the degree and slope classification on a geometric Fargues–Fontaine curve (Fargues and Scholze 2026, Proposition II.2.10 and Theorem II.2.14). Fiberwise surjectivity is the relative subbundle condition; the local forms in Lemma 9 and descent give the corresponding relative statements. ◻

Proof. We construct the quotient frame, normalize its determinant, and then recover the markings from an arbitrary surjection. The reconstruction will show that every surjection arises from an integral connected marking.

Constructing the quotient frame. Over \(\mathcal T_{\mathrm{et}}\), Proposition 8 identifies the chosen Tate generator with a nonzero section \(\mathcal O\to V_3\). After adding a connected marking, there is also a quotient frame. To construct it, orient the formal marking as an isomorphism from the specialized formal law to \(\Gamma_2\). Its coefficients are bounded by one on perfectoid tests. Indeed, its leading coefficient has norm \(|t|^{1/(d-1)}<1\), and the same coefficient comparison gives \(b_n^d-t^n b_n=Q_n(b_1,\ldots,b_{n-1})\) with integral coefficients. If all earlier coefficients have norm at most one, a root with \(|b_n|>1\) would make the monic term strictly larger than every other term. Induction gives the assertion.

Consequently the formal marking can be evaluated on all small points. Compose it with the zeroth projection in Lemma 7. This gives an additive map \(\mathcal H_3\to\mathcal H_2\). It is \(\mathbb Q_p\)-linear because multiplication by \(p\) is invertible on the Honda target. It kills the Tate generator and hence its \(\mathbb Q_p\)-span. By Lemma 9 it comes from a unique bundle map \(V_3\to V_2\). On a geometric fiber this map is nonzero: near the origin the formal marking has nonzero linear coefficient, and the zeroth projection is surjective.

Lemma 10 identifies the quotient by the Tate section with \(V_2\) locally. The induced nonzero endomorphism of this quotient is invertible, since its endomorphism algebra is the division algebra \(D_2\). The resulting quotient frame is therefore an isomorphism, also relatively. Uniqueness in relative full faithfulness makes the construction equivariant and compatible with all base changes.

Normalizing the determinant. An initial determinant identification \(\iota_0\) exists over \(k\). In the cyclic Honda isocrystal the determinant Frobenius differs from the rank-one degree-one isocrystal by the cyclic permutation sign \((-1)^{i-1}\). The possible minus sign is removed over \(\mathbb F_{p^2}\) by a Teichmüller unit \(u\) with \(u^p=-u\). Our field contains \(\mathbb F_{p^6}\), so the required identifications are defined over \(k\). Automorphisms act on these determinant lines by their reduced norms.

Connected markings form a \(J\)-torsor over \(\mathcal T_{\mathrm{et}}\) and change the quotient frames through precisely \(J\). On this torsor define the determinant isomorphism and its discrepancy by \[\begin{align*} \Delta(e,f)(e\wedge v_2\wedge v_3)&=f(v_2)\wedge f(v_3),\\ r(e,f)&=\Delta(e,f)\iota_0^{-1}\in\underline{\mathbb Q_p^\times}. \end{align*}\] Changing the marking by \(j\in J\) multiplies \(r\) by \(\mathop{\mathrm{Nrd}}(j)\in\mathbb Z_p^\times\). Consequently \(\nu=v_p(r)\) descends through this torsor to a locally constant function \(\mathcal T_{\mathrm{et}}\to\underline{\mathbb Z}\). After extension to the field \(C\) used in Proposition 8, the space \(\mathcal T_{\mathrm{et},C}\) is a punctured perfected open disc. Nested connected closed annuli exhaust it, and perfection preserves their topology; thus \(\nu_C\) is a single integer \(m\). The covers \(S_C\to S\) constructed there also show that \(\mathop{\mathrm{Spa}}(C,C^\circ)\to\operatorname{Spd}(k)\) is \(v\)-surjective. Hence \(\mathcal T_{\mathrm{et},C}\to\mathcal T_{\mathrm{et}}\) is a \(v\)-cover and \(\nu=m\) descends globally. Set \(\iota=p^m\iota_0\). The new discrepancy is \(p^{-m}r\) and is everywhere a unit.

Each pair of connected and étale markings therefore gives an exact sequence \[0\longrightarrow\mathcal O\longrightarrow V_3 \longrightarrow V_2\longrightarrow0\] whose determinant identification is a unit multiple of \(\iota\). For a fixed connected marking, multiply its étale generator by this unit to obtain determinant equality. There is exactly one such generator; uniqueness glues it over \(\widetilde X\). We have thus constructed a map from \(\widetilde X\) to \(\operatorname{Surj}(V_3,V_2)\), with a determinant-normalized integral kernel frame.

Recovering the markings. To prove that this map is an isomorphism, start with an arbitrary surjection \(f:V_3\twoheadrightarrow V_2\). Determinant equality supplies its kernel generator \(e_f\). Proposition 8 reconstructs from \(e_f\) a deformation and a primitive integral Tate generator. Choose a connected marking locally and let \(f_0\) be the quotient frame it constructs. Write \(f=b f_0\) with \(b\in D_2^\times\). Put \(r_0=\Delta(e_f,f_0)\iota^{-1}\), which is a unit by the normalization above. Since \(\Delta(e_f,f)=\iota\), we have \(1=\mathop{\mathrm{Nrd}}(b)r_0\). Thus \(v_p(\mathop{\mathrm{Nrd}}b)=0\), so \(b\in J\) and the required change of marking is integral. It is unique because the marking torsor and the integral quotient-frame torsor have the same group \(J\). Indeed, replacing the local marking by \(j\) replaces \(f_0\) by \(jf_0\) and \(b\) by \(bj^{-1}\), so the adjusted markings agree and glue. The resulting connected marking maps back to \(f\). Conversely, starting with a connected marking, Proposition 8 recovers its deformation from the normalized integral generator, and the \(J\)-torsor uniqueness recovers its marking. These constructions commute with base change, so they prove the relative isomorphism (16). In particular, the normalized kernel frame of every surjection is the integral Tate generator of its reconstructed deformation.

A \(p\)-power rescaling of \(e_f\) can change the reconstructed deformation; the coordinate comparison still returns a primitive basis for that deformation. No primitivity claim about the old deformation is needed.

The frame actions. The actions can be recorded without an implicit choice of a section of the norm map. In the frame convention \[(g,j)\cdot f=j f g^{-1},\qquad(g,j)\in P\times J,\] the normalized kernel frame satisfies \[ e_{jfg^{-1}}=\mathop{\mathrm{Nrd}}(j)\mathop{\mathrm{Nrd}}(g)^{-1}\,g e_f. \tag{17}\] This is the determinant identity for the exact sequence. Reversing all frame conventions inverts the displayed norm characters. In either convention \(H\) fixes the normalized étale frame under transport, and the \(P\) and \(J\) adjustments are continuous constant units. ◻

Proof. For \(V\triangleleft U\) open, \(L^V/L^U\) is a finite étale \(U/V\)-torsor. Perfection commutes with finite étale base change. After completion it remains the finite étale torsor \[\mathop{\mathrm{Spa}}((L^V)^\flat)\longrightarrow\mathop{\mathrm{Spa}}(B^\flat).\] One may check the completed algebra explicitly in a finite field basis: completion of its scalar extension from \(B^{\mathrm{perf}}\) is \((L^V)^\flat\), since finite-dimensional valued spaces are complete and their norms are equivalent to the coordinate norms. This works componentwise for field products and preserves the torsor identities.

The completed direct limit of these perfected finite-stage algebras is uniform, perfect, and Tate, with the common pseudouniformizer \(t\). It is therefore perfectoid. Its associated space is their affinoid inverse limit. The finite-level torsor identities pass to this limit and identify its relation with \(\underline U\times\widetilde X\). Surjectivity can be checked on a geometric point of \(\mathop{\mathrm{Spa}}(B^\flat)\): choose compatible lifts through the surjective finite étale covers, using compactness of their finite fibers, and extend the resulting bounded map to the completion. This also proves the covering assertion. Every construction commutes with restrictions and the transported \(P\)-action. ◻

Proof. By Proposition 11, \(\widetilde X\) is the space of surjections \(V_3\to V_2\) with their uniquely normalized kernel frames. Quotienting the middle frame by \(H\) allows the middle bundle to vary while preserving its determinant frame. Lemma 10 identifies precisely the resulting objects with the nowhere-split extensions of \(V_2\) by \(\mathcal O\), namely \(N^*\). There is no automorphism group left over an extension with its two ends fixed, since \(\mathcal H^0(V_2^\vee)=0\).

To see that this is a torsor presentation relatively, first choose a local basic frame of the middle bundle. Its determinant frame can be corrected to the prescribed one: reduced norm \(D_3^\times\to\mathbb Q_p^\times\) is surjective. The compatible frames form a torsor under its norm-one subgroup, which is exactly \(H\), because norm one has valuation zero. On units, surjectivity of \(P\to\mathbb Z_p^\times\) already follows from the norm of the maximal unramified cubic subfield. Local basic triviality then gives the asserted \(H\)-torsor on the site.

Combine it with the \(U\)-torsor in Lemma 12. The actions commute, so the two iterated quotient presentations give (18). Tracking determinants gives the claimed actions. For example, in the frame convention of (17), the residual \(g\in P\) acts on extension classes by the scalar \(\mathop{\mathrm{Nrd}}(g)\); a target change \(j\) acts by pullback along \(j^{-1}\) together with kernel scalar \(\mathop{\mathrm{Nrd}}(j)^{-1}\). This states the actual action and does not choose a group-theoretic splitting of reduced norm. Naturality and the parameter assertion follow from the natural torsor maps and the uniform coordinate comparison. ◻

Proof. Each \(B_i^{1/p^r}\) is a finite product of complete valued fields. First let \(e\in C\) be an idempotent. Approximate it by \(b\in B_i^{1/p^r}\) with \(\lVert b-e\rVert<1\). Then \(\lVert b^2-b\rVert<1\) and \(2b-1\) is a unit with inverse of norm at most one. Hensel’s lemma in that complete finite product gives an idempotent \(e_i\) at distance less than one from \(e\). Two such idempotents are equal: both \(e(1-e_i)\) and \(e_i(1-e)\) are idempotents of norm less than one, hence zero. Purely inseparable extensions create no idempotents, so \(e=e_i\) belongs to \(B_i\). Finitely many idempotents descend to a common stage.

The algebra \(E\) descends to a finite étale algebra over some \(B_i\). After a finite idempotent partition it has a monogenic étale presentation; each component field of \(B_i\) is infinite, so the primitive-element argument applies also to its finite étale products. Descend the idempotents selected by the proposed section. It remains to descend finitely many simple roots.

Let \(f(z)=0\) with \(f\in B_i[T]\) and \(f'(z)\) invertible in \(C\). Approximate \(z\) and \(f'(z)^{-1}\) simultaneously by \(b,c\) in a single \(B_j^{1/p^r}\), so closely that \[\lVert1-cf'(b)\rVert<1.\] The geometric series in this complete algebra makes \(f'(b)\) invertible with inverse norm at most \(\lVert c\rVert\). Let \(M\geq1\) bound the quadratic Taylor remainder of \(f\) on the unit ball about \(b\), and put \(C_0=\max(1,\lVert c\rVert)\). The approximations may also be chosen so that \[\lVert f(b)\rVert C_0^2M<1.\] Newton iteration remains in \(B_j^{1/p^r}\), has uniformly bounded inverse derivatives, and converges there to a root \(z_j\). Choose the initial approximation inside a simple-root uniqueness ball about \(z\); the divided difference is a unit on that ball, so \(z_j=z\). Componentwise, this root is separable over \(B_j\) but belongs to its purely inseparable extension, and hence was already in \(B_j\). All the finitely many roots use a common stage. This proves surjectivity, and injectivity follows from \(L\hookrightarrow C\). ◻

Proof. The integral Tate frame constructed in Proposition 11 gives, at each level \(l\), a section of the finite étale generator algebra of \(G_L^{\mathrm{et}}[p^l]\) after passage to \(\widehat{L^{\mathrm{perf}}}\). Lemma 14 descends this section to some finite separable marking stage. The descended sections are compatible: their transition identities hold in the completion and hence in \(L\) by injectivity. The same argument descends the action identities, including the \(H\)-invariance. The finite stage can depend on \(l\); no single finite stage containing the entire Tate generator is required. ◻

Proof. The fiber over the marked étale generator is a torsor for the connected kernel. The connected marking identifies that kernel with \(\Gamma_2[p^l]\). By Lemma 6, its algebra before base change along the chosen generator is \[k[[s_l]][1/t] \quad\text{over}\quad k[[s_l^{d^l}]][1/t],\] and this is the full lift algebra of rank \(d^l\). The generator marking of Lemma 15 gives its base change to \(L\).

Every finite separable extension of \(K\) has one-element \(p\)-basis \(t\), and the same is true of their ind-étale union. A finite purely inseparable extension of degree \(d^l\) in its perfection is therefore \(L^{1/d^l}\). The displayed lift algebra retains that degree after the separable base change: componentwise, \(s_l^{d^l}\) is a \(p\)-basis of its Laurent-series field and remains a \(p\)-basis after any separable extension. This proves (19), also for products by a common finite-stage argument. Purely inseparable embeddings are unique, so compatibility of the marked generators identifies the transition maps with the inclusions inside \(R\).

The group \(H\) fixes both the connected frame and the normalized étale generator. Its transported action on each lift torsor therefore commutes with translations in the named Honda group. For the remaining actions, transport followed by restoration of the normalized étale generator rescales by its constant norm unit; changing the connected frame also applies the named Honda automorphism. These are continuous constant group automorphisms and satisfy the action identities because the markings do. At a fixed level all these statements involve finite torsion algebras and finitely many coefficients. Those coefficients lie in a finite separable stage, and the resulting maps between complete finite-dimensional valued spaces are continuous. Enlarging that stage handles any specified finite list of levels and identities.

For later use, the order-two action has a direct sign check. The central elements \(\tau=-1\in P\) and \(j=-1\in J\) both act on surjections by \(f\mapsto-f\). Since \(V_2\) has rank two, \(\Delta(e,-f)=\Delta(e,f)\), so the normalized generator \(e\), its reconstructed deformation, and its étale frame are unchanged. The connected marking \(\phi\) is replaced by \(-\phi\). A translation labeled by \(c\in\Gamma_2[p^l]\) uses \(\phi^{-1}(c)\); after this change it uses \(\phi^{-1}(-c)\). Thus \(\tau\), whose reduced norm is \(-1\), conjugates the named translation group by \([-1]\). Its action on the one-dimensional tangent of the Cartier dual is \(-1\), as required in Lemma 32. ◻

Proof. On the Fargues–Fontaine curve with coefficient field \(F\), the chosen untilt divisor gives (Fargues and Scholze 2026, Proposition II.2.3) \[0\longrightarrow\mathcal O_F(-1) \longrightarrow\mathcal O_F \longrightarrow i_*C\longrightarrow0.\] The degree-zero section sheaf of \(\mathcal O_F\) is \(\underline F\), its higher relative cohomology sheaves vanish, and the negative line has no sections. The cohomology sequence therefore identifies \[\mathcal H^1(\mathcal O_F(-1)) \simeq (\mathbb G_{a,C})^\diamond/\underline F.\] Finite pushforward along the unramified coefficient extension takes \(\mathcal O_F(-1)\) to \(V_2^\vee\), giving the first isomorphism. The preimage of the zero extension class is exactly the sheaf \(\underline F\). Its complement consists precisely of classes nonzero on every geometric fiber, proving the second isomorphism as an open subquotient.

The sequence is equivariant for multiplication by \(F^\times\). The embedding and frame conventions may conjugate or invert this action. The independent determinant normalization of the kernel contributes exactly the scalar described in Theorem 13. For later use, a scalar \(u\in\mathbb Z_p^\times\) has norm \(u^2\) in \(F/\mathbb Q_p\), which is trivial modulo \(3\); thus this adjustment does not change the nontrivial residue-norm character on \(\mathcal O_F^\times\). ◻

Proof. Fix a right-hand side \((y_n)\) and positive real numbers \(\varepsilon_n\to0\). Suppose that a finite tuple \(x_0,\ldots,x_n\) solves the first \(n\) equations. By density choose \(x_{n+1}\) such that \[d_n=y_n+f_n(x_{n+1})-x_n,\qquad \|d_n\|<\varepsilon_n.\] Replace \(x_n\) by \(x_n+d_n\), and replace each earlier \(x_i\) by \[x_i+f_i f_{i+1}\cdots f_{n-1}(d_n).\] The enlarged tuple now solves the first \(n+1\) equations. Each previously chosen coordinate has changed by at most \(\varepsilon_n\). Starting with any \(x_0\), this procedure produces a Cauchy sequence in every fixed coordinate. Completeness gives a solution of all equations.

For the parameter assertion, the mapping spaces are complete and the induced maps are contractions. They have dense image: a continuous map from \(T\) to \(M_n\) is uniformly approximated by a map constant on a finite clopen partition, and its finitely many values can be approximated in the image of \(M_{n+1}\). Apply the preceding argument to these mapping spaces. Finally, the two-term Roos complex (23) computes the derived inverse limit of a countable system. ◻

Proof. For \(u\neq0\), write \(u=p^v a\), where \(a\) is an integer prime to \(p\). Then \[ |\epsilon^u-1|=\theta^{p^v}, \qquad p^v\leq |u|_{\mathbb R}. \tag{28}\] In particular, multiplication by \(\epsilon^u-1\) has zero kernel. The invariants are therefore exactly the coefficient at \(u=0\).

For \(u>0\), the ratio of the target outer weight to the source outer weight, after division, is bounded by \[(b'/b)^u\theta^{-p^v} \leq\theta^{u-p^v}\leq1.\] For \(u<0\), the same calculation using the inner weights gives \[(r'/r)^u\theta^{-p^v} \leq\theta^{|u|-p^v}\leq1.\] These inequalities prove both convergence and the asserted norm bound, even for exponents with arbitrarily large \(p\)-power denominators. They preserve uniform \(c_0\)-tails with parameter \(T\). The constant summand of the two-term complex has zero differential. Its degree-one generator is the reduction modulo \(p\) of the Kummer torsor of the coordinate, mapped into \(\mathcal O^\flat\). ◻

Proof. First consider a closed disc \(D\). Pullback to its perfectoid power-map cover is injective on sections, by the sheaf property. The pullback of a section is invariant under the root-of-unity automorphisms. On the punctured disc, the fractional monomial description and (28) force all nonconstant coefficients to vanish. The same is then true on the whole perfectoid disc. Thus \(H^0(D^\diamond,\mathcal O^\flat)\) consists of constants. With parameter \(T\), these constants are precisely \(R_T\). This argument uses the power-map cover only for sections; it does not treat the branched cover of a disc as a Kummer torsor.

Fix a disc \(D=\{|z|\leq R\}\). Choose \(e\in C\) with \[ |e|>R,\qquad R/|e|<|p|^{p/(p-1)}. \tag{30}\] There are nested annuli about \(e\), both containing \(D\), whose radii satisfy (27). They are contained in some larger disc \(D'\) about zero. Restriction from \(D'\) to \(D\) factors through these two annuli. Since an annulus has cohomology only in degrees \(0,1\), the factorization kills degrees greater than one. In degree one, Lemma 20 leaves only multiples of the Kummer class of \(z-e\).

That remaining class vanishes on \(D\). Indeed, choose a \(p\)-th root of \(-e\) in \(C\). The binomial series for \((1-z/e)^{1/p}\) converges under (30): its \(n\)-th coefficient has \(p\)-adic valuation \(-n-v_p(n!)\), so its radius of convergence is \(|p|^{p/(p-1)}\). It supplies a first \(p\)-th root of \(z-e\) on \(D\). This trivializes the mod-\(p\) Kummer torsor, hence also its image in \(\mathcal O^\flat\)-cohomology. The root and the annular radius choices are independent of \(T\).

Now exhaust \(\mathbb A^1_C\) by a countable increasing sequence of closed discs. Derived global sections on the union are the derived inverse limit of those on the discs. In degree zero the restriction system is the constant system \(R_T\). In every positive degree it is pro-zero by the preceding factorization. The derived inverse limit of a countable system has dimension at most one, so its spectral sequence gives (29). ◻

Proof. The invariants in (31) are \(R_T\). They agree with the affine-line constants. The support triangle and Lemma 21 therefore identify the sole support group with \(Q(c,r;T)\) in degree \(2\). The coefficient of \((z-c)^0\) vanishes on the image of \(\gamma-1\); extraction is onto, with a lift of \(1\) given by the Kummer class of \(z-c\).

If \(f\in\mathcal M(c,r;T)\) has constant coefficient zero, form \[ h=\sum_{u\neq0} \frac{f_u}{\epsilon^u-1}(z-c)^u. \tag{33}\] For any target outer radius \(b\geq r'\), the negative exponents are controlled by the inner-radius gap \(r'\geq r/\theta\); the positive exponents are controlled by using the source outer radius \(b/\theta\). The calculation in Lemma 20 gives \[ \|h\|_{[r',b]}\leq \|f\|_{[r,b/\theta]}. \tag{34}\] The source is available for every \(b\), so \(h\) converges on the whole enlarged exterior, with continuous parameter \(T\). It satisfies \((\gamma-1)h=f\) there. This proves the uniform vanishing of the residue kernel.

For completeness, the independence of the center is an assertion about residues, and can be checked after making a further enlargement. If the centers are \(c,c'\), then sufficiently far out \[\frac{z-c'}{z-c}=1+\frac{c-c'}{z-c}\] has a first \(p\)-th root by the binomial estimate used in (30). The two mod-\(p\) Kummer classes then coincide. Enlargement preserves the constant-coefficient residue, and its kernels are killed by (34); hence the residue identifications agree already before the further enlargement. Multiplying a coordinate by a constant also preserves its Kummer class, since a constant has a \(p\)-th root in \(C\). The normal Kummer generator is thus preserved by translations and by multiplication by \(F^\times\). ◻

Proof. Choose \(b_+\) in the value group with \(b_0<b_+<p\). At tube level \(n\), use the larger closed discs of radius \(r_n^+=|p|^n b_+\) about all the \(c_{n,\alpha}\). These discs and \(E_n\) form an open cover of the affine line in the adic topology; the closed discs and annular boundaries here are rational open subspaces. The strict gap \(r_n<r_n^+\) includes all boundary points. In each bounded region only finitely many discs occur. Denote these pairwise disjoint larger discs by \(D_\alpha\), and put \(V_n=\coprod_\alpha D_\alpha\). For this paragraph abbreviate \(R\Gamma_v(T\times Y^\diamond,\mathcal O^\flat)\) by \(R\Gamma(Y)\). The two-open cover \(\mathbb A^1_C=E_n\cup V_n\) gives \[R\Gamma(\mathbb A^1_C)\simeq \operatorname{fib}\left( R\Gamma(E_n)\oplus\prod_\alpha R\Gamma(D_\alpha) \longrightarrow \prod_\alpha R\Gamma(D_\alpha\cap E_n)\right).\] Taking the fiber of restriction to \(R\Gamma(E_n)\) yields \[\prod_\alpha\operatorname{fib}\left( R\Gamma(D_\alpha)\longrightarrow R\Gamma(D_\alpha\cap E_n)\right).\] Here products are ordinary products of complexes of \(\mathbb F_p\)-vector spaces and are exact. For a single hole, the two-open cover by \(D_\alpha\) and its full exterior identifies the corresponding factor with the support complex of Lemma 22. Thus the sole cohomology group at this stage is \[ Q_n(T)= \prod_{\alpha\in F/p^n\mathcal O_F} Q(c_{n,\alpha},r_n;T) \quad\text{in degree }2. \tag{37}\] This argument uses an actual open cover and ordinary sheaf cohomology; in particular the product at infinity has no imposed uniform norm bound.

The inclusions \(E_n\subset E_{n+1}\) induce an inverse system on the support complexes. A parent receives the sum of the support maps from its finitely many children. By Lemma 22, the residue of this map is literally the sum of their residues. We obtain short exact sequences of inverse systems \[ 0\longrightarrow K_n(T)\longrightarrow Q_n(T) \longrightarrow D_n(T):=\prod_{\alpha\in F/p^n\mathcal O_F}R_T \longrightarrow0. \tag{38}\] It is not necessary to choose Kummer lifts commuting with refinement: their residues, rather than their lifts, are canonical.

Choose once and for all an integer \(s\geq1\) such that \(p^s\geq\theta^{-1}\). A level-\(n+s\) child has radius \(r_{n+s}\), and its parent’s radius is \(r_n=p^s r_{n+s}\). The child center lies strictly inside the parent disc. The parent exterior is therefore the same exterior when recentered at that child. In the child-centered Kummer complex for this same parent exterior, a residue-kernel representative has a primitive given by (34). It is therefore an actual boundary in the ordinary algebraic cokernel. Its vanishing is intrinsic to the parent exterior and does not require the parent’s original centered Kummer cover to agree with the child’s. Consequently the following map is zero: \[K_{n+s}(T)\longrightarrow K_n(T).\] This same \(s\) works for all \(n\), all centers, all parents in the product, and all \(T\). Thus the kernel system is uniformly pro-zero.

The transition \(D_{n+1}(T)\to D_n(T)\) is split surjective: choose one child of each parent, put the parent coefficient in that child, and put zero in its other children. This defines a continuous section for the product topologies. Hence its derived inverse limit is its ordinary inverse limit \(\mathcal D_{\mathrm{lf}}(F,R_T)\), whereas the derived inverse limit of \(K_n(T)\) is zero. Formula (35), the support triangles, and exactness of limits in the derived category now prove the assertion. The one-hole annular exhaustions involved in this argument have no additional derived-limit contribution by Lemma 19 and (31).

Translations and scalar multiplication send tube systems to cofinal tube systems. Their action on the residue targets is the one proved in Lemma 22; consequently the limit action is the natural action on distributions. ◻

Proof. Write \(F=\bigcup_{m\geq0}p^{-m}\mathcal O_F\). Each lattice is isomorphic to \(\mathbb Z_p^2\). The completed group-ring resolution in two coordinates computes its continuous cohomology with trivial \(R_0\)-coefficients as \(\bigwedge^\ast(R_0^2)\). Restriction from one lattice to its \(p\)-multiple is zero in every positive degree: on degree one it is multiplication by \(p=0\), and the higher-degree maps are its exterior powers.

At the level of bar cochains, restriction between these compact open lattices is surjective, by extending a continuous function by zero on the clopen complement. Consequently continuous cochains on \(F\) are their derived inverse limit, including parameter \(T\). The degree-zero cohomology system is constant and the positive systems are pro-zero. This proves (39).

Set \(\Lambda=\mathcal O_F\), considered additively. By the product over \(\Lambda\)-cosets, \(\mathcal D_{\mathrm{lf}}(F,R_0)\) is the continuous coinduction from \(\Lambda\) of \(\mathcal D(\Lambda,R_0)\). The quotient \(F/\Lambda\) is discrete; any set of representatives is therefore a continuous section. The usual bar comparison for Shapiro’s lemma, and its simplicial contraction, are continuous with this section and the product topology. It reduces (40) to \(\Lambda\)-cohomology.

Choose a \(\mathbb Z_p\)-basis of \(\Lambda\). At finite quotient level the group-ring coordinate maps \(\gamma_i-1\) identify its distribution module with \[R_0[X_1,X_2]/(X_1^{p^n},X_2^{p^n}).\] The transition is the quotient map, because it sums coefficients over fibers of the finite group quotient. Inverse limit gives \[\mathcal D(\Lambda,R_0)=R_0[[X_1,X_2]]\] with the product topology on its coefficients. This is the formal power-series module, without restrictions on its coefficient sequence.

The two-coordinate completed resolution now gives the cohomological Koszul complex \[ 0\longrightarrow M \xrightarrow{(X_1,X_2)} M^2 \xrightarrow{(-X_2,X_1)} M\longrightarrow0,\qquad M=R_0[[X_1,X_2]]. \tag{41}\] Multiplication by \(X_1\) is injective, and multiplication by \(X_2\) is injective after reduction modulo \(X_1\). Thus the only cohomology is \(M/(X_1,X_2)=R_0\) in degree \(2\). Coefficient deletion, insertion, and shifts give continuous maps in this argument, so it also proves the statement with \(R_T\) and with further compact profinite parameters.

Finally, a lattice automorphism with matrix \(A\in \operatorname{GL}_2(\mathbb Z_p)\) induces on the linear terms of the completed group coordinates the matrix \(A\bmod p\). Its action on the top exterior generator of the resolution is \(\det A\bmod p\). Applying \(\mathop{\mathrm{Hom}}\), with the contravariant cochain convention, gives the inverse determinant on the augmentation quotient. For multiplication by \(u\), this determinant is \(N_{F/\mathbb Q_p}(u)\). The construction is independent of the chosen lattice basis and agrees under Shapiro’s comparison. ◻

Proof. By Lemma 17, \[(N^\ast)_{C^\flat} \simeq [\,(\mathbb A^1_C\setminus\underline F)^\diamond/ \underline F\,].\] The Čech spectral sequence of the translation quotient may be computed by continuous \(F\)-cochains. Indeed, in each bar degree its locally profinite parameter \(F^r\) is a disjoint union of compact open pieces. The preceding analytic calculations apply on every such piece, and ordinary products give their global cochains.

Apply continuous \(F\)-cohomology to the support triangle of Proposition 24. Lemma 21 and (39) put the affine-line term in degree zero, equal to \(R_T\). The support term is in degree \(2\) before group cohomology, and (40) puts it in degree \(4\) afterward. The support triangle therefore puts the additional class of the complement quotient in degree \(3\). This proves the degrees and dimensions in (42).

Scalar multiplication preserves the normal Kummer residue. The extra action is consequently the rank-two lattice orientation of Lemma 25. At \(p=3\) its inverse equals itself, giving \(\delta\). The residue norm \(\mathbb F_9^\times\to\mathbb F_3^\times\) is surjective, so this character is nontrivial. A scalar \(a\in\mathbb Z_p^\times\), regarded as an element of \(F^\times\), has norm \(a^2\), which is \(1\bmod3\). Thus the determinant adjustments of the kernel frame in Lemma 17 act trivially on the line. Inversion of frame conventions and the other unramified embedding give the same order-two character.

All coefficient contractions and all lattice calculations were made with \(R_T\), with their supremum and product topologies. Evaluation on \(T\) therefore identifies the results with continuous functions to the displayed lines. No equivariant splitting of the complex between its two cohomology degrees is asserted or needed. ◻

Proof. Begin with \(B=k((s))\). Elements of \(\mathcal A_B\) are fractional Laurent series \(\sum_{u\in\mathbb Z[1/p]}c_u s^u\) converging on every smaller closed annulus in \(0<|s|<1\). Frobenius sends \(c_u\) to \(c_u^q\) and leaves \(s\) fixed. Its fixed coefficients belong to \(k\). The annular \(c_0\)-condition for finite-field coefficients says exactly that the support is bounded below and has only finitely many exponents below any fixed real bound. These are the series in \(\widehat{k((s))^{\mathrm{perf}}}=B^\flat\). Moreover, the topology induced by the annular seminorms is its valuation topology: for nonzero finite-field coefficients the norm of a series is determined by its least exponent. This identifies the kernel in (43), including its topology.

We prove surjectivity with the parameter \(Q\) present. A continuous \(Q\)-family has continuous coefficient functions \(c_u:Q\to C^\flat\) and uniformly small weighted tails in each annulus seminorm. For each \(c\in C^\flat\), there is \(b\in C^\flat\) satisfying \[ b^q-b=c. \tag{45}\] If \(|c|<1\), the unique solution with \(|b|<1\) is \[b=-\sum_{j\geq0}c^{q^j},\qquad |b|=|c|.\] If \(|c|>1\), every solution has \(|b|=|c|^{1/q}\); if \(|c|=1\), a solution has norm at most one. In every case we can choose \[ |b|\leq |c|. \tag{46}\] The polynomial in (45) has derivative \(-1\). Near each value \(c_0\), a chosen root therefore extends to a continuous local branch, obtained by adding the small solution for \(c-c_0\). For a continuous coefficient \(c_u:Q\to C^\flat\), compactness of \(Q\) and a finite clopen refinement give a continuous choice \(b_u\) with (46). The partitions may depend on \(u\). The uniform bound preserves the weighted \(c_0\)-tails in every annulus seminorm. More explicitly, fix one such seminorm, write its monomial weight as \(w(u)\), and fix \(q_0\in Q\) and \(\varepsilon>0\). Outside a finite set of exponents \(I\), we have \(\sup_{q\in Q}|b_u(q)|w(u)<\varepsilon\). The same bound holds for \(|b_u(q)-b_u(q_0)|w(u)\) by ultrametricity. For the finitely many \(u\in I\), intersect the continuity neighborhoods of the coefficient functions. This proves continuity in the chosen annular seminorm; applying it to each seminorm proves continuity into \(\mathcal A_B\). Thus the assembled series is a continuous solution of (43), although the partitions chosen for its different coefficients need not agree.

For \(B=k'((s))\), put \(f=[k':k]\). Geometric base change has one factor for each \(k\)-embedding of \(k'\) into \(C^\flat\). Index these factors modulo \(f\), so that \(\Phi(x)_i=F(x_{i-1})\), where \(F\) raises each Laurent coefficient to its \(q\)-th power. To solve \(\Phi(x)-x=y\), first solve \[(F^f-1)x_0 =y_0+\sum_{j=1}^{f-1}F^{f-j}y_j,\] using the preceding argument with \(q\) replaced by \(q^f\), and then put \(x_i=F(x_{i-1})-y_i\) for \(1\leq i<f\). These operations preserve the Fréchet algebra and all continuous parameter families, since \[\|F^j a\|_{[r,b]} =\|a\|_{[r^{1/q^j},b^{1/q^j}]}^{q^j}.\] All transformed radii remain inside the open unit interval. In the kernel, \(F^f x_0=x_0\) and \(x_i=F^i x_0\), which recovers the original finite constant field. Every nonzero finite-field coefficient has norm one, so the annular norms on this kernel depend only on the least exponent and induce exactly the valuation topology of \(B^\flat\). Finite cycles and finite products preserve this identification. The resulting kernel inclusion is intrinsic to \(B\), and is independent of the uniformizer and of the chosen decomposition of the constants.

The auxiliary root choices prove surjectivity; they are not claimed to be equivariant. The kernel inclusion and \(\Phi-1\) themselves are canonical and natural.

Finally apply (43) with \(Q=T\times G^r\) in every bar degree. The maps commute with the bar faces, since the action is over \(k\) and commutes with base Frobenius. Termwise exactness gives (44). ◻

Proof. The fixed-stage comparison. At a fixed stage \(B=L^U\), Theorem 13 gives \[ [X_B/H]\simeq[N^\ast/U]. \tag{52}\] It is an identity of quotient v-stacks with their actual restriction and frame-change maps. After geometric base change, take derived global sections, including the product with \(T\), and then the fiber of \(\Phi-1\). On the left, Lemma 27 in every \(H\)-bar degree gives exactly \(C^\bullet_{\mathrm{cts}}(H,B^\flat;T)\). On the right, fibers commute with totalization, giving the \(U\)-bar totalization of the complexes \(K_k(T\times U^r)\). Thus there is a natural comparison \[ C^\bullet_{\mathrm{cts}}(H,B^\flat;T) \simeq \mathop{\mathrm{Tot}}\bigl([r]\longmapsto K_k(T\times U^r)\bigr). \tag{53}\] Every operation here is at the fixed stage \(B\). The comparison uses the pro-étale torsor in Theorem 13; it involves no interchange between continuous cochains and completion of the entire marking union.

Finiteness and profinite families. Choose \(U\) small enough to be torsion-free uniform and to fix the finite lines in (48). Such \(U\) form a cofinal family; they may be chosen normal in \(J\). Their Lie dimension is \(4\). Continuous cohomology with finite characteristic-\(p\) coefficients is finite and vanishes above degree \(4\), by the finite analytic resolution; see Lemma 18. The hypercohomology spectral sequence of (53) has only the rows \[ E_2^{r,0}=H^r_{\mathrm{cts}}(U,k),\qquad E_2^{r,3}=H^r_{\mathrm{cts}}(U,\ell), \tag{54}\] and \(0\leq r\leq4\). It is first-quadrant with a finite possible range. Therefore the abutment is finite and vanishes above degree \(7\). No splitting between the two rows is needed for this conclusion.

We verify the family assertion in this spectral sequence. For finite discrete coefficients \(V\), a continuous function \(T\times U^r\to V\) factors through a finite clopen partition of \(T\): cover the compact product by finitely many clopen rectangles on which the function is constant, and refine the partition of \(T\). Thus cycles and boundaries in these row complexes may be computed on finitely many \(T\)-pieces. The same applies to primitives, and gives \[H^r_{\mathrm{cts}}\! \left(U,\operatorname{Map}_{\mathrm{cts}}(T,V)\right) = \operatorname{Map}_{\mathrm{cts}} \left(T,H^r_{\mathrm{cts}}(U,V)\right)\] in the indicated parameter convention. Together with (48), this gives \(E_2^{r,j}(T)=\operatorname{Map}_{\mathrm{cts}} (T,E_2^{r,j}(*))\). All these groups at a point are finite. At every subsequent page, naturality under point evaluations makes the differential pointwise equal to the differential at a point: evaluation in its target is injective by the preceding-page identification. Continuous functions on \(T\) are exact on finite discrete groups, by lifting on a finite clopen partition. Taking kernels and cokernels therefore proves the same identification on the next page. The fixed finite rectangle of possible bidegrees gives a uniform limiting page and its natural finite abutment filtration. Here is also a direct verification for the abutment extensions. Suppose a functorial extension has outer terms \(\operatorname{Map}_{\mathrm{cts}}(T,A)\) and \(\operatorname{Map}_{\mathrm{cts}}(T,B)\), with \(A,B\) finite, and its middle functor commutes with finite clopen decompositions of \(T\). On a partition where the image in \(B\) is constant, lift that constant using an element of the middle group at a point. Subtracting its constant pullback leaves a continuous \(A\)-valued function. Evaluation in the middle group is therefore locally constant. The same construction realizes every locally constant function into the middle group at a point, and evaluation is injective by the two outer terms. All the filtration functors here commute with finite clopen decompositions, since the original complexes do. Induction through the finite filtration gives (49). This argument establishes the finite-stage parameter assertion before any use of strictness in the uncompleted calculation.

Invariants and the marking colimit. In total degree zero, the sole term of (54) is \(H^0(U,k)=k\). It is induced by the constant unit. Hence \((B^\flat)^H=k\). The inclusions \(B\hookrightarrow B^\flat\) are injective and \(H\)-equivariant, so \(B^H=k\) as well. Every element of \(L\) lies in some stage and then in a sufficiently large stage in the chosen cofinal family. This proves \(L^H=k\) without any decompletion argument.

It remains to pass to the marking colimit. For a finite smooth coefficient module \(V\), every positive-degree continuous cohomology class of an open subgroup becomes zero after restriction to a sufficiently small open subgroup. One may use normalized inhomogeneous cochains: a continuous cocycle is zero at the identity tuple and has discrete values, hence is zero on the power of a sufficiently small open subgroup. Degree-zero invariants have colimit \(V\). The spectral sequences (54) are uniformly bounded, and filtered colimits of abelian groups are exact. Their colimit therefore leaves only \(k\) in degree zero and \(\ell\) in degree three, proving (51) for a point.

For general \(T\), use (49). Continuous maps to a discrete group have finite image, so \(\operatorname{Map}_{\mathrm{cts}}(T,-)\) commutes with a filtered colimit of discrete groups: finitely many values and finitely many equalities are realized at one common stage. This gives (51) with all \(T\).

Finally, (52) and (53) are natural for all frame changes. Their colimit action is therefore the action on the two lines in (48). By Proposition 26, the residual determinant acts through scalar multiplication by a \(\mathbb Z_p^\times\)-unit on the negative space, with trivial action on both lines at \(p=3\). The \(J\)-action restricts on \(\mathcal O_F^\times\) to \(\delta\) on \(\ell\), as claimed. ◻

Proof. Use the finite resolution in Lemma 18. Its Hom complex on \(M\) has compact terms and closed boundaries, so its cohomology is compact Hausdorff. The continuous comparisons with bar cochains identify the resulting topology with the bar-cochain quotient topology. This does not assert compactness of the bar-cochain spaces.

Continuous \(\mathbb F_p\)-duality is exact on compact vector spaces: a functional on a closed subspace extends after passing to a suitable finite-dimensional quotient. If \(Q\) is finite projective over \(\Lambda=\mathbb F_p[[H]]\), put \(Q^*=\mathop{\mathrm{Hom}}_\Lambda(Q,\Lambda)\) and turn this right module into a left module using the involution \(h\mapsto h^{-1}\). Finite-projective evaluation gives \[\mathop{\mathrm{Hom}}_{\Lambda,\mathrm{cts}}(Q,M)^\vee \cong \mathop{\mathrm{Hom}}_{\Lambda,\mathrm{cts}}(Q^*,M^\vee).\] For a finite free module this is coordinatewise evaluation; taking projective summands proves the general case. The identification is compatible with precomposition and coefficient maps. Apply it to each term, reverse degrees about eight, and use the reversed dual resolution in Lemma 18. Exactness of compact duality proves (55) and its naturality. Dualizing again proves the discrete assertion. ◻

Proof. Put \(E=A^{a-1}\), \(Z=\ker d^a\), and \(f=d^{a-1}:E\to Z\). The closed subspace \(Z\) is complete and metrizable. Choose decreasing bases of open linear subspaces \(E=E_0\supset E_1\supset\cdots\) in \(E\) and \(W_0\supset W_1\supset\cdots\) in \(Z\). Separability makes \(E/E_n\) countable. Since \(Z/f(E)\) is finite, \(Z/f(E_n)\) is countable as well. Set \[H_n=\overline{f(E_n)}\subset Z.\] Countably many cosets of this closed subgroup cover \(Z\). Baire’s theorem gives \(H_n\) nonempty interior, so it is open.

We show that \(f(E_n)=H_n\) for every \(n\). Given \(z\in H_n\), put \(r_n=z\). If \(r_j\in H_j\), density of \(f(E_j)\) in \(H_j\) and openness of \(H_{j+1}\cap W_{j+1}\) allow a choice \(e_j\in E_j\) with \[r_{j+1}=r_j-f(e_j)\in H_{j+1}\cap W_{j+1}.\] Indeed the translated neighborhood of \(r_j\) lies in \(H_j\) and meets the dense image. The series \(\sum_{j\ge n}e_j\) is Cauchy because its tails lie in the decreasing open subspaces \(E_j\). Completeness gives its sum \(e\in E_n\), since \(E_n\) is closed. The residuals tend to zero, so continuity gives \(f(e)=z\). Thus every \(f(E_n)=H_n\) is open. Taking \(n=0\) proves that the boundary subgroup is open and closed in \(Z\); the other values of \(n\) prove that the differential onto it is open. Choosing \(E_n\) inside a prescribed neighborhood gives the last assertion. ◻

Proof. Choose a decreasing basis of open subgroups \(E_n\) of \(E\) tending to zero, small enough that their images also tend to zero in \(F\). Openness makes each \(q(E_n)\) open. Approximate the given function on a finite clopen partition of \(T\) by finitely many values and lift those values to \(E\), leaving an error in \(q(E_1)\). Refine the partition and approximate that error by values with lifts in \(E_1\), leaving an error in \(q(E_2)\). Continue. The corrections form a uniformly Cauchy sequence because all later corrections lie in \(E_n\); completeness produces a continuous lift. No metrizability of \(T\) is used.

For the second assertion, a continuous family of cycles has a continuous image in the finite, discrete cohomology group by Lemma 30. A family mapping to zero lifts through the open differential onto boundaries by the first assertion. Conversely, choose a cycle representative for each of the finitely many values of a family of cohomology classes. This proves (56). Finally, a continuous map from \(T\) to a discrete filtered colimit has finite image. Its finitely many values, and every equality needed between them, occur at one stage. Hence \[\mathop{\mathrm{colim}}_i C_{\mathrm{cts}}(T,V_i) \cong C_{\mathrm{cts}}(T,\mathop{\mathrm{colim}}_i V_i)\] for discrete vector spaces \(V_i\), even when their transition maps are not injective. Exactness of filtered colimits proves the last claim. ◻

Proof. The Cartier dual of the height-two, dimension-one Honda group is again connected of height two and dimension one; see (Tate 1967, sec. 2.2, Proposition 1, and Section 2.3, Proposition 3) and the Frobenius–Verschiebung calculation below. A coordinate on that dual identifies the finite distribution algebra with \(k[D]/(D^{p^{2l}})\); the transition maps are restriction to the dual torsion kernels. Their inverse limit is \(k[[D]]\).

After faithfully flat trivialization of a finite torsor, its function module is the dual regular module of the truncated polynomial ring. The pairing that extracts the coefficient of \(D^{p^{2l}-1}\) from a product identifies that dual module with a single regular nilpotent block. Consequently \(\ker D^e\) on \(R_l\) has rank \(e\) over \(L\) for \(1\le e\le p^{2l}\). These ranks descend under faithful flatness. For \(q=p^i\), the formal-group coproduct satisfies \[\Delta(D^q)\in(D^q\otimes1,1\otimes D^q).\] Hence \(\ker D^q\) is a subalgebra. On a field factor it is an intermediate field of purely inseparable degree \(q\).

The element \(t\) is a \(p\)-basis of \(K\) and remains a \(p\)-basis under separable extension. An intermediate field of degree \(p^i\) in a one-generator purely inseparable extension is contained in \(L^{1/p^i}\): every element has purely inseparable degree at most \(p^i\). Equality follows from the degrees. This proves the kernel formula on field factors, and hence on \(L\). To justify the last passage without requiring a chosen global field factor, note that \(L\) is an ind-étale algebra over a field and is absolutely flat. The finite free module identities in question can be checked at its residue fields, which are separable algebraic extensions of \(K\). In a larger nilpotent block, the old kernel is contained in the image of \(D\). This proves surjectivity on the union and nonvanishing of the top operator on every finite kernel.

By Lemma 15, a norm unit rescales the chosen étale generator by that unit or its inverse. It therefore conjugates the translation group by the corresponding scalar. Lemma 16 checks directly that the central order-two unit acts by \([-1]\) on the named translations. On the tangent of the Cartier dual it therefore has character \(-1\). If \(D_0\) is any parameter and \(\tau\) denotes that order-two action, replace it by \((D_0-\tau(D_0))/2\). Its linear coefficient is unchanged, so it is a parameter and satisfies \(\tau(D)=-D\). Continuity of the marking action gives continuity on every finite jet of this parameter algebra.

For the Frobenius identities, use the Honda model over \(\mathbb F_p\). Its Frobenius \(\Phi\) satisfies \(\Phi^2=[p]\). Since \(V\Phi=[p]=\Phi^2\) on the divisible group and \(\Phi\) is faithfully flat, cancellation gives \(V=\Phi\). Cartier duality exchanges these operators. For the indicated even \(i\), their \(i\)th iterates are \([p^{i/2}]\). If \(A_l=\mathcal O(\Gamma_2[p^l])\) and \(\langle b,\xi\rangle=\xi(b)\) is its pairing with distributions, Cartier-dual naturality gives, because \(q\) fixes \(k\), \[\langle b^q,D\rangle =\langle b,(V^i)^*D\rangle =\langle b,D^q\rangle.\] For the last equality, an \(\mathbb F_p\)-coordinate \(z\) on the dual has \((V^i)^*z=z^q\); writing \(D=\sum c_nz^n\) over \(k\) gives \((V^i)^*D=D^q\). If a torsor coaction is \(\rho(f)=\sum_j f_j\otimes b_j\), it follows that \[D(f^q)=\sum_j f_j^q\langle b_j^q,D\rangle =\left(\sum_j f_j\langle b_j,D^q\rangle\right)^q =(D^qf)^q.\] The coefficients \(f_j\) retain their \(q\)th powers; no splitting of the torsor is used. Apply this identity to \(af\) with \(a^q\in L\) and use \(L\)-linearity of \(D\) to obtain the second identity in (58).

On a fixed root level only finitely many powers of \(D\) act. Their matrices in the fractional-power basis of the \(p\)-basis have finitely many coefficients in \(L\), hence are defined at one finite separable stage. They are continuous maps between finite-dimensional spaces over complete local fields. For a finite target stage, choose a \(D\)-preimage of each member of a finite basis over that stage’s separable coefficient algebra. Put these finitely many preimages in one larger stage and extend linearly. This is a continuous additive section. At a finite root stage the intersection with \(\ker D=L\) is its separable stage, with the induced topology, by separability versus pure inseparability. The coefficient exactness criterion above proves the final assertion, including parameters. ◻

Proof. By Lemma 4, ordinary differentials of \(A\) are freely generated by \(da,dt\) and agree with continuous differentials. The Kodaira–Spencer homomorphism for the universal \(p\)-divisible group is the functorial isomorphism \[ \mathop{\mathrm{Hom}}_A(\mathop{\mathrm{Lie}}\mathcal G,\omega_{\mathcal G^\vee}) \xrightarrow{\ \sim\ }\Omega^1_{A/k}. \tag{59}\] For a \(p\)-divisible group \(G\) over any ring \(B\) on which \(p\) is nilpotent, isomorphism classes of lifts across \(B=B'/I\) with \(I^2=0\) form a torsor under \(\mathop{\mathrm{Hom}}_B(\omega_{G^\vee},I\otimes_B\mathop{\mathrm{Lie}}G)\). The associated functorial Kodaira–Spencer map \(\mathop{\mathrm{Hom}}_B(\mathop{\mathrm{Lie}}G,\omega_{G^\vee})\to\Omega^1_{B/\mathbb Z}\) controls changes of a lifting ring map (Lau 2010, Theorem 5.1 and Equation (5.1), p. 226). The isomorphism in (59) uses the separate universal-deformation case (Lau 2010, paragraph following Equation (5.2), p. 227): \(A=k[[a,t]]\) is a complete local Noetherian \(k\)-algebra with residue field \(k\), the deformation is universal, and \(k\) is perfect. In this case Lau identifies completed relative differentials with ordinary absolute differentials. Since every absolute derivation kills the perfect field \(k\), the latter are \(\Omega^1_{A/k}\), in agreement with the finite \(p\)-basis calculation above. Universality makes the closed-point map bijective; both modules are free of rank two, so the map is an isomorphism over \(A\).

Over \(L\) the two markings give \[0\longrightarrow\Gamma_{2,L} \longrightarrow\mathcal G_L \longrightarrow(\mathbb Q_p/\mathbb Z_p)_L\longrightarrow0.\] Thus \(\mathop{\mathrm{Lie}}\mathcal G_L=\mathop{\mathrm{Lie}}\Gamma_{2,L}\) has an \(H\)-fixed frame. Dualizing gives the exact sequence of invariant-form modules \[ 0\longrightarrow\omega_{\Gamma_2^\vee,L} \longrightarrow\omega_{\mathcal G^\vee,L} \longrightarrow\omega_{\mu_{p^\infty},L} \longrightarrow0. \tag{60}\] Both end lines have \(H\)-fixed frames. Exactness may be checked after the faithfully flat extension to perfect coefficients, where the marked connected–étale extension splits, and then descended. The determinant of the middle module is therefore equivariantly trivial even when the extension over \(L\) is nonsplit. Taking determinants in (59) shows that \(\det(\Omega^1_{A/k}\otimes_A L)\) has an \(H\)-fixed basis.

The conormal line of \(a=0\) also has an \(H\)-fixed frame on this marking cover. Explicitly, comparison of the coefficient of \(T^p\) under a coordinate change with leading coefficient \(c\) gives \(a'=c^{1-p}a\) in the corresponding coordinate convention. Thus \(a(dT)^{p-1}\) is the first Hasse section, and its conormal has the inverse periodicity twist. The connected marking frames \(dT\) and trivializes that twist. More explicitly, if the fixed cotangent frame is \(e=u\,dT\), then \(T'=cT+\cdots\) gives \(u'=u/c\) and \(a'=c^{1-p}a\). Hence \([a'](u')^{1-p}=[a]u^{1-p}\) is an invariant conormal frame. Since \(L/K\) is ind-étale, the conormal sequence is \[0\longrightarrow L\,da \longrightarrow\Omega^1_{A/k}\otimes_A L \longrightarrow\Omega^1_{L/k}\longrightarrow0.\] Taking the determinant quotient produces the desired \(H\)-fixed basis \(\eta\). This construction uses differentials over \(A\) and \(L\), not differentials of a perfection. Writing \(\eta=f\,dt\), both \(f\) and \(f^{-1}\) belong to a common finite marking stage; enlarging that stage if necessary makes \(\eta\) a basis on every field factor. ◻

Proof. The functional \(I_i(f)=D^{q-1}(f^{1/q})\) takes values in \(L\) by (57); it is nonzero on every field factor by the nilpotent-block calculation. It satisfies \(I_i(a^qf)=aI_i(f)\) and is \(H\)-equivariant. The same semilinearity and equivariance hold for \(C_0^i\). The Cartier pairing gives an isomorphism \[ \mathrm{Fr}_*^iL\longrightarrow \mathop{\mathrm{Hom}}_L(\mathrm{Fr}_*^iL,L),\qquad b\longmapsto\bigl[f\longmapsto C_0^i(bf)\bigr]. \tag{67}\] Here \(\mathrm{Fr}_*^iL\) has the scalar action \(a\cdot f=a^qf\). Both modules are free of rank \(q\) in the common \(p\)-basis. On a field factor the pairing is nondegenerate: for \(b\ne0\), choose \(z\) with \(C_0^i(z)\ne0\) and evaluate at \(b^{-1}z\). This proves (67), also for finite products and the ind-étale union. Thus a unique multiplier \(b_i\) satisfies \(I_i(f)=C_0^i(b_if)\). It is a unit by nonvanishing on each factor. Equivariance and uniqueness give \(b_i\in L^H=k\) by (61). Taking \(c_i=b_i^{1/q}\) proves (63).

This argument works for \(i=1\) without the even-iterate restriction in (58). Since \(D\) kills \(L\), it gives \(c_1C_0(f^p)=D^{p-1}f=0\). In particular \(C_0\mathrm{Fr}=0\) is a consequence of the invariant torsor and differential; it was not assumed for an arbitrary choice of \(\eta\).

On a Laurent field the Cartier formula is \[\operatorname{Car}\left(\sum_n a_ns^n\frac{ds}{s}\right) =\sum_n a_{pn}^{1/p}s^n\frac{ds}{s}.\] It proves continuity and surjectivity. Its kernel consists of differentials whose logarithmic coefficients at indices divisible by \(p\) vanish; a continuous additive primitive is \(\sum_{p\nmid n}(a_n/n)s^n\). The kernel of \(d\) is \(B^p\), and its root map is continuous. A continuous section of Cartier sends \(\sum a_ns^n ds/s\) to \(\sum a_n^ps^{pn}ds/s\). Multiplication or division by \(\eta\) gives the sections and exactness in (64).

The logarithmic order of a differential is independent of the local uniformizer, since \(ds'/s'\) is a unit multiple of \(ds/s\). Invariance of \(\eta\) therefore makes \(M\) and \(M^0\) invariant under \(H\), including permutations of the field factors. The Cartier formula preserves positive and nonnegative logarithmic order. Furthermore \(M^0/M\cong\prod_j\kappa_j\) as additive groups.

Residue is invariant under coordinate change, and the sum of traces is invariant under residue-field automorphisms and permutations. Thus (65) is \(H\)-invariant. A continuous functional on \(M\) uses finitely many Laurent coefficients; it is represented by an element of \(B/\mathcal O_B\). Conversely every negative principal part is detected by pairing against a suitable positive logarithmic coefficient. This proves the first dual identification. An arbitrary functional on the discrete space \(B/M\) is a sequence of coefficient functionals, represented by a power series in \(\mathcal O_B\), which proves the second. The classical residue adjointness includes a Frobenius twist (Serre 1958, sec. 10, Proposition 9). Here the local normalization gives \[\operatorname{Res}\bigl(g\operatorname{Car}(f\eta)\bigr) =\operatorname{Res}\bigl(\operatorname{Car}(g^pf\eta)\bigr).\] Taking the finite-field trace removes the \(p\)th-root operation on the residue and proves the last identity in (66). ◻

Proof. In a regular nilpotent block, a preimage \(\theta_0\) of \(1\) is killed by \(D^2\) and hence by \(D^p\). The kernel formula puts it in \(L^{1/p}\). Its \(p\)th power and the finite coefficients of \(\eta\) and its inverse belong to one finite separable stage. The action on finite distribution jets is continuous, so the kernel of its action modulo \(D^{p+2}\) is open. Intersect these finite-data stabilizers, take an open normal core, and pass to a still smaller uniform subgroup in the cofinal collection of Theorem 28. This gives all three conditions, cofinally.

If \(b\in B^{1/p}\), then \(D^pb=0\). For \(\sigma\in U\), (68) gives \(\sigma(Db)=Db\); moreover \(Db\) is still killed by \(D^p\). Uniqueness of purely inseparable roots identifies \((L^{1/p})^U=B^{1/p}\), so \(D\) preserves this stage. Continuity follows from Lemma 32 and the induced valuation topology. ◻

Proof. Write \(\partial\) for the cochain differential and \(\mathrm{Fr}\) for coefficient Frobenius. A continuous positive-valued cochain on the compact space \(H^r\) has valuations bounded below by some \(\epsilon>0\). Its iterates under \(\mathrm{Fr}\) therefore tend uniformly to zero. In positive degree, apply Lemma 30 to \(C^\bullet_{\mathrm{cts}}(H,B^\flat)\), whose cohomology is finite by Theorem 28. A sufficiently large Frobenius iterate of a positive cocycle has a primitive with positive values. Frobenius is a homeomorphism of \(B^\flat\) and of its positive lattice; its inverse carries that primitive back to a positive primitive of the original cocycle. In degree zero the finite invariant group is discrete in the cycle topology. A sufficiently large Frobenius iterate of a positive invariant is therefore zero, and injectivity of Frobenius makes the invariant zero.

The residue algebra \(\mathcal O_{B^\flat}/(B^\flat)^{++}\) is a finite product of finite fields. Its cohomology is finite by Lemma 18. The valuation quotients have continuous sections, so the long exact sequences for the positive, integral, and full coefficients prove both finiteness assertions. ◻

Proof. Compactness of the nested sets of preimages of an element of \(I\) gives \(CI=I\). On \(Q=N/I\), the compact images \(C^nQ\) have intersection zero. They eventually lie in any specified neighborhood of zero: otherwise their intersections with the closed complement of that neighborhood would have the finite intersection property. Thus \(C\) is uniformly topologically nilpotent on \(Q\). The formula \[\left(\sum_{n\ge0}a_nz^n\right)x =\sum_{n\ge0}a_nC^nx\] defines a continuous \(\mathbb F_p[[z]]\)-module structure on \(Q\), with \(z=C\). The series converges uniformly, and this also proves continuity.

Choose lifts \(x_1,\ldots,x_r\) of a basis of the finite space \(Q/CQ\). Successively express a vector modulo \(CQ\), then its remainder modulo \(C^2Q\), and so on. The remainder tends uniformly to zero, so every vector is an \(\mathbb F_p[[z]]\)-linear combination of the \(x_j\). Hence \(Q\) is a finitely generated module over the discrete valuation ring \(\mathbb F_p[[z]]\). Its structure theorem makes its \(z\)-power torsion finite. In particular \(\ker(C:Q\to Q)\) is finite, and the finite kernel of \(N\to Q\) then makes \(\ker(C:N\to N)\) finite.

If an element has finite forward orbit, its image in \(Q\) tends to zero through a finite set and is eventually zero. Such images belong to the finite \(z\)-power torsion of \(Q\). The fibers of \(N\to Q\) are finite, proving the orbit assertion.

For the last claim, fix \(x\) in the left side of (69). The sets \(f^{-1}(x)\cap C^nN\) are nonempty nested compact sets: each is a closed subset of the finite fiber \(f^{-1}(x)\). Their intersection is nonempty and lies in \(I\). The reverse inclusion is immediate. ◻

Proof. We first prove countability at the separable stage \(B\). The finite cohomological range lets us argue at a largest degree where countability could fail. At that degree, Cartier will force Frobenius to have countable image; an explicit distribution lift will show that its kernel is countable as well. Once countability is known in every degree, compact duality will give finiteness.

Fix a good stage \(B\) and set \[ N_a=H^a(H,M),\qquad X_a=H^a(H,B),\qquad f_a:N_a\longrightarrow X_a. \tag{70}\] The spaces \(N_a\) are compact Hausdorff by Lemma 29; no separatedness is yet asserted for \(X_a\). Cartier acts on both and commutes with \(f_a\). Since \(B/M\) is countable discrete, its cochain groups are countable. The long exact sequence therefore gives \[ \ker f_a\ \text{and}\ \mathop{\mathrm{coker}}f_a\ \text{are countable}. \tag{71}\]

Step 1: the compact Cartier model. For every degree \(a\), put \[I_a=\bigcap_{n\ge0}C_0^nN_a.\] We claim that \(I_a\) is finite. The completed calculation supplies finite cohomology of the discrete principal-part quotient.

Compact duality and the residue pairing identify \[N_a^\vee=H^{8-a}(H,B/\mathcal O_B),\] and transpose \(C_0\) to Frobenius. There is an identification of discrete \(H\)-modules \[ \mathop{\mathrm{colim}}_{\mathrm{Fr}} B/\mathcal O_B =B^{\mathrm{perf}}/\mathcal O_{B^{\mathrm{perf}}} =B^\flat/\mathcal O_{B^\flat}. \tag{72}\] At the \(e\)th copy of the left side the first map sends \(f\) to \(f^{1/p^e}\); its compatibility is precisely the Frobenius transition. The second equality follows by density: every completed element differs from an element of the root-stage union by an integral element, and integrality on that union is the induced valuation condition.

Continuous cochains into discrete modules commute with filtered colimits, since they have finite image. Consequently \(\mathop{\mathrm{colim}}_{\mathrm{Fr}}N_a^\vee\) is finite by Lemma 36. Dualizing makes \(\varprojlim_{C_0}N_a\) finite. Its projection onto its zeroth coordinate is exactly \(I_a\). To see surjectivity onto \(I_a\), the finite strings of compatible predecessors of any element of \(I_a\) form nonempty compact sets; compactness supplies an infinite compatible string. Thus \(I_a\) is finite.

Step 2: Cartier at a largest uncountable degree. Suppose that some \(X_a\) is uncountable and choose the largest such \(a\); such an \(a\) exists because cohomology vanishes above eight. Split (64) into the two short exact sequences with intermediate module \(V=\mathop{\mathrm{im}}(d/\eta)\). They are exact on cochains. The first sequence shows that \(H^{a+1}(H,V)\) is countable, since it lies between quotients or subgroups of the countable groups \(X_{a+1}\) and \(X_{a+2}\). The second sequence embeds \(X_a/C_0X_a\) in \(H^{a+1}(H,V)\), so this quotient is countable.

Put \(W=\ker f_a\), \(A_a=\mathop{\mathrm{im}}f_a\), and \(E_a=\mathop{\mathrm{coker}}f_a\). The exact sequences for \(C_0\) on \(0\to W\to N_a\to A_a\to0\) and \(0\to A_a\to X_a\to E_a\to0\) show that \(N_a/C_0N_a\) is countable: its successive possible contributions are \(W/C_0W\), \(E_a[C_0]\), and \(X_a/C_0X_a\), all countable by (71) and the countability of \(X_a/C_0X_a\) just proved. This quotient is compact Hausdorff. A countable compact Hausdorff group is finite by Baire’s theorem: if no singleton were open, its countable cover by singletons would be meagre. Lemma 37 now applies to \(N_a\), since Step 1 made its stable Cartier image \(I_a\) finite.

Every element of \(W\) has finite forward Cartier orbit. In degree zero \(W=0\). In positive degree represent it by the connecting image of a cocycle with values in \(B/M\). That cochain has finite image. In the logarithmic Laurent expansion of any of its finitely many values, only finitely many negative indices occur. Repeated Cartier eventually removes all of them, leaving values in the finite module \(M^0/M\). The resulting cohomology classes belong to the image of the finite group \(H^{a-1}(H,M^0/M)\). Cartier preserves this image. Hence the forward orbit is finite, and Lemma 37 makes \(W\) finite.

The same lemma gives a finite kernel of \(C_0\) on \(N_a\). The kernel on \(A_a\) is then finite, by the exact sequence with the now finite group \(W/C_0W\). The kernel on \(X_a\) maps into the countable group \(E_a[C_0]\), so it is countable. Finally, the finite kernel of \(f_a\) allows us to apply (69), giving \[ \bigcap_{n\ge0}f_a(C_0^nN_a)=f_a(I_a), \tag{73}\] a finite group. This will control the Frobenius-boundary presentations in the next step.

Step 3: the Frobenius kernel at that degree. We now prove that \(\ker(\mathrm{Fr}:X_a\to X_a)\) is countable. A Frobenius boundary has a presentation in \(B^{1/p}\); applying \(D\) turns it into a cocycle. Its principal part has only countably many possibilities. The remaining positive-valued presentations will all map into the fixed finite group in (73).

In degree zero the kernel is zero by injectivity of coefficient Frobenius. For \(a>0\), consider the additive presentation group \[\mathcal P_a= \{b\in C^{a-1}_{\mathrm{cts}}(H,B^{1/p}): \partial b\in C^a_{\mathrm{cts}}(H,B)\}.\] The map \(b\mapsto[\partial b]\) takes values in the Frobenius kernel, since \((\partial b)^p=\partial(b^p)\) and \(b^p\) is \(B\)-valued. It surjects onto that kernel: if \(\mathrm{Fr}(\alpha)=\partial c\) for a \(B\)-valued cocycle \(\alpha\), take \(b=c^{1/p}\). For any presentation put \(w=Db\). The good-stage property puts \(w\) in \(C^{a-1}_{\mathrm{cts}}(H,B^{1/p})\), and \(\partial w=0\) because \(D\) kills \(B\) and commutes with \(H\).

The quotient \(B^{1/p}/(B^{1/p})^{++}\) is countable discrete. Thus the map \[\mathcal P_a\longrightarrow C^{a-1}_{\mathrm{cts}} (H,B^{1/p}/(B^{1/p})^{++}),\qquad b\longmapsto Db\] has countable image. Its kernel \(\mathcal P_a^+\), consisting of presentations with positive \(w\), has countable index. It suffices to show that \(\mathcal P_a^+\) has finite image in \(X_a\).

Fix \(b\in\mathcal P_a^+\). If \(w=0\), then \(b\) is \(B\)-valued by (57), and \([\partial b]=0\). Otherwise compactness gives a uniform lower bound \(v(w)\ge\epsilon>0\) on all field factors and cochain values. Let \(i\) be positive, even, and divisible by \([k:\mathbb F_p]\), and put \(q=p^i\). With the good-stage element \(\theta_0\) define \[ b'=\theta_0^{1/q}w \in C^{a-1}_{\mathrm{cts}}(H,B^{1/(pq)}). \tag{74}\] Since \(w\in L^{1/q}\), the Frobenius identities give \(D^qb'=w\). Therefore \(\partial b'\) is killed by \(D^q\), so it belongs to \(L^{1/q}\). It is also fixed by \(U\), hence is \(B^{1/q}\)-valued.

Similarly \(e=D^{q-1}b'-b\) is killed by \(D\). It is fixed by \(U\) as well. Indeed (68) gives \[\sigma(D)=D(1+h),\quad h\in D^{p+1}k[[D]], \qquad \sigma(D^{q-1})-D^{q-1}\in D^{q+p}k[[D]].\] The last ideal kills \(b'\), since \(D^{q+p}b'=D^pw=0\). Thus \(e\) is \(B\)-valued. Every operation here is continuous at a finite marking and root stage by Lemma 32; once its values have been shown to lie in \(B\) or \(B^{1/q}\), the induced topology makes it a continuous cochain there. The auxiliary larger stage may depend on \(i\).

Set \(\alpha_i=(\partial b')^q\), a \(B\)-valued cocycle. The equation \(\partial e=D^{q-1}\partial b'-\partial b\) and Lemma 34 give \[ [\partial b]=c_i C_0^i[\alpha_i]\quad\text{in }X_a. \tag{75}\] Normalize valuations by their restriction to \(K\) and take their minimum over the factors. The \(H\)-action preserves these valuations, so the cochain differential does not lower the minimum. Formula (74) yields \[v(\alpha_i)\ge q\epsilon+v(\theta_0).\] For every sufficiently large admissible \(i\) this exceeds the fixed bounds defining \(M\), and \(\alpha_i\) is \(M\)-valued. Since \(c_i\in k^\times\), \(c_iC_0^i(\alpha_i)=C_0^i(c_i^q\alpha_i)\), with \(c_i^q\alpha_i\) still in \(M\). It follows that \([\partial b]\in f_a(C_0^iN_a)\) for a cofinal sequence of \(i\). These images are nested, so (73) puts every such class in the fixed finite group \(f_a(I_a)\). This proves that the positive presentation subgroup has finite image. Its countable index therefore makes the Frobenius kernel countable, as required.

Step 4: from countability to finiteness. At the degree chosen in Step 2, \(C_0\mathrm{Fr}=0\) puts the Frobenius image inside the countable group \(\ker(C_0:X_a\to X_a)\). Step 3 makes its kernel countable. Thus \(X_a\) is countable, contrary to its choice. Every \(X_a\) is therefore countable.

For every degree \(a\), (71) now makes \(N_a\) countable. It is compact Hausdorff, hence finite. Any \(H\)-stable valuation-order lattice is commensurable with \(M\), with finite quotients after intersecting the two lattices. Long exact sequences and finite-coefficient cohomology therefore give finite cohomology for every such lattice, including \(\mathcal O_B\). The second residue duality in (66), followed by Lemma 29, makes \(H^a(H,B/M)\) finite. The long exact sequence for \(0\to M\to B\to B/M\to0\) now makes each \(X_a\) finite.

Finally \(p^e\)th power is an additive \(H\)-equivariant homeomorphism \(B^{1/p^e}\to B\). It takes valuation-order lattices to valuation-order lattices and transports all the asserted finiteness statements. ◻

Proof. First fix a good stage \(B\) and take \(T\) to be a point. Proposition 38 gives finite cohomology of \(B^{++}\). A positive cocycle tends uniformly to zero under successive Frobenius powers. By Lemma 30, its sufficiently large iterates are boundaries in the positive complex. Thus Frobenius acts nilpotently on each finite group \(H^a(H,B^{++})\). The same conclusion in degree zero follows from discreteness of the finite invariant group and injectivity of Frobenius. Transporting root-stage inclusions by their power homeomorphisms identifies their action on cohomology with Frobenius. It follows that \[ \mathop{\mathrm{colim}}_eC^\bullet_{\mathrm{cts}}(H,(B^{1/p^e})^{++}) \quad\text{is acyclic}. \tag{78}\] The completed positive complex is acyclic by Lemma 36.

Density gives an isomorphism of discrete \(H\)-modules \[ B^{\mathrm{perf}}/(B^{\mathrm{perf}})^{++} \xrightarrow{\ \sim\ } B^\flat/(B^\flat)^{++}. \tag{79}\] Surjectivity follows by approximating a completed element with error of strictly positive valuation; injectivity is the induced valuation condition. A continuous map from a compact space into either quotient has finite image. Its finitely many values lift to one root stage on the left and to the completed algebra on the right. Hence the coefficient sequences with positive kernels are exact on the prescribed cochains. The acyclicity of their kernels and (79) prove the fixed-\(B\) comparison.

Now allow arbitrary compact profinite \(T\). All finite-root positive complexes have finite cohomology, so Lemma 31 upgrades (78) to the corresponding parameterized assertion. The same lemma upgrades completed positive acyclicity. The finite-image lifting argument for (79) applies to \(T\times H^r\) without any countability assumption on \(T\). It proves the fixed-\(B\) comparison with these parameters.

Taking the filtered colimit over the cofinal family of good stages proves (76). The comparison itself is the coefficient inclusion on the full diagram of marking and root stages. The differential \(\eta\), the parameter \(D\), and the good subfamily were used only to prove it is a quasi-isomorphism. Consequently its naturality for the full \(P\)- and \(J\)-actions is unaffected by these choices or by conjugating to another cofinal family. Finally apply Theorem 28 to obtain (77) and all the stated actions. ◻

Proof. We give a bar-resolution construction that works on each fixed coefficient stage. Put \[I^j(M)=C_{\mathrm{cts}}(G^{j+1},M),\qquad (g f)(g_0,\ldots,g_j) =g\,f(g^{-1}g_0,\ldots,g^{-1}g_j),\] with the alternating omission differential. A section \(s:Q\to G\), chosen with \(s(1)=1\), gives an \(N\)-equivariant continuous retraction \(r(g)=g\,s(\overline g)^{-1}\) from \(G\) to \(N\). Restriction to \(N^{j+1}\) and pullback along \(r\) compare \(I^\bullet(M)^N\) with homogeneous \(N\)-cochains. They are mutually inverse up to a continuous cochain homotopy: the usual prism inserts, one vertex at a time, the vertices \(r(g_i)\) in place of \(g_i\). In degree \(j\) this is the finite alternating sum obtained by evaluating at \[(g_0,\ldots,g_i,r(g_i),\ldots,r(g_{j-1})), \qquad 0\le i<j.\] Cancellation of adjacent omissions gives the homotopy identity. It preserves a fixed coefficient stage and commutes with a profinite parameter. Thus \(I^\bullet(M)^N\) computes the required \(N\)-cochains.

Each \(I^j(M)^N\) is coinduced as a \(Q\)-module. Explicitly, the function \[F(\overline g_0;r_1,\ldots,r_j) =g_0^{-1}f(g_0,g_0r_1,\ldots,g_0r_j)\] identifies it with \(C_{\mathrm{cts}}(Q,C_{\mathrm{cts}}(G^j,M))\). The expression is independent of the lift \(g_0\) by \(N\)-invariance; \(Q\) acts by left translation on its first variable. Continuity in both directions follows from the chosen section. The positive \(Q\)-cohomology of this module vanishes: in the homogeneous bar complex, evaluation after inserting the identity in the free \(Q\)-coordinate is a continuous contracting homotopy. The degree-zero invariants are \(I^j(M)^G\).

Consider the first-quadrant double complex \[C^s_{\mathrm{cts}}(Q,I^t(M)^N).\] Taking horizontal cohomology first therefore identifies its total cohomology with that of \(I^\bullet(M)^G\), namely continuous \(G\)-cohomology. Taking vertical cohomology first uses the stipulated family property, with parameter \(Q^s\), and gives \(C^s_{\mathrm{cts}}(Q,V_t)\). This proves (80). In each total degree there are only finitely many bidegrees, so the standard filtration converges.

All comparisons, prism sums, and contractions use continuous evaluation, group operations, and finite sums at one coefficient stage. For a stage union, perform them before taking the filtered colimit; this preserves their identities and exactness. Replace every parameter \(Q^s\) by \(T\times Q^s\) to obtain the parameterized assertion. A map that is a quasi-isomorphism for all these \(N\)-parameters is an isomorphism on the vertical cohomology pages, hence on total cohomology. This proves the final assertion. ◻

Proof. The transformation law of the distribution parameter gives the \(H'\)-equivariant coefficient sequence \[ 0\longrightarrow L\longrightarrow R \xrightarrow{D}R(\chi)\longrightarrow0. \tag{83}\] Here \(R(\chi)\) means that \(h\in H'\) acts as \(\chi(h)\) times its action on \(R\). The kernel, surjectivity, and continuous lifts at finite larger stages, including arbitrary profinite parameters, were proved in Lemma 32. Thus (83) is exact on our \(H'\)-cochain complexes.

By Theorem 39, the two nonzero \(H\)-cohomology groups of \(R\) have trivial \(\mu_2\)-action. After twisting by \(\chi\) they have the nontrivial order-two action. Averaging over \(\mu_2\) is exact because \(2\) is invertible in \(k\). Applying Lemma 40 to \(H'=H\times\mu_2\), with all profinite parameters, shows that \(C^\bullet_{\mathrm{cts}}(H',R(\chi);T)\) is acyclic. The long exact sequence of (83) therefore makes \[C^\bullet_{\mathrm{cts}}(H',L;T)\longrightarrow C^\bullet_{\mathrm{cts}}(H',R;T)\] a quasi-isomorphism for every \(T\). Its map is the natural inclusion, so it retains the commuting \(P\)- and \(J\)-actions although the auxiliary operator \(D\) need not do so. Apply the comparison assertion of Lemma 40 to \(H'\subset P\) to obtain (81).

It remains to calculate the cohomology of \(R\). Exact averaging gives \(H^tC^\bullet_{\mathrm{cts}}(H',R)=k\) for \(t=0\), \(\ell\) for \(t=3\), and zero otherwise. The quotient \(Q=P/H'\cong\mathbb Z_3\) acts trivially on both lines by Theorem 39. The two-term completed-group-ring resolution \[0\longrightarrow\mathbb F_p[[Q]] \xrightarrow{\gamma-1}\mathbb F_p[[Q]] \longrightarrow\mathbb F_p\longrightarrow0\] for a topological generator \(\gamma\) computes \(H^s(Q,V)=V\) in degrees \(s=0,1\) and zero otherwise when \(Q\) acts trivially on \(V\). The spectral sequence (80) thus has exactly the four terms in (82); no differential is possible, and each total degree contains one term. The same resolution and the parameterized version of Lemma 40 give the formula for arbitrary \(T\).

Finally \(J\) commutes with \(P\) and hence acts trivially on the quotient \(Q\) and its degree-one cohomology line. The low-degree coefficient line consists of constants. The upper coefficient line has the action specified in Theorem 39. This proves all assertions about the \(J\)-action. ◻

Proof. Finite constants and the selected Morava factor. The classification of étale algebras over a ring spectrum (Mathew 2016, Definition 2.31 and Theorem 2.32) lifts \(W(k)/\mathbb Z_p\) and gives the first identity in (84). Represent a \(\mathbb Z_p\)-basis of \(W(k)\) by elements of \(\pi_0S_k\). The resulting map \(S^{\oplus d}\to S_k\) is an isomorphism on every homotopy group, proving the second assertion. The étale mapping property stated immediately after Theorem 2.32 in Mathew (2016) lifts the coefficient inclusion into \(\pi_0E_i(k)\) uniquely and supplies its compatibility with automorphisms. In particular, \(Z_k\simeq (L_{K(3)}S)^{\oplus d}\) as an underlying spectrum.

Here is the constant-field argument for descent. Put \(k_0=\mathbb F_{p^3}\), \(E^{(0)}=E_3(k_0)\), and \(T=L_{K(3)}S\). The algebra \(E^{(0)}\) is descendable over \(T\) (Mathew 2016, Theorem 4.18 and Proposition 10.10). The images of \(S\) and of the sphere in this category agree, since completion of the sphere is a mod-\(p\) equivalence. Base change preserves descendability (Mathew 2016, Corollary 3.21); hence \[\widetilde E=Z_k\otimes^{K(3)}_T E^{(0)}\] is descendable over \(Z_k\). Finite étale scalar extension and \[W(k_0)\otimes_{\mathbb Z_p}W(k) =\prod_{\alpha:k_0\hookrightarrow k}W(k)\] give a product decomposition \[\widetilde E\simeq\prod_{\alpha:k_0\hookrightarrow k}E_\alpha, \qquad E_\alpha\simeq E_3(k).\] Each factor realizes the universal deformation after the indicated residue-field extension. Frobenius in the extended stabilizer of \(E^{(0)}\) induces a \(Z_k\)-linear automorphism of \(\widetilde E\) that cycles its three factors. Consequently \(\widetilde E\) and any selected \(E_\alpha\) generate the same thick tensor ideal of \(Z_k\)-modules: one direction uses the product projections, and the other uses the equivalences of the three factors. This proves descendability of the selected \(E_3(k)\). In particular, no averaging over a group of order three is involved.

The unextended cooperation maps. The factor selection also determines the group and the maps in (87). Write \(G=P\rtimes\operatorname{Gal}(k_0/\mathbb F_p)\). The completed Morava cooperation theorem (Devinatz and Hopkins 2004, Proposition 2.2 and Equations (2.3) and (2.5)), followed by the finite scalar extension above, gives \[\pi_*\bigl(\widetilde E^{\otimes^{K(3)}_{Z_k}(s+1)}\bigr) =\operatorname{Map}_{\mathrm{cts}}(G^s,\widetilde E_*).\] Finite free scalar extension commutes with these continuous-function modules. We spell out the conversion from the cited inverse-action coordinates. Write \(\mu\) for iterated multiplication. In the coordinates \(h_1,\ldots,h_s\) of Devinatz and Hopkins (2004, Equation (2.5)), evaluation is \[\mu\circ(h_1^{-1}\otimes\cdots\otimes h_s^{-1}\otimes1),\] where \(1\) denotes the identity action. For \(a_r=g_1\cdots g_r\) and \(a_0=1\), put \(h_i=a_{i-1}^{-1}a_s=g_i\cdots g_s\) and postcompose by \(a_s\). Multiplicativity of the action gives the identity of evaluation composites \[ a_s\circ\mu\circ(h_1^{-1}\otimes\cdots\otimes h_s^{-1}\otimes1) =\mu\circ(1\otimes a_1\otimes\cdots\otimes a_s). \tag{88}\] This is a continuous invertible coordinate change, with \(g_i=h_i h_{i+1}^{-1}\) and \(h_{s+1}=1\); it moves the coefficient action from the last factor to the first. A pure tensor is therefore sent to \[(g_1,\ldots,g_s)\longmapsto b_0\,g_1(b_1)\,(g_1g_2)(b_2)\cdots(g_1\cdots g_s)(b_s).\] The same identity of evaluation composites checks the bar operations. For a cochain \(f\), unit insertion in the first slot gives \(g_1 f(g_2,\ldots,g_{s+1})\); an interior insertion gives \(f(g_1,\ldots,g_i g_{i+1},\ldots,g_{s+1})\); and insertion in the last slot gives \(f(g_1,\ldots,g_s)\). Multiplication of neighboring slots gives insertion of the identity among the group variables. These are the actual cofaces and codegeneracies after the coordinate change. Selecting the same factor idempotent in all \(s+1\) slots first selects that value factor and then requires every cumulative product \(g_1\cdots g_r\) to preserve it. The stabilizer of the factor is \(P\); these conditions are equivalent to \(g_r\in P\) for every \(r\). This yields (87). The selection commutes with multiplication, units, and evaluation, so the resulting faces are the actual bar faces, not merely maps with the same groups of coefficients. ◻

Proof. The assertion about the indexing functor is the left-cofinality theorem of Sinha (2009, Definitions 6.2–6.3 and Theorem 6.7). We include a proof that records the role of codegeneracies. For \(0\leq m\leq r\), the comma category \((g_r\downarrow[m])\) consists of nonempty subsets \(S\subseteq[r]\) with weakly increasing labelings \(S\to[m]\); its arrows extend labelings. It is the nonempty face poset of the finite simplicial complex \(K_{r,m}\) with simplices \[\{(i_0,j_0),\ldots,(i_q,j_q)\},\qquad i_0<\cdots<i_q,\quad j_0\leq\cdots\leq j_q.\] This complex is flag. In a flag complex one may delete a vertex \(v\) if a neighboring vertex \(w\) is adjacent to every other neighbor of \(v\): sending \(v\) to \(w\) and fixing the other vertices gives a simplicial retraction whose composite with the inclusion is contiguous to the identity.

For \(j=m,m-1,\ldots,1\), delete \((i,j)\) for \(i=0,\ldots,j-1\) in that order, using \(w=(i+1,j)\). This vertex exists since \(j\leq m\leq r\). The only neighbors of \((i,j)\) possibly not adjacent to \(w\) are \((i+1,k)\) with \(k>j\); they have already been deleted at the earlier value \(k\). Next, for \(j=0,1,\ldots,m\), delete \((i,j)\) for \(i=r,r-1,\ldots,j+1\), using \(w=(i-1,j)\). The only possible exceptions are now \((i-1,k)\) with \(k<j\); they were deleted at the earlier value \(k\). Each chosen \(w\) is still present. The vertices left are \((0,0),\ldots,(m,m)\), which span a simplex. Thus \(K_{r,m}\), and hence the nerve of \((g_r\downarrow[m])\), is contractible. The weak label inequalities here include noninjective simplex maps; this is the check for the full truncation.

The opposite form of the \(\infty\)-categorical cofinality criterion (Lurie 2009, Theorem 4.1.3.1) makes \(g_r\) initial for limits. The dual of (Lurie 2009, Proposition 4.1.1.8) gives (89). The poset on its right has a finite nerve, since strict chains of its subsets have length at most \(r\). For the inclusions of initial segments, the squares formed by \(g_r\) and \(g_{r+1}\) commute strictly. The displayed equivalences, being restriction maps on cones, consequently commute coherently with the tower maps. Adding the empty subset, sent to \([-1]\) in the augmented simplex category, shows the same compatibility with the augmentation. Finally, an exact functor preserves the finite limit on the right, naturally in these diagrams. Applying the comparison in source and target proves the last assertion. ◻

Proof. The Amitsur error tower. Let \(T_r=\mathop{\mathrm{Tot}}_r C^\bullet\). In the stable category of \(Z_k\)-modules in \(K(3)\)-local spectra, form the external tensor cube \[Q_r(S)=\bigotimes_{i=0}^{r} Z_i(S),\qquad Z_i(S)= \begin{cases}E_3,&i\in S,\\ Z_k,&i\notin S,\end{cases} \qquad S\subseteq[r],\] where the tensors are the \(K(3)\)-local relative tensors over \(Z_k\). An edge applies the unit \(Z_k\to E_3\) in its ordered slot. More generally, the Amitsur map for a weakly increasing simplex map multiplies the factors in each ordered fiber and inserts a unit for an empty fiber; its codegeneracies are the multiplication maps. The restriction along \(g_r\) uses precisely the unit insertions. The associativity and unit equivalences therefore identify it with the punctured cube \(Q_r|_{\mathcal P_0([r])}\) as a coherent diagram, and the empty vertex is the actual augmentation \(Z_k\). Lemma 43 gives \[T_r\simeq\lim_{\varnothing\ne S\subseteq[r]}Q_r(S)\] as an augmented tower. For \(r=0\) the limit is \(E_3\); for \(r=1\) it is the homotopy pullback of \(E_3\to E_3\otimes^{K(3)}_{Z_k}E_3\leftarrow E_3\), with ordered maps \(b\mapsto b\otimes1\) and \(b\mapsto1\otimes b\).

Put \(I=\operatorname{fib}(Z_k\to E_3)\). Iterating fibers in the tensor cube, using exactness of the tensor in each factor, gives \[ \operatorname{fib}(Z_k\to T_r) \simeq I^{\otimes(r+1)}. \tag{92}\] Indeed, after separating the last coordinate, the total fibers of the two faces are \(I^{\otimes r}\) and \(E_3\otimes I^{\otimes r}\); their fiber is \(I^{\otimes(r+1)}\). The case \(r=0\) is the definition of \(I\). Naturality of this iterated-fiber construction identifies a tower transition with applying \(I\to Z_k\) in the last slot.

Take the cofiber of the augmentation in the category of towers, \[Z_k\longrightarrow T_r\longrightarrow U_r.\] Thus \(U_r\simeq\Sigma I^{\otimes(r+1)}\) compatibly with transitions. Multiplication retracts \(E_3\simeq E_3\otimes Z_k\to E_3\otimes E_3\). Applying that retraction to the defining fiber nullhomotopy shows that \(E_3\otimes I\to E_3\) is null. Therefore each transition \(U_r\to U_{r-1}\) becomes null after tensoring with \(E_3\). Descendability supplies a bound on the length of any composite of such maps (Mathew 2016, Proposition 3.27 and Corollary 4.4). Thus there is \(N\) such that \(U_{r+N}\to U_r\) is null for every \(r\).

Passing the tower through the exact test. This uniform bound is the descent input that will survive the change of height. The test need only preserve the finite limits defining the partial totalizations. Lemma 43 identifies \(\mathcal T(T_r)\) with \(\mathop{\mathrm{Tot}}_r\mathcal T(C^\bullet)\) as augmented towers: exactness moves \(\mathcal T\) through the finite punctured-cube limit. Here \(\mathcal T\) is applied to the already formed source cube; no monoidal property of the test is needed. Exactness then gives the cofiber tower \[\mathcal T(Z_k)\longrightarrow \mathop{\mathrm{Tot}}_r\mathcal T(C^\bullet)\longrightarrow\mathcal T(U_r),\] and every composite \(\mathcal T(U_{r+N})\to\mathcal T(U_r)\) is still null. Increase \(N\) to at least one if necessary, and write \[X=\mathcal T(Z_k),\qquad V_s=\mathop{\mathrm{Tot}}_s\mathcal T(C^\bullet),\qquad W_s=\mathcal T(U_s).\] The inverse limit of \(W_s\) is zero: applying mapping spectra gives pro-zero homotopy groups and zero \(\lim^1\). Taking the inverse limit of the displayed cofiber sequences gives \(X\simeq\lim_sV_s\) and proves (90). The tower \(\{V_s\}\) is strongly constant in the sense of (Mathew 2016, Definition 4.2), and (Mathew 2016, Proposition 4.3) supplies a horizontal vanishing line at a finite page.

The filtration of actual homotopy. We derive the abutment filtration and strong convergence directly from the same nilpotence bound. This will let us use a surviving term as a subquotient of the tested homotopy group. For any homotopy degree \(q\) and \(u\ge s+N\), the cofiber triangles give \[\operatorname{im}(\pi_qV_u\longrightarrow\pi_qV_s) =\operatorname{im}(\pi_qX\longrightarrow\pi_qV_s)=:M_s^q.\] Indeed, the image of a class from \(\pi_qV_u\) in \(\pi_qW_s\) is zero, because it factors through the null transition \(W_u\to W_s\); exactness then puts its image in the image of \(\pi_qX\). Conversely, the augmentation to \(V_s\) factors through every \(V_u\). Hence each homotopy tower is Mittag–Leffler, and its \(\lim^1\) vanishes. The Milnor exact sequence therefore identifies \(\pi_qX\) with \(\lim_s\pi_qV_s\).

Extend \(V_s=0\) for \(s<0\) and give this actual homotopy group the filtration \[F^s\pi_qX=\ker(\pi_qX\longrightarrow\pi_qV_{s-1})\qquad(s\ge0).\] Here \(F^0\pi_qX=\pi_qX\). Exactness identifies \(F^{s+1}\pi_qX\) with the image of the boundary \(\delta_s:\pi_{q+1}W_s\to\pi_qX\). Naturality gives \(\delta_N=\delta_0\circ\pi_{q+1}(W_N\to W_0)=0\), so \(F^{N+1}\pi_qX=0\) for every \(q\).

Use total-degree grading \(\widetilde E_r^{s,q}=\mathsf E_r^{s,q+s}\) in the exact couple of the tower. Its first couple has \[D_1^{s,q}=\pi_qV_s,\qquad \widetilde E_1^{s,q}= \pi_q\operatorname{fib}(V_s\longrightarrow V_{s-1}).\] The long exact fiber sequences give maps \(i_1:D_1^{s,q}\to D_1^{s-1,q}\), \(j_1:D_1^{s,q}\to\widetilde E_1^{s+1,q-1}\), and \(\epsilon_1:\widetilde E_1^{s,q}\to D_1^{s,q}\). Deriving the couple replaces \(D\) by the image of \(i\). Induction therefore gives \[D_r^{s,q}=\operatorname{im}(\pi_qV_{s+r-1}\longrightarrow\pi_qV_s),\] with maps \[i_r:D_r^{s,q}\to D_r^{s-1,q},\qquad j_r:D_r^{s,q}\to\widetilde E_r^{s+r,q-1},\qquad \epsilon_r:\widetilde E_r^{s,q}\to D_r^{s,q},\] and differential \(\widetilde d_r=j_r\epsilon_r\). In the original grading this has bidegree \((r,r-1)\).

Put \(M_s^q=0\) for \(s<0\). For \(r\ge N+1\), the stable-image equality says \(D_r^{s,q}=M_s^q\) for every \(s\). The transitions \(M_s^q\to M_{s-1}^q\) are surjective because both images come from \(\pi_qX\). Exactness of the derived couple now makes every \(j_r\) zero and identifies \[\widetilde E_r^{s,q}\xrightarrow[\epsilon_r]{\ \sim\ } \ker(M_s^q\longrightarrow M_{s-1}^q).\] Indeed, in the exact segment \(D_r^{s-r,q+1}\to\widetilde E_r^{s,q}\to D_r^{s,q}\to D_r^{s-1,q}\), the first arrow is zero since the preceding \(i_r\) is surjective. Thus the pages stabilize from \(r=N+1\). The augmentation maps \(F^s\pi_qX\) onto the displayed kernel with kernel \(F^{s+1}\pi_qX\), so in the original grading \[\mathsf E_\infty^{s,t} \cong F^s\pi_{t-s}X/F^{s+1}\pi_{t-s}X\qquad(s\ge0).\] This is an exhaustive, separated, and complete finite filtration of the actual group \(\pi_{t-s}(Y/p)\): it starts at \(F^0\) and ends at \(F^{N+1}=0\). The stable page consequently vanishes for \(s\ge N+1\), and the Milnor term above is zero. This proves strong convergence with the asserted actual abutment filtration. The argument uses only exactness of \(\mathcal T\) through the finite cube, not preservation of arbitrary inverse limits.

Coefficient actions and the diagonal summand. The maps in the tested cosimplicial object are \(E_2\)-linear and compatible with the action of \(J\). Lemma 2 makes their homotopy groups, and hence the spectral sequence, \(k[[x]]\)-linear. The second coefficient action comes from \(S_k\) on \(C^\bullet\) and commutes with the first. After reduction modulo \(p\) the two actions give an action of \[k\otimes_{\mathbb F_p}k\simeq \prod_{\sigma\in\operatorname{Gal}(k/\mathbb F_p)}k.\] Its idempotent for \(\sigma=\mathop{\mathrm{id}}\) is fixed by \(P\) and \(J\). It commutes with the tower maps, all differentials, and the maps defining the abutment filtration. Taking its image proves the diagonal assertions. This construction only needs an idempotent on mod-\(p\) homotopy groups; it does not identify the two spherical scalar actions on an ordinary smash product before reduction. ◻

Proof. Evenness and flatness before localization. The Landweber exact cooperation formula expresses \((E_2)_*(E_3)\) as the two-sided base change of the formal-group isomorphism Hopf algebroid. It is even and flat over either coefficient ring. These are the usual ordinary Landweber cooperations; see (Hovey and Strickland 1999, Theorem 2.7 and Propositions 2.12 and 2.16). For flatness over both coefficient rings, apply Proposition 2.16 with the two Morava spectra in each order and use smash symmetry. We justify this ordinary tensor calculation by a finite resolution. By (Hovey and Strickland 1999, Proposition 2.12), \(E_2\) is evenly generated; (Hovey and Strickland 1999, Proposition 2.16), applied with ring spectrum \(E_3\), then gives a graded projective resolution \[0\longrightarrow P_1\longrightarrow P_0 \longrightarrow\pi_*(E_2\wedge E_3)\longrightarrow0.\] The projectives may be taken in even degrees. Realize \(P_0\) as a retract \(Q_0\) of a wedge of even suspensions of \(E_3\), and realize the surjection by an \(E_3\)-module map \(Q_0\to E_2\wedge E_3\). Its fiber has homotopy \(P_1\), so realizing that projective similarly identifies the fiber with a projective \(E_3\)-module \(Q_1\). Thus \(Q_1\to Q_0\to E_2\wedge E_3\) is a finite cofiber sequence. Tensor this sequence over \(E_3\) with \(C^s\). For each \(Q_i\), its homotopy is the algebraic tensor product \(P_i\otimes_{E_{3*}}\pi_*C^s\), since \(Q_i\) is a retract of a wedge of suspensions of \(E_3\). Flatness of \(\pi_*(E_2\wedge E_3)\) over \(E_{3*}\) makes the map between these two tensor products injective. We also have \[E_2\wedge C^s\simeq(E_2\wedge E_3)\otimes_{E_3}C^s.\] The long exact homotopy sequence therefore gives (94) and evenness. This finite argument supplies the tensor calculation without a convergence assumption on an unbounded periodic Künneth spectral sequence.

We also need flatness of the second factor in (94). The compact analytic space \(P^s\) has a countable cofinal system of finite clopen partitions. The modules of functions constant on these partitions are finite free over \(E_{3,0}\). A refinement map is a split inclusion with free cokernel, so their union is free. Its maximal-ideal completion is \(\operatorname{Map}_{\mathrm{cts}}(P^s,E_{3,0})\): reduction modulo any power of the maximal ideal is a locally constant function, and compatible reductions recover a continuous function. Completion of a flat module over a Noetherian ring is flat. Thus this continuous-function module is flat over \(E_{3,0}\).

To see flatness of the full tensor product over \(E_{2*}\), first tensor an exact sequence of \(E_{2*}\)-modules with \((E_2)_*(E_3)\), and then tensor the resulting exact sequence of \(E_{3*}\)-modules with the continuous-function module. Both operations are exact. It follows that \(p,x^m\) is a regular quotient calculation on (94). The same holds after taking the diagonal idempotent. Flatness over \(A_m\) follows by base change.

The residue algebra as a marking algebra. Choose complex orientations. Quillen’s universality theorem identifies the complex-cobordism formal group with the universal formal group (Quillen 1969, sec. 3, Theorem 2). We use its strict-isomorphism Hopf algebroid as in (Powell 2012, sec. 3.1), with the inverse series when the direction of the isomorphism is reversed. The ordinary cooperation identity is \[(E_2)_*E_3=(E_2)_*\otimes_{MU_*}MU_*MU \otimes_{MU_*}(E_3)_*, \qquad MU_*MU=MU_*[b_1,b_2,\ldots],\] with the two unit maps of the formal-group Hopf algebroid. Both tensor products are algebraic. This follows by applying Landweber exactness (Hovey and Strickland 1999, Theorem 2.7) first to \((E_2)_*(E_3)\) and then, using symmetry, to \(MU_*(E_3)=(E_3)_*(MU)\).

Choose periodicity generators \(u_i\in\pi_2E_i\). If \(b(Z)=Z+\sum_{r\geq1}b_rZ^{r+1}\) is the strict graded isomorphism from the right formal law to the left, its degree-zero form is \[f(X)=u_2b(u_3^{-1}X),\qquad f'(0)=c=u_2/u_3.\] Conversely, any degree-zero isomorphism with invertible derivative \(c\) recovers \(u_3=u_2/c\) and the strict isomorphism \(u_2^{-1}f(u_3Z)\). The periodicity ratio therefore supplies exactly the arbitrary linear coefficient of a formal-law isomorphism.

Reduce the left coefficients by \((p,x)\) and select the diagonal copy of \(k\). The left law is \(\Gamma_2\), with \([p]_{\Gamma_2}(X)=X^{p^2}\). For the right law \(F_3\) over \(A=k[[a,t]]\), compare coefficients in \[f([p]_{F_3}(X))=f(X)^{p^2}.\] The coefficient of \(X^p\) gives \(ca=0\), hence \(a=0\). The coefficient of \(X^{p^2}\) then gives \(ct=c^{p^2}\), so \(t=c^{p^2-1}\) is invertible. The right coefficient ring is consequently \(A/(a)[1/t]=K\). The remaining coefficient relations classify exactly the isomorphisms from the connected formal law of \(\mathcal G_K\) to \(\Gamma_2\). By the construction of the connected-marking torsor, their algebra is \(L=\mathop{\mathrm{colim}}_U L^U\): each element uses finitely many marking coefficients and lies in one finite étale marking algebra. This is the algebraic union, with no additional completion.

Continuous functions at a finite marking stage. On the continuous-function factor, reduction modulo \((p,a)\) gives the continuous functions into \(k[[t]]\). Indeed coefficientwise lifts give surjectivity, and division of an element in the kernel by \(p\) or \(a\) is continuous with a bounded shift of adic order. After inverting \(t\) the result is \(\operatorname{Map}_{\mathrm{cts}}(P^s,K)\): the compact image of a continuous \(K\)-valued function has bounded pole order, giving one common power of \(t\) as denominator.

Finally, every finite étale marking stage is a finite-dimensional \(K\)-algebra with its usual finite-dimensional topology. A choice of \(K\)-basis therefore identifies continuous functions into that stage with its tensor product with \(\operatorname{Map}_{\mathrm{cts}}(P^s,K)\). Passing to the algebraic union proves (95). Every cooperation element uses only finitely many marking coefficients, so this is a union of complete finite stages, with no completion of the infinite extension. Evaluation and transport of the height-three law are exactly the cooperation faces, giving the asserted compatibility. ◻

Proof. For \(E_2\)-modules, \(K(2)\)-localization is derived completion at \((p,x)\): it is the inverse limit of the successive regular-ideal quotients (Hovey 2004, Proposition 2.2 and Theorem 2.3); compare (Hovey and Strickland 1999, Proposition 7.10(e)). Here is the comparison for the finite extension of constants. In Section 3 the height-two parameter was chosen already over \(\mathbb F_{p^2}=\mathbb F_9\), and \(E_2(k)\) uses its scalar-extended universal deformation. The map \(E_2(\mathbb F_9)\to E_2(k)\) therefore sends the chosen regular system \((p,x)\) to the same named system. Apply the standard-coefficient theorem to an \(E_2(k)\)-module restricted along this map, using the regular ideals \((p^r,x^r)\), whose intersection is zero. The underlying spectrum and its \(K(2)\)-localization do not change under restriction. Multiplication by \(p^r\) and \(x^r\) is the same map on that spectrum, so its derived quotients and their transition maps are also the same. This proves the same completion description for \(E_2(k)\)-modules. Exactness first permits reduction modulo \(p\) before localization. On this reduction \(p\) acts null by Lemma 2; hence derived \((p,x)\)-completion is derived \(x\)-completion. Equivalently, tensoring the usual completion tower with the finite \(E_2\)-module \(E_2/p\) commutes with its inverse limit, and the extra \(p\)-power tower has pro-zero error. Thus, for \(M=E_2\wedge C^s\), \[L_{K(2)}M/p\simeq\lim_m M/(p,x^m).\] Lemma 45 computes every term on the right by the ordinary even quotient of (94). The homotopy transition maps are surjective, so the Milnor \(\lim^1\) term vanishes. Periodicity on the height-two factor then gives (96). Its line tensor commutes with the limit because it is finite free as an underlying \(k[[x]]\)-module. These constructions are natural in all bar maps. ◻

Proof. The split special fiber. The two markings of Lemma 15 identify the connected part of \(\mathcal G_R\) with \(\Gamma_2\) and its étale quotient with \(\mathbb Q_p/\mathbb Z_p\). Their extension splits canonically over \(R\). This also follows directly from Lemma 16: the compatible inclusions \(L^{1/p^{2l}}\to R\) give the unique lifts of the marked étale generators over the perfect ring. Thus \[\mathcal G_R\simeq\Gamma_2\oplus\mathbb Q_p/\mathbb Z_p.\] Let \(\mathcal H_m\) be the universal height-two deformation over \(A_m\), pulled back to \(R[x]/x^m\), and put \[\mathcal D_m=\mathcal H_m\oplus\mathbb Q_p/\mathbb Z_p.\] The displayed splitting specifies the initial identification with \(\mathcal G_R\).

Square-zero deformation theory. The parameter map used here is nonlocal: it sends \(t\) to a unit. We use square-zero deformation theory formulated for arbitrary base ring maps. This is Grothendieck’s lifting theory for Barsotti–Tate groups, as presented in (Illusie 1985, Theorem 4.4 and Corollary 4.7); the Hodge filtration formulation for locally liftable groups appears in (Messing 1972, V, Theorem (1.6)). The precise square-zero interface used below is Lau’s formulation.

We recall precisely the deformation-theoretic input. For a \(p\)-divisible group \(G\) over a ring on which \(p\) is nilpotent, isomorphism classes of lifts across a square-zero ideal \(I\) form a torsor under \[\mathop{\mathrm{Hom}}(\omega_{G^\vee},I\otimes\mathop{\mathrm{Lie}}G).\] Changing a lifting ring map acts through the Kodaira–Spencer homomorphism (Lau 2010, Theorem 5.1 and Equation (5.1)). For the universal deformation \(\mathcal G/A\), where \(A=k[[a,t]]\) is a complete local Noetherian \(k\)-algebra with residue field \(k\) and \(k\) is perfect, we use Lau’s separate universal-deformation statement (Lau 2010, paragraph following Equation (5.2), p. 227). As in Equation (59), its closed-point criterion and Lemma 4 give \[\kappa_{\mathcal G}: \mathop{\mathrm{Hom}}_A(\mathop{\mathrm{Lie}}\mathcal G,\omega_{\mathcal G^\vee}) \xrightarrow{\ \sim\ }\Omega^1_{A/\mathbb Z} \simeq\Omega^1_{A/k}=A\,da\oplus A\,dt.\] At a square-zero jet stage, the ideal \(I\) is an \(A\)-module through \(\varphi_m\). Applying \(\mathop{\mathrm{Hom}}_A(-,I)\) to this isomorphism over \(A\) gives the isomorphism of torsor-action groups used in Lau’s Equation (5.1).

Parameter lifts and their uniqueness. We must first exhibit ring lifts for the nonlocal specialization. For nilpotent corrections, prescribe \[a\longmapsto\alpha,\qquad t\longmapsto t+\beta,\qquad \alpha,\beta\in xR[x]/x^m.\] For \(f\in A=k[[a,t]]\), define its Hasse derivatives using independent formal variables \(z,w\) by \[f(a+z,t+w)=\sum_{r,s\geq0} (\partial_a^{[r]}\partial_t^{[s]}f)(a,t)\,z^r w^s \quad\text{in }k[[a,t,z,w]]=A[[z,w]].\] These prescriptions extend uniquely to a ring map by the finite Hasse–Taylor expression \[ f\longmapsto \sum_{\substack{r,s\geq0\\r+s<m}} (\partial_a^{[r]}\partial_t^{[s]}f)(0,t)\, \alpha^r\beta^s. \tag{98}\] One may obtain the same formula without topology by expanding \(A\) over \(A^{p^e}\) in the finitely many monomials \(a^r t^s\) with \(0\leq r,s<p^e\), where \(p^e\geq m\); the nilpotent corrections then disappear on \(p^e\)-th powers. This proves both well-definedness and uniqueness of the substitution. In particular, arbitrary lifts of the coefficients of \(\alpha\) and \(\beta\) from one jet to the next give a ring lift.

Suppose the map and identification have been chosen modulo \(x^m\). The kernel of \(R[x]/x^{m+1}\to R[x]/x^m\) is square-zero. The ring lifts just constructed form a torsor under the corresponding derivations from \(A\). The map from this torsor to the torsor of deformations is equivariant for the Kodaira–Spencer isomorphism. It is therefore a bijection. Exactly one parameter lift realizes \(\mathcal D_{m+1}\) with the prescribed identification modulo \(x^m\).

There is also at most one isomorphism lifting the prescribed one. Here the needed rigidity has a short square-zero proof. If a homomorphism of \(p\)-divisible groups reduces to zero across a square-zero ideal in characteristic \(p\), its values lie in the infinitesimal kernel at the identity. That kernel is an additive group of derivations into the square-zero ideal and is killed by \(p\). Indeed, on any test algebra and at each finite torsion level, a point reducing to the identity has the form \(\varepsilon+d\), where \(\varepsilon\) is the identity point and \(d\) is such a derivation; the group law adds these derivations. The homomorphism is consequently killed by \(p\); since multiplication by \(p\) on its source \(p\)-divisible group is an epimorphism for the flat topology, the homomorphism is zero. Applying this to the difference of two lifts proves uniqueness. Induction proves compatibility of all maps and isomorphisms in (97).

Compatibility with the two stabilizer actions. The unique lifts must respect both the action of \(P\) on the deformation and the marking-change action of \(J\). We check the special-fiber action and then use uniqueness at each square-zero step. On the perfect special fiber there are no cross homomorphisms between the marked connected and étale summands. A section of connected Honda torsion over the reduced ring \(R\) is zero, and a map in the opposite direction vanishes by connectedness. The transported action is therefore block diagonal. By Lemma 15, \(P\) preserves the connected frame and changes the étale frame by a constant norm unit. The action of \(J\) changes the connected frame by a constant Honda automorphism and the étale frame by a constant unit. These actions extend to \(\mathcal D_m\): use the universal height-two deformation action on its connected factor and the same units on the constant factor.

More explicitly, let \(f:\mathcal G_R^0\to\Gamma_2\) be the connected marking and let \(s_f:\Gamma_2\oplus\mathbb Q_p/\mathbb Z_p\to\mathcal G_R\) be the splitting supplied by \(f\) and the uniquely lifted normalized generator \(e_f\). In the convention of (17), \[f'=jfg^{-1},\qquad s_{f'}=g s_f \bigl(j^{-1}\oplus[\mathop{\mathrm{Nrd}}(j)\mathop{\mathrm{Nrd}}(g)^{-1}]\bigr).\] The equality is checked on the connected marking and on the normalized étale generator, which determine both summands. This is the block action that we lift to \(\mathcal D_m\), using the universal height-two action and the same constant unit on the étale factor.

Functoriality of the two lifting torsors, followed by uniqueness, makes (97) and its group isomorphism equivariant.

Control at a finite coefficient stage. Write \(\varphi_m(a)=\alpha\) and \(\varphi_m(t)=t+\beta\) for the selected map. For fixed \(m\), the finitely many coefficients of \(\alpha\) and \(\beta\) belong to a common finite root/marking stage. The Hasse derivatives in (98) are elements of \(k[[t]]\) and depend continuously on \(f\) in its \((a,t)\)-adic topology. For fixed \(m\) their orders in \(t\) tend to infinity with the \((a,t)\)-adic order of \(f\), up to a fixed finite shift. Formula (98) therefore gives continuity into that complete stage. Every individual coefficient of the formal-group isomorphism is an element of \(R[x]/x^m\). A finite set of them lies in a common larger stage. This last assertion is deliberately about finite sets of coefficients; no single finite stage for the whole marking is required. ◻

Proof. Recovering the formal isomorphism. Apply Lemma 47. We first show directly that its isomorphism of algebraic \(p\)-divisible groups supplies the formal-law isomorphism required by ordinary cooperations. Fix \(m\), put \(\Lambda=R[x]/x^m\), and write \(I=(x)\). Let \(\mathcal F_3\) be the formal law over \(A\). The preparation used to form the finite kernel over \(A\) is \[[p^l]_{\mathcal F_3}(T)=U_l(T)W_l(T),\qquad U_l\in A[[T]]^\times,\] where \(W_l\) is monic. Set \[\mathcal B_l=\Lambda[T]/\varphi_m(W_l),\qquad f_l(T)=\varphi_m([p^l]_{\mathcal F_3}(T))\in\Lambda[[T]].\] Coefficientwise application of \(\varphi_m\) preserves the displayed identity. Moreover \(\varphi_m(U_l)\) is still a unit, because its constant coefficient is the image of an \(A\)-unit. Thus \((f_l)=(\varphi_m(W_l))\) in \(\Lambda[[T]]\).

Put \(n_l=p^{2l}\). Modulo \(I\), the law has height two and \[\overline f_l=T^{n_l}v_l(T), \qquad v_l(0)\in R^\times.\] In \(Q_l=\Lambda[[T]]/(f_l)\) this gives \(T^{n_l}\in IQ_l\); hence \(T^{mn_l}=0\), since \(I^m=0\). In the other direction, the invariant-differential formula in characteristic \(p\) gives \([p]'(T)=0\). The series \([p](T)\) therefore factors through \(T^p\), and its \(l\)-fold composite belongs to \((T^{p^l})\). Consequently \[ (T^{mp^{2l}})\subseteq(f_l)\subseteq(T^{p^l}). \tag{102}\] These nested ideals are cofinal with the powers of \((T)\).

We also identify the finite algebraic factor explicitly. The reduced prepared polynomial has the factorization \[\overline{\varphi_m(W_l)}=T^{n_l}h_l(T), \qquad h_l(0)\in R^\times.\] Its factors are relatively prime, so the Chinese remainder theorem gives \[\mathcal B_l/I\mathcal B_l \cong R[T]/T^{n_l}\times R[T]/h_l.\] The identity idempotent lifts uniquely across the nilpotent ideal \(I\mathcal B_l\), giving \[\mathcal B_l=\mathcal B_l^0 \times\mathcal B_l^{\mathrm{away}}.\] On \(\mathcal B_l^0\), \(T^{n_l}\in I\mathcal B_l^0\) and thus \(T^{mn_l}=0\). On \(\mathcal B_l^{\mathrm{away}}\), \(T\) is invertible modulo \(I\) and hence invertible. The polynomial map \(\mathcal B_l\to Q_l\) kills the away factor, because \(T\) is nilpotent in \(Q_l\). Conversely, evaluation at the nilpotent element \(T\in\mathcal B_l^0\) defines a map \(Q_l\to\mathcal B_l^0\). These two maps are inverse on coefficients and \(T\); both algebras are generated by polynomials in \(T\). Therefore \[\Lambda[[T]]/(f_l)\cong\mathcal B_l^0.\] The \(T\)-adic completion of \(\mathcal B_l\) stabilizes at this factor. In particular this identification uses no Noetherian hypothesis on \(\Lambda\).

The factor \(\mathcal B_l^0\) is the relative identity component of the finite kernel. Its unique idempotent construction is natural under base change and frame isomorphisms. The isomorphism of \(p\)-divisible groups therefore restricts compatibly to these factors. By (102), their inverse limit recovers \(\Lambda[[T]]\). To recover the group law as well, put \(N_l=mp^{2l}\). The same bounds give \[(T,S)^{2N_l-1} \subseteq(f_l(T),f_l(S)) \subseteq(T,S)^{p^l} \quad\text{in }\Lambda[[T,S]].\] Thus every finite order in the two coordinates is seen at a sufficiently large common torsion level. The compatible homomorphism identities on the products of the finite kernels determine the identity on the full formal laws. We have obtained the required continuous formal-law isomorphism and its inverse, naturally under all the marking actions.

Evaluating the cooperation complex. This isomorphism evaluates the degree-zero cooperation algebra in \(R[x]/x^m\). To include the other factor in (94), apply \(\varphi_m\) pointwise to its characteristic-\(p\) height-three continuous functions. Formula (98) makes this a continuous evaluation into a complete finite root/marking stage. More explicitly, an element of the ordinary tensor product is a finite sum of cooperation elements times continuous coefficient functions. For this element and this \(m\), choose a stage containing the parameter corrections and the finitely many formal-isomorphism coefficients occurring in those cooperation elements. All its evaluated functions then take values in that one stage. This proves the finite-stage requirement for (99); it does not demand one stage for all cooperation elements at once. The same argument retains arbitrary compact profinite parameters, using the same finite Taylor formula and supremum topology.

Agreement with the realized stabilizer action. To use this coefficient map in the homotopy test, we must identify its \(J\)-action with the action induced by the actual automorphisms of \(E_2\). This requires the full marking isomorphism through every jet. We make that agreement explicit. Let \(\mathcal F_2\) be the chosen height-two law over \(B_2=W(k)[[x]]\). For \(j\in J\), write its marked-lift transport as \[\rho_j:B_2\longrightarrow B_2,\qquad h_j:\rho_j^*\mathcal F_2\xrightarrow{\sim}\mathcal F_2,\qquad h_j\bmod(p,x)=j.\] The last equality retains the entire Honda series. These are the coefficient action and universal transport of Devinatz and Hopkins (1995, sec. 1, Equation (1.4), pp. 673–674), in the chosen coordinate. Uniqueness identifies their reductions with the height-two action through every jet and gives the group law for the transports.

The geometric marking action \(\mu_j(f)=jf\) induces the left coefficient action \(\sigma_j^R=(\mu_{j^{-1}})^*\). Let \(\iota_m:B_2\to\Lambda=R[x]/x^m\) be the coefficient map and \(\sigma_j\) the diagonal action, so \(\sigma_j\iota_m=\iota_m\rho_j\). Inverse pullback changes the displayed special-fibre splitting to \(s_f(j\oplus[\mathop{\mathrm{Nrd}}(j)^{-1}])\). Replacing this connected block by \(\iota_m(h_j)\), with the same étale unit, restores the same marked deformation at each lifting step. The two uniqueness statements in Lemma 47 therefore give, for the recovered formal marking \(\Phi_m:\varphi_m^*\mathcal F_3\to\iota_m^*\mathcal F_2\), \[\sigma_j\varphi_m=\varphi_m,\qquad \sigma_j^*\Phi_m=\iota_m(h_j^{-1})\circ\Phi_m: \varphi_m^*\mathcal F_3\longrightarrow(\iota_m\rho_j)^*\mathcal F_2.\] Here the superscript \(\sigma_j^*\) means coefficientwise base change.

After diagonal characteristic-\(p\) reduction, this is the same formula as for \(\Phi_m\), on all isomorphism coefficients; equivalently \(\psi_2(h_j^{-1})=(\psi_2\rho_j)(h_{j^{-1}})\). For \(d_j=h_j'(0)\), a compatible degree-two cotangent generator transforms by \(d_j^{-1}\), and the derivative \(c=u_2/u_3\) of the marking transforms by the same factor. This identifies the action on the actual line \(\Omega\): a compatible frame change with linear term \(a\) replaces \(d_j\) by \(a d_j\rho_j(a)^{-1}\). The entire series \(j\) is retained, so this includes every principal unit, even when its closed-fibre derivative is one. Naturality of the external product in (94) carries this calculation to every \(C^s\), with \(J\) fixing the height-three function factor. The bar faces are evaluation, multiplication, and transport by \(P\). Equivariance and uniqueness in Lemma 47 show that evaluation commutes with the action face; the remaining faces and degeneracies commute by their formulas. The preceding calculation gives \(J\)-equivariance. We have therefore constructed actual maps of the homotopy cochain complexes in (99). Their construction in nonzero even internal degrees uses only the height-two periodicity line and hence gives its stated tensor power.

Quasi-isomorphism at each finite jet. At \(m=1\), Lemma 45 identifies this map with \[C^\bullet_{\mathrm{cts}}(P,L)\longrightarrow C^\bullet_{\mathrm{cts}}(P,R).\] It is the natural inclusion and is a quasi-isomorphism by Theorem 41. Filter (99) by powers of \(x\). Flatness in Lemma 45 identifies each source associated-graded layer with \(D_1^\bullet\), and each target layer with \(C^\bullet_{\mathrm{cts}}(P,R)\). The induced map is the same residue inclusion, multiplied by the corresponding formal power of \(x\). The finite filtration thus shows that (99) is a quasi-isomorphism for every \(m\).

Passing to the inverse limit. Both inverse systems of cochain terms have surjective transition maps. Their strict inverse limits consequently compute their derived inverse limits, and the latter preserve the levelwise quasi-isomorphisms. Combining this with Lemma 46 proves (100). On the target the differential acts coefficientwise in \(x\). Products of complexes of \(k\)-vector spaces are exact, or equivalently the finite-jet cohomology maps are surjective. Thus its cohomology is \(H^s(P,R)[[x]]\), giving (101) and its asserted coefficient lines.

The action on completed homotopy. We identify the action above with the action on actual completed homotopy. The original tested spectrum carries \(J\), even if the cofibers by \(x^m\) were chosen without equivariance. Fix a cosimplicial degree \(s\), and put \[D_{\mathrm{ord}}= \bigl(\pi_0((E_2\wedge C^s)/p)\bigr)_\Delta, \qquad \widehat D=\lim_m D_m^s.\] By Lemma 45, \(D_{\mathrm{ord}}\) is flat over \(k[[x]]\) and \(D_m^s=D_{\mathrm{ord}}/x^mD_{\mathrm{ord}}\). Lemma 46 identifies \(\widehat D\) with the actual degree-zero diagonal completed homotopy and its projections with reduction maps. For every \(m\geq1\), \[\ker(\widehat D\longrightarrow D_m^s)=x^m\widehat D.\] Indeed, the component modulo \(x^{m+r}\) of an element in this kernel is \(x^m\) times a unique component modulo \(x^r\), because multiplication by \(x\) is injective on \(D_{\mathrm{ord}}\). These components are compatible as \(r\) varies, proving one inclusion; the reverse inclusion is immediate. The image of \(D_{\mathrm{ord}}\) is dense for the separated topology defined by these projection kernels. The natural map \((E_2\wedge C^s)/p\to\mathcal T(C^s)\) is \(J\)-equivariant, so the actual action agrees with the ordinary action on that image. For every \(g\in J\), semilinearity and \(g(x)\in x\,k[[x]]^\times\) give \(g(x^m\widehat D)=x^m\widehat D\). The kernel identity therefore makes the actual \(g\)-action continuous for this topology. The action obtained from the compatible \(D_m^s\)-actions is also continuous and agrees on the same dense image, so the two actions coincide. Tensoring with the actual finite free line \(\Omega^{\otimes j}\) gives the same conclusion in every even internal degree \(2j\). All diagonal projections commute with this action. We have used the new coefficient maps to identify the existing spectral sequence, without requiring a separate spectral realization of those maps. ◻

Proof. We first show that the action of \(G_F\) on \(Q_x\) has infinite image. Let \(\mathcal F\) be the characteristic-\(p\) universal height-two formal group over \(\mathcal O_x\). If \(g\in G_F\) acts trivially on \(\mathcal O_x\), its marking-change isomorphism is an automorphism of \(\mathcal F\) over that ring. Over \(Q_x\), the first Hasse coefficient is invertible, so \(\mathcal F\) has height one. After extending \(Q_x\) to an algebraic closure, the height classification identifies it with the multiplicative formal group (Lazard 1955, Theorem IV).

The automorphisms of the multiplicative formal group are precisely the intrinsic scalar automorphisms \([c]\), \(c\in\mathbb Z_p^\times\). Indeed, restriction to its finite \(p^r\)-torsion group schemes identifies an automorphism with a compatible system in \(\lim_r(\mathbb Z/p^r)^\times=\mathbb Z_p^\times\): the character group of \(\mu_{p^r}\) is \(\mathbb Z/p^r\), and the ideals \((T^{p^r})\) are cofinal in the formal-coordinate topology. Thus these restrictions determine one scalar automorphism of the entire formal group.

Every \([c]\) is already defined on \(\mathcal F\) over \(\mathcal O_x\). For example, integer approximations to \(c\) modulo \(p^r\) give compatible formal power series because the order of \([p^r](T)\) tends to infinity. The injection of \(\mathcal O_x\) into the algebraic closure of \(Q_x\) therefore shows, coefficient by coefficient, that the marking-change automorphism equals \([c]\) over \(\mathcal O_x\). Specializing at \(x=0\) shows that \(g\) is scalar on the Honda group, or that \(g^{-1}\) is scalar under the opposite action convention. Thus the kernel of the action on \(Q_x\) is contained in the central subgroup \(\mathbb Z_p^\times\). The quotient \(\mathcal O_F^\times/\mathbb Z_p^\times\) is infinite: on principal units, the \(3\)-adic logarithm identifies the relevant ranks with those of \(3\mathcal O_F\) and \(3\mathbb Z_p\), namely two and one. The action on \(Q_x\) has infinite image.

Every element of \(G_F\) fixes \(k\) and preserves the \(x\)-adic valuation. Suppose \(f\in Q_x^{G_F}\) is not in \(k\). If its valuation is negative, replace it by \(f^{-1}\); if its valuation is zero, subtract its residue in \(k\). We obtain a nonzero invariant \(z\) with strictly positive valuation \(d=v_x(z)\). The substitution \(k[[Z]]\to k[[x]]\), \(Z\mapsto z\), makes \(k[[x]]\) finite free of rank \(d\) over \(k[[z]]\), by the one-variable Weierstrass theorem. Hence \(Q_x/k((z))\) is a finite field extension. Continuity implies that every element fixing \(z\) fixes \(k((z))\), so the image of \(G_F\) lies in the finite automorphism group of this extension. This contradicts its infinite image. Separability is not needed: the automorphism group of any finite field extension has cardinality at most its degree. We have proved \[ Q_x^{G_F}=k. \tag{103}\]

Next consider the first Hasse invariant. For an even-periodic theory the cotangent line of its formal group identifies with \(\pi_2\): restriction of the augmentation ideal to \(\mathbb{CP}^1\) identifies its quotient by its square with \(\widetilde E_2^0(\mathbb{CP}^1)=\pi_2E_2\). Thus \(\Omega=\omega/p\) is the cotangent line of \(\mathcal F\). In a coordinate \(T\), let \(a_1\) be the coefficient of \(T^p\) in \([p]_{\mathcal F}(T)\). Under \(T'=cT+O(T^2)\), that coefficient becomes \(c^{1-p}a_1\), while \(dT'=c\,dT\) at the identity. Therefore \[h=a_1(dT)^{\otimes(p-1)}\] is an intrinsic, hence \(G_F\)-invariant, section of \(\Omega^{\otimes(p-1)}=\Omega^{\otimes2}\). The universal height-two parameter is its simple zero: the transported Lubin–Tate normalization in Section 3 gives \(a_1=x\) in the chosen coordinate. In any integral frame \(e\) of \(\Omega\), write \[ h=h_e e^{\otimes2},\qquad v_x(h_e)=1. \tag{104}\] Thus \(h\) is an invariant basis of \(\overline\omega^{\otimes2}\) over \(Q_x\).

Suppose now that \(Q_x(\delta)\simeq\overline\omega^{\otimes b}\). The image of the distinguished basis of \(Q_x(\delta)\) is a basis \(v=f e^{\otimes b}\) with \(f\in Q_x^\times\) and \(g(v)=\delta(g)v\). Since \(\delta^2=1\), the ratio \[\frac{v^{\otimes2}}{h^{\otimes b}} =\frac{f^2}{h_e^b}\] is a nonzero invariant function. By (103), it lies in \(k^\times\). Taking valuations in (104) gives the exact equality \[ 2v_x(f)=b. \tag{105}\] Hence \(b\) is even. But \(h^{\otimes b/2}\) is then an invariant basis of \(\overline\omega^{\otimes b}\), so this line is trivial.

Finally, \(Q_x(\delta)\) is not trivial. Otherwise there would be \(u\in Q_x^\times\) with \(g(u)=\delta(g)u\) for every \(g\in G_F\); the inverse convention is the same because \(\delta^{-1}=\delta\). Then \(u^2\in k^\times\) by (103), so \(u\) is algebraic over \(k\). The finite field \(k\) is relatively algebraically closed in \(k((x))\). To see this, an element algebraic over \(k\) satisfies \(u^{|k|^r}=u\) for some \(r>0\). If nonzero it has valuation zero; subtract its residue and compare positive valuations in the same equation to see that the difference is zero. Thus \(u\in k^\times\), contradicting the nontriviality of \(\delta\). This proves the lemma. ◻

Proof. Apply Proposition 44 to \(Y/p=\mathcal T(Z_k)\), take the diagonal constant-field summand, and localize homotopy groups from \(\mathcal O_x\) to \(Q_x\). Restrict the actions to \(G_F\). Theorem 48 gives the complete list of nonzero terms on the second page: \[ \begin{array}{c|c} \text{cohomological degree }s & \mathsf E_2^{s,2j},\quad j\in\mathbb Z \\\hline 0,1 & \overline\omega^{\otimes j}\\[2pt] 3,4 & Q_x(\delta)\otimes_{Q_x}\overline\omega^{\otimes j}. \end{array} \tag{108}\] All odd internal degrees vanish. Here \(\mathsf E_r\) denotes the diagonal, localized spectral sequence. Its differentials have bidegree \[d_r:\mathsf E_r^{s,t}\longrightarrow\mathsf E_r^{s+r,t+r-1}.\] An even \(r\) changes internal parity, so its differential is zero. For odd \(r\geq3\), the four cohomological degrees in (108) leave only \(r=3\), from \(s=0\) to \(s=3\) or from \(s=1\) to \(s=4\). Every such differential would have the form \[ \overline\omega^{\otimes j}\xrightarrow{\ d_3\ } Q_x(\delta)\otimes_{Q_x}\overline\omega^{\otimes(j+1)}. \tag{109}\] It is a \(Q_x\)-linear, \(G_F\)-equivariant map. A nonzero map between these one-dimensional representations is an isomorphism; after tensoring with the inverse periodicity line, it would identify \(Q_x(\delta)\) with a periodicity power. Lemma 50 excludes that possibility. Both possible \(d_3\) families therefore vanish, and no later differential is possible.

The diagonal idempotent commutes with the actual finite filtration supplied by Proposition 44, and localization from \(\mathcal O_x\) to \(Q_x\) is exact. Thus \[F^s\mathcal V_d^\Delta(Z_k)/F^{s+1}\mathcal V_d^\Delta(Z_k) \simeq\mathsf E_\infty^{s,d+s}.\] For \(d=2r\), the only nonzero quotients occur at \(s=0,4\); they are \(\overline\omega^{\otimes r}\) and \(Q_x(\delta)\otimes\overline\omega^{\otimes(r+2)}\), respectively. For \(d=2r-1\), they occur at \(s=1,3\), with lines \(\overline\omega^{\otimes r}\) and \(Q_x(\delta)\otimes\overline\omega^{\otimes(r+1)}\). The filtration starts with the whole homotopy group and ends at zero, so the higher-filtration line is a subrepresentation and the lower one is its quotient. This proves the two exact sequences and the dimension assertion. Taking \(r=-1\) in (107) gives \[0\longrightarrow Q_x(\delta) \longrightarrow\mathcal V_{-3}^\Delta(Z_k) \longrightarrow\overline\omega^{\otimes(-1)}\longrightarrow0.\] ◻

Proof of Theorem 1. Suppose, to the contrary, that \[L_2L_{K(3)}S\in\mathop{\mathrm{Thick}}\{L_0S,L_1S,L_2S\}\] in the category specified in the Theorem. Forgetting to spectra and applying exact \(K(2)\)-localization sends \(L_0S\) and \(L_1S\) to zero, and sends \(L_2S\) to \(\mathbf 1_2\). Moreover, \(L_{K(2)}L_2L_{K(3)}S\simeq L_{K(2)}L_{K(3)}S\). These are the Bousfield-class relations in Ravenel (1984, Theorem 2.1(d),(i)), together with the smashing property of \(E(i)\)-localization (Hovey and Strickland 1999, Theorem 5.1 and Proposition 5.3). Thus \(L_{K(2)}L_{K(3)}S\) lies in the ordinary thick subcategory generated by \(\mathbf 1_2\).

By (84), the underlying \(S\)-module \(S_k\) is finite free. Consequently \(Z_k=L_{K(3)}S_k\), as an underlying spectrum, is a finite direct sum of copies of \(L_{K(3)}S\). The same thick-generation conclusion holds for \(L_{K(2)}Z_k\). No Galois-equivariant choice of basis of \(S_k\) is required: the detector action comes from its \(E_2\) factor, so every underlying spectrum map induces an equivariant detector map. Proposition 3 implies that \[ \mathcal V_d(Z_k)\text{ has only constituents } \overline\omega^{\otimes b},\quad b\in\mathbb Z, \quad\text{for every }d\in\mathbb Z. \tag{110}\] But Proposition 51 supplies the simple subrepresentation \(Q_x(\delta)\) of the diagonal summand of \(\mathcal V_{-3}(Z_k)\), hence a simple constituent of that whole test. Lemma 50 excludes it from (110). This contradiction proves the Theorem. ◻