Finite generation for the K(n)-local sphere

Proof. The kernel assertion is (i). A cochain in the quotient has compact image T in a single step, and composing it with the section in (ii) lifts that cochain. The section is not required to commute with G. Thus every degree of the cochain sequence is exact.

For compact vector spaces, continuous duality with discrete vector spaces turns a strict surjection into an injection. Splitting that injection as a map of vector spaces and dualizing gives a continuous \kk-linear section. In the Noetherian local case the modules are compact. A submodule is closed, and its induced topology is its adic topology by Artin–Rees. This proves (i), while the compact splitting proves (ii). For a localization, a fixed quotient lattice is the image of a compact source lattice: choose lifts of finitely many module generators and a common denominator. Its map from that source lattice is a strict compact surjection and has a continuous linear section. The kernel in a fixed lattice is again a finite module, so Artin–Rees gives (i) after a common enlargement of the denominator. This also proves the assertions for finite twists and finite direct sums.□

Proof. Let T\subset M_a be compact. The set GT is compact in some M_b. Let N be its closed linear span there. Modulo any open linear subspace of M_b, the image of GT is finite. Since \kk is finite, its linear span in that discrete quotient is finite as well. Thus N is totally bounded and complete, hence compact. For any g\in G, the action map M_b\to M_c is continuous after enlarging c uniformly in g. The image of N in the Hausdorff space M_c is compact and closed. It contains the image of the linear span of GT, and continuity therefore shows gN\subset N. The injection N\to M_c is a homeomorphism onto its image, so the restricted action is continuous. A finite sum of such spaces is again compact in a common step. Every element, and every compact cochain image, is contained in one of them. A map from a compact domain whose image lies in N is continuous into N, by its subspace topology in M_b. This proves (i), including the assertion about cochains.

Choose a normal uniform open subgroup K\subset G. For the augmentation filtration of \kk[[K]], the associated graded ring is a polynomial ring on \dim G variables. Thus \kk[[K]] is Noetherian, and the finite crossed-product extension \kk[[G]] is Noetherian as well. Successively resolving the finitely generated kernels of a finite free presentation gives P_\bullet. For a compact module N, compare it with the compact bar resolution. Its terms are induced from compact \kk-spaces and are projective: every compact \kk-space is projective by continuous linear duality, and completed induction preserves that lifting property. Both are projective resolutions, and their augmentations and acyclic kernels split as compact \kk-spaces by continuous duality. The lifting construction for resolutions gives chain maps and chain homotopies in the compact category. Applying continuous Hom gives the usual continuous bar cochains. For comparison with the ind-profinite formulation under its countability hypotheses, see Boggi and Corob Cook [6, Proposition 8.2 and Corollary 8.4]. The compact splitting argument just given does not impose second countability on N and proves that all its comparison homotopies are continuous.

There is a well-defined \kk[[G]]-action on M: integrate on a compact G-stable envelope of a given element. It is independent of that envelope. Since every P_j is finite free, its Hom into M is the filtered union of its Hom into the N. Filtered colimits of vector spaces are exact. Passing the compact comparison and its homotopies through this colimit proves (ii); in particular no new completion is introduced. The same argument applies to the finite free right resolution and tensor products. For compact N, every differential has compact, hence closed, image in a finite power of N. Continuous duality is exact on compact vector spaces and takes these finite-power complexes to the corresponding dual complexes. This proves (iii). Finally, \kk[[\Zp]]=\kk[[\gamma-1]], and multiplication by \gamma-1 gives the stated two-term free resolution.□

Proof. An element of H_y has a lift taking x into its K-orbit. Correcting that lift by the unique element of K gives an element fixing x. Since K acts freely, the resulting homomorphism G_x\to H_y is bijective. Write F=G_x. A continuous section with value x at y exists by the assumed trivialization. Its failure to be F-equivariant is a continuous nonabelian cocycle c_f\in C(V,K) for the action combining conjugation by F and its action on V. All c_f(y) equal 1. Shrinking to an F-invariant clopen neighborhood makes all these cocycles arbitrarily close to 1.

Here is the required small-cocycle argument, including the case p\mid |F|. Choose an F-stable sufficiently small Lie lattice in \Lie(K) on which exponential, logarithm, and the Campbell–Hausdorff formula converge. In the Banach space of continuous functions on V with values in \Lie(K), use the supremum norm of an F-invariant norm. Put \lambda_f=\log c_f and \epsilon=\max_f\|\lambda_f\|. Convergence of the Campbell–Hausdorff series gives, for a fixed constant c, the bound

With b=|F|^{-1}\sum_f\lambda_f, summing this identity over g gives

Changing the section by \exp(-b) changes its cocycle to \exp(-b)c_f f(\exp b). Another application of the same convergence estimate bounds the new logarithms by c'\epsilon^2, for a constant c' depending only on F and the chosen Lie lattice. Choose the initial neighborhood small enough that these exponentials stay in the convergence lattice and c'\epsilon<1. Iteration then gives uniformly convergent continuous changes of section, since the successive logarithms tend to zero at least quadratically. The limit has trivial cocycle. Every change is the identity at y, so the limiting section still takes y to x. Division by |F| was only performed in \Qp; its fixed norm loss was absorbed in the initial choice of neighborhood.□

Proof. First consider a finite subgroup F and a semilinear module L with a trace element t=t_F. The map

is F-linear when F acts on the first factor of the target. Its composite with f\otimes v\mapsto fv is multiplication by \sum_f f(t)=1. Thus L is a direct summand of an induced module. The same formulas on continuous cochains, which are finite sums of bounded continuous operators, show H^{>0}(F,L)=0 in our convention. They apply also to continuous function modules with pointwise multiplication by t, and to any finite-dimensional twist.

The trivial \kk[[K]]-module has a finite free resolution of length d_G. For finite cohomological dimension and uniform-group Poincare duality, see Lazard [20, Chapter V, Proposition 2.5.7.1 and Theorem 2.5.8] and Beaudry et al. [4, Lemma 4.15 and Proposition 4.16]. Thus \kk has projective dimension d_G, and the Noetherian local ring \kk[[K]] has finitely generated free terms in a minimal resolution. A sufficiently deep uniform subgroup is used also when p=2. Tensor-induce this resolution from K to G: take its completed tensor product over the a_G cosets and let G permute the factors, using the usual signs of complexes and the conjugate K-actions. Denote the resulting augmented complex by T_\bullet\to\kk. It is concentrated in degrees 0 through a_Gd_G. The original augmented resolution is split exact as a complex of compact \kk-spaces. Its tensor product is therefore again a split resolution of \kk in that category. This is the tensor-induction construction of Symonds [35, Theorem 5.2].

Each term of T_\bullet is a finite sum of signed permutation modules \kk[[X]]. Here X is a finite disjoint union of products of copies of K, one for each tensor factor. The diagonal K-action is free, and setting the first coordinate equal to 1 gives a continuous trivialization of X\to K\backslash X. A possible sign is a locally constant rank-one coefficient and will be carried along. We claim that

The Hom module is C_{\cts}(X,M), with complete uniform-convergence steps. A K-trivialization identifies it, after untwisting the diagonal action on M, with a continuously coinduced K-module. Evaluation in the first coordinate contracts the coinduced bar complex, so its positive K-cohomology vanishes. All operators in that contraction are bounded continuous operators on function steps. The Hochschild–Serre double complex therefore reduces its G-cohomology to the cohomology of H=G/K on the K-invariants.

Apply Lemma 2.5 at a point of Y=K\backslash X. After shrinking its neighborhood V further, its translates by elements outside H_y are disjoint: first separate y from the finitely many distinct points hy, and then intersect the resulting clopen neighborhoods. On the induced neighborhood HV, the K-invariant functions are induced from H_y\simeq G_x acting on C_{\cts}(V,M), with its possible rank-one sign. The equivariant section makes this the ordinary semilinear action on values together with the action on V. Its higher H_y-cohomology vanishes by the trace-element splitting. These induced neighborhoods cover the compact space Y. Choose a finite such cover and take successive differences of its H-invariant clopen members to obtain a disjoint finite invariant clopen partition. Each piece retains its local description and splitting. Finite additivity and finite-group Shapiro now prove (2.1).

Finally form the first-quadrant double complex

The compact \kk-linear contracting homotopy of the augmented T_\bullet acts on these Hom modules: a continuous map from a compact domain into a step has image in a compact envelope, so integration and the contracting homotopy take place in complete steps by Lemma 2.3. Computing first in the v direction identifies the total cohomology with H^*(G,M). Computing first in the u direction and using (2.1) leaves only u=0 and 0\leq v\leq a_Gd_G. This proves the bound. Tensoring M with V preserves every step condition and the trace elements, so the same construction proves the last assertion with the identical bound.□

Proof. Write R=\kk[[G]], R_K=\kk[[K]], H=G/K, and \Sigma=\kk[H]. Regard \Sigma as the quotient bimodule of R. It is perfect as a left or right R-module, since it is induced from the length-d_G free resolution over R_K. Uniform-group duality, with its conjugation action retained, gives

as a (\Sigma,R)-bimodule. To recall the equivariance in this formula, restrict R to R_K and decompose it into its finitely many cosets. The dual of the uniform resolution has cohomology only in degree d_G, with one orientation line on each coset. Right translation permutes those lines by the right regular action; precomposition by a quotient element gives the left regular action and its action on the orientation line. The latter is trivial by hypothesis. This proves exactly the two commuting actions asserted in (2.2). The uniform duality, orientation identification, and compatibility with conjugation are also recorded in Beaudry et al. [4, Propositions 4.16 and 4.36, Theorem 4.19 and Remark 4.23].

Put

Perfectness permits evaluation of the dual resolution against M, so (2.2) gives a natural equivalence C\simeq B[-d_G] in the derived category of \Sigma-modules. The cohomology of C lies in degrees 0,\ldots,d_G. The finite free comparisons in Lemma 2.3 show that these derived expressions compute precisely the specified continuous cochains and homology, even when M is a union of steps.

We prove that the finite-group norm

is an equivalence. This is the point where finite subgroup cohomology cannot be discarded. For any finite-dimensional left \Sigma-module V, derived adjunction gives

By Proposition 2.6 these groups vanish for j>a_Gd_G, uniformly in V. Choose a bounded-below injective resolution J^\bullet of C. If N is greater than both d_G and a_Gd_G, dimension shifting along its exact tail gives

Every quotient \Sigma/I by a left ideal is finite-dimensional. Baer’s criterion therefore says that this last kernel is injective. Truncating there gives a bounded complex of injective modules representing C.

For a finite group algebra, every injective module, including an infinite-dimensional one, is projective. Indeed, the canonical injection of any module L into \Hom_\kk(\Sigma,L) splits if L is injective; the target is free because the symmetric pairing on \kk[H] identifies coinduction with induction. On a free module of arbitrary rank the ordinary norm is an isomorphism from coinvariants to invariants: on each copy of \kk[H] it sends the class of 1 to \sum_{h\in H}h. It is consequently an isomorphism on projective summands. Applying this degreewise to the bounded injective-projective model of C proves (2.3), with no infinite totalization or completion.

The left-hand side of (2.3), shifted by d_G, is

the right-hand side is \RGamma(G,M). Taking cohomology in degree -i of the shifted equivalence gives H_i(G,M)=H^{d_G-i}(G,M), with the stated grading.

For G=P_n, its Lie algebra is the central division algebra D_n of degree n over \Qp. Conjugation by a\in D_n^\times has determinant

The orientation character is the reduction of this determinant character, hence is trivial. The dimension of D_n is n^2.□

Proof. Write [p^\ell]_{\cF}(T)=f(T^{p^b}). Modulo the maximal ideal of A_m, the series f(Z) is Z^{p^{n\ell-b}}. Weierstrass preparation therefore gives

where v is a unit and all c_i lie in the maximal ideal. Consequently [p^\ell](T) generates the same ideal as W(T^{p^b}). Division by this monic polynomial shows both that A_m[Z]/(W(Z))\longrightarrow A_m[[T]]/([p^\ell](T)), Z\mapsto T^{p^b}, is injective and that its image has basis 1,T^{p^b},\ldots,T^{p^b(p^{n\ell-b}-1)}. This proves the rank assertion.

A coordinate change T\mapsto a(T) carries T^{p^b} to a(T)^{p^b}, a series in T^{p^b}. Thus the subalgebra is preserved; it is the relative Frobenius image of the division scheme on coordinate rings. The same monic-division calculation after base change identifies the resulting algebra with the corresponding powered division algebra of the restricted group. In particular it commutes with passage to A_{m+1}=A_m/(x). The numerical restriction on b persists, since b\leq m\ell\leq(m+1)\ell.

The basic modules are finite free. An extension with finite free quotient splits as an underlying A_m-module, so its middle term is finite free. A finite direct summand is projective and hence free over the local ring A_m. These facts also show that reduction modulo x preserves the exact filtrations in the definition. The periodicity line restricts to the corresponding line at height m+1. The last assertion now follows by induction over the finite constructions defining the class.□

Proof. Write an isomorphism f:\cF\longrightarrow\Gamma_m as an ordinary power series. Its coefficient at T^{p^i} is a_i. At a degree j which is not a power of p, some intermediate binomial coefficient \binom{j}{v} is nonzero modulo p. In the coefficient of X^vY^{j-v} in f(\cF(X,Y))=\Gamma_m(f(X),f(Y)), the unknown coefficient of T^j therefore has a nonzero constant coefficient. All other terms involve earlier coefficients. It follows inductively that every non-p-power coefficient is a polynomial over A_m in the preceding a_i.

Now compare f([p]_{\cF}(T)) with f(T)^{p^m}. In degree p^m this gives a_0x=a_0^{p^m}, hence the first equation, since a_0 is invertible on the open marking space. In degree p^{m+i}, the only term involving a_i on the left is x^{p^i}a_i; all remaining terms involve earlier coefficients. This proves Equation (3.3), with polynomials over A_m. The recursive comparison is also the usual proof of the Honda classification; see Ravenel [28, Theorem A2.2.12, its proof, and Lemmas A2.2.20–A2.2.21].

Let H_{m,h} be the successive monic algebra defined by these equations. Monic division gives the displayed basis, so it is finite free of rank (p^m-1)p^{mh}. After inverting x, a_0 is invertible, the derivative of its equation is invertible, and the derivative of the ith subsequent equation is -x^{p^i}. Thus H_{m,h}[1/x] is finite étale.

The actual marking quotient is finite étale by the ring-level classification above. After a strictly henselian base change its fiber is the corresponding quotient of P_m. A Honda automorphism has p^m-1 possible nonzero linear coefficients, and p^m choices at each succeeding p-power; the other coefficients are then determined. This follows either from the Honda coefficient calculation or from the order filtration on its endomorphism ring. Hence that quotient has rank (p^m-1)p^{mh}. The equations define a map from H_{m,h}[1/x] to its coordinate ring. Every jet coefficient is a polynomial in a_0,\ldots,a_h, so these coordinates distinguish geometric jets. The map on geometric function algebras is therefore surjective, and finite étaleness gives surjectivity of the map itself. A surjection between finite étale algebras of the same rank is an isomorphism, as can be checked after strict henselization. This identifies the algebra, including over nonreduced test bases, with the connected marking quotient.

Precomposing f with \alpha_g gives

Every coefficient in this formula lies in A_m; only finitely many coefficients of \alpha_g are used at a fixed jet order. The formulas preserve the defining relations after inverting x. Since H_{m,h} is x-torsion-free, they preserve them before localization as well. The inverse coordinate change supplies the inverse action.

Order the displayed basis lexicographically starting with e_h, then e_{h-1}, and so on. In expanding a transformed monomial, any use of an earlier-variable term in Equation (3.4) lowers this order. Reducing a power a_i^{p^m} by its monic equation lowers it again. The diagonal term on a basis vector is therefore \chi_g^{e_0+pe_1+\cdots+p^he_h}. The initial spans in this order are stable, with the claimed line quotients. All these spans and quotients are free over A_m, so the filtration specializes exactly to every deeper height locus. Finally the formulas are continuous in g and in the compact coefficient topology, because every fixed finite coefficient quotient uses finitely many coefficients of the continuous universal coordinate change.□

Proof. Choose the connected jet order h and the étale level \ell defining U. Relative Frobenius of exponent m\ell kills precisely the connected part of \cF[p^\ell] on the exact-height open. Its image is the étale quotient, whose coordinate algebra is D_m(\ell,m\ell)[1/x]. One can check étaleness directly: writing [p^\ell](T) in the variable Z=T^{p^{m\ell}}, its linear coefficient is invertible when x is invertible. Equivalently its geometric fibers have p^{r\ell} distinct points, and the resulting finite flat group scheme is étale.

The set of ordered bases in the r-fold power of this finite étale group is an open and closed subscheme. Indeed it can be checked after an étale trivialization of the group, where it is a subset of a finite constant set. Its characteristic idempotent is preserved by every isomorphism of the group, in particular by G. Take

and extend that idempotent over the connected factor. This gives the asserted e and the identification with the two marking torsors. Both factors of B' are finite étale over B_m.

Tensor the monomial filtration of Proposition 3.3 with the whole powered tuple algebra. Its quotients are precisely of the form Equation (3.2). The same holds after reduction modulo x by Lemma 3.2. Tensoring the finite free bases gives a basis of A', and all multiplication constants lie in A_m. This construction uses the entire tuple algebra at the boundary; no extension or specialization of e across x=0 is involved.

The base action and the action on a powered division coordinate are substitution by the universal formal coordinate change and its appropriate Frobenius power. Their reduction in a fixed finite free basis converges in the parameter topology. Modulo any power of the maximal ideal, only a finite truncation is needed, so its matrix entries vary continuously in g. The connected-jet entries have the same property by Equation (3.4). Thus all matrices have integral entries and zero pole loss in the whole lattice, uniformly in g. Preservation of (x) is Equation (3.1).□

Proof. Strauch describes the corresponding torsion stratum over the completed maximal unramified coefficient base. There the full Drinfeld level-p^\ell deformation ring \widetilde R_\ell is finite flat over the Lubin–Tate deformation ring and regular, with the universal torsion coordinates \phi_1,\ldots,\phi_n as regular parameters. The quotient

is a regular ring of dimension r over the height-at-least-m base; its generic torsion coordinates \phi_{m+1},\ldots,\phi_n give the lifted étale bases. The induced generic Galois action is the full \GL_r(\Z/p^\ell) action [34, §3, Proposition 4.1, and the proof of Theorem 2.1]. Over our finite Honda field we establish the whole reduced closure directly, using the division divisor.

Let C_\ell be the coordinate algebra of the whole r-fold division tuple scheme over A_m. Over B_m, let Q_\ell be the finite étale algebra of ordered bases of the étale division group, and let L_\ell be its full generator-lift algebra. The map Q_\ell\to L_\ell is finite faithfully flat: its fibers are torsors under the product of the r connected division groups. The basis condition selects an open and closed subscheme of the powered tuple scheme, so C_\ell[1/x]\to L_\ell is surjective. Define

This is the reduced closure of the entire lifted basis locus. It is finite over A_m, as a quotient of C_\ell, and R[1/x]=(L_\ell)_{\mathrm{red}}. Every component dominates B_m: the finite flat algebra L_\ell/B_m has its minimal primes over the generic point. Taking their closures adds no vertical component. Consequently A_m\to R is injective and integrality gives \dim R=r. The closed fiber of C_\ell is supported at the tuple of origins, so R is complete local with residue field \kk. Denote the universal lifted coordinates by t_i=t_{i,\ell}, and put t_{i,1}=[p^{\ell-1}]_{\cF}(t_i). At geometric generic points, the p^r combinations of the t_{i,1} are all the points of \cF[p], each with multiplicity p^m. Weierstrass preparation therefore gives the divisor identity

where v is a unit. Equality holds in R, since R is reduced and all its components meet the generic basis locus. Modulo (t_1,\ldots,t_r), the right side is a unit times Z^{p^n}. Successively comparing the Hasse coefficients forces every u_m,\ldots,u_{n-1} to vanish in that quotient. Consequently the maximal ideal of R is generated by the r elements t_i. Completeness gives a surjection \kk[[T_1,\ldots,T_r]]\twoheadrightarrow R. A nonzero ideal of the power-series domain has quotient of dimension at most r-1; since \dim R=r, the kernel is zero. This proves the asserted description of R_{m,\ell} over \kk.

For the powered assertion, set h=p^{m\ell} and let S=A_m[s_1,\ldots,s_r]\subset R, where s_i=t_i^h. This is a finite A_m-module: it is a submodule of a finite module over the Noetherian ring A_m. It is therefore complete; it is local with residue field \kk, and \dim S=r by integrality. The first-level coordinates raised to p^m belong to S. Indeed, [p^{\ell-1}](T) is a series in T^{p^{m(\ell-1)}}, so raising [p^{\ell-1}](t_i) to p^m gives an integral series in s_i. The same is true of all their formal-group combinations.

To apply Equation (3.5) inside S, write [p]_{\cF}(Z)=V(Z^{p^m}). The monic factor on its right is a polynomial in Z^{p^m} whose coefficients lie in S. Division of V(W) by this monic polynomial in W is performed in S[[W]]; the remainder is zero after embedding in R[[W]], hence is already zero in S[[W]]. The quotient is a unit, as its reduction at the closed point is a unit. Reducing now modulo (s_1,\ldots,s_r) again gives a unit times W^{p^r}. The successive Hasse coefficients therefore put all the base parameters in (s_1,\ldots,s_r)S. Thus S is complete with maximal ideal generated by the r elements s_i. The dimension argument just used gives S=\kk[[s_1,\ldots,s_r]]. In particular the inclusion of the base factors through this power-series subring.

The powered coordinates on the exact-height open are the étale division coordinates. Their basis condition gives a surjection Q_\ell\to S[1/x], since the s_i generate the target over B_m. The composite into R[1/x]=(L_\ell)_{\mathrm{red}} is injective: Q_\ell is reduced, and the faithfully flat map Q_\ell\to L_\ell cannot send a nonzero element of Q_\ell to a nilpotent. Thus Q_\ell\simeq S[1/x], identifying the basis algebra on the whole open.

The extension

is free with basis \prod_i t_i^{e_i}, 0\leq e_i<h; this follows by partitioning the monomials in a power series by their exponents modulo h. Its rank is h^r. After localization, R[1/x] is a closed subscheme of the generator-lift scheme over S[1/x]. That lift scheme is a torsor under the product of r connected p^\ell-division groups, so it too is finite locally free of rank p^{m\ell r}=h^r. On coordinate rings this is a surjection of finite locally free modules of equal rank, hence an isomorphism. This proves the assertion about the full scheme, including its nonreduced geometric fibers.

All maps used to define s_i are relative Frobenius maps, and all transition maps are multiplication by powers of p. They commute with group isomorphisms. Changing a basis replaces the t_i by formal integral linear combinations, whose hth powers are the corresponding combinations of the s_i. Hence the descriptions are natural for G and both marking actions. For an étale algebra, relative Frobenius is an isomorphism [37, Lemma 41.14.3, Tag 0EBS]. Root extension therefore commutes with any subsequent finite étale marking extension: if E/S[1/x] is étale, then

The same identity gives descent through marking levels, with their natural change-of-marking actions retained. Finally t_{i,j}=[p^{\ell-j}](t_{i,\ell}) is an integral series in t_{i,\ell}^{p^{m(\ell-j)}} with coefficients in A_m. Substituting the already proved expressions for those coefficients puts it in \kk[[t_{1,\ell}^{p^{m(\ell-j)}},\ldots, t_{r,\ell}^{p^{m(\ell-j)}}]]. At level one, each base parameter lies in the maximal ideal of \kk[[t_{1,1}^{p^m},\ldots,t_{r,1}^{p^m}]], which gives the stated order bound.□

Proof. Choose a level L at least as large as all the \ell_j. A lift of the marked étale generators in \cF[p^L] supplies a splitting of the connected–étale extension at that level: send a standard basis vector of (\Z/p^L)^r to its lift and extend by the group law. The lifts are p^L-torsion, so this is well defined. The resulting product decomposition identifies the division group with

Changing the lifts translates this splitting by \Gamma_m[p^L]^r. The coordinate algebra of the displayed group is a finite-dimensional \kk-space; its restrictions to the lower levels and their powered-coordinate images are functorial under relative Frobenius. They therefore give finite-dimensional representations of the same finite translation quotient. Torsor descent identifies their associated bundles with the original powered division coefficients. Tensor products and finite sums preserve this identification.

The derivative of the connected marking identifies the intrinsic linear coefficient line with the Honda line. This is an equivariant trivialization and uses only a_0, which is invertible on the open. It treats every positive or negative periodicity power. The schemes and their coactions use finite division level, finitely many coefficients, and a finite-dimensional vector space over the finite field \kk. The continuous change-of-marking action consequently factors through a finite quotient on each fixed datum. Finite presentation of the torsor descent and its maps puts them at a finite marking stage; the maximum of finitely many stages works for a finite collection. This assertion is about the basic coefficients. Arbitrary extensions and summands in \mathscr C_m will be handled by exact sequences and retracts, without asserting that every such extension is itself a constant translation representation.□

Proof. Let g\ne1 have finite order N, and suppose it fixes a geometric point z of \Spec B_U. Its image on the base factors through

Every ideal of a Noetherian complete local ring is closed, so R is complete local. The framing action becomes an actual automorphism \gamma of the formal group over R. Its reduction at the closed point is the nonidentity Honda automorphism g. The difference endomorphism

therefore has nonzero reduction, of some finite order d in T. Weierstrass preparation over R makes H=\Spec R[[T]]/(f(T)) finite free of rank d. For every \ell the fixed subscheme of the finite division scheme is

a closed subscheme of H. Every geometric fiber of H_\ell thus has coordinate dimension at most d. This preparation and these finite quotients were formed over the complete base before localization.

Lift z to a full marking t over its algebraically closed residue field. Such a lift exists by the marking torsors: the inverse system of nonempty finite fibers has surjective transitions. As g fixes the U-coset, gt=tu for some u\in U. Commutation of the actions gives u^N=1. Since U is torsion-free, u=1. In particular \gamma induces the identity on the connected height-m formal group, hence on its finite p^\ell-division subgroup inside the specialized division scheme. That subgroup has rank p^{m\ell} and is contained in (H_\ell)_z. Taking p^{m\ell}>d is a contradiction. Notice that only the connected marking is needed for this last contradiction; no splitting of the connected–étale extension at z is being used. To spell out the specialization, its connected division coordinate is nilpotent, so evaluation of each original formal series there is a finite sum. It gives a surjection from the specialized prepared algebra defining H_\ell onto that connected coordinate algebra, since both f and [p^\ell] vanish there. This uses only the finite algebra formed over R and never identifies R[[T]]\otimes_R\Omega with \Omega[[T]].

We give the trace argument for an arbitrary commutative algebra D with a geometrically free action of a finite group F; this avoids a finite-type assumption on B_U. Put C=D^F. The image I=\Tr_F(D) is an ideal of C, because trace is C-linear. If I were contained in a maximal ideal \mathfrak m of C, the integrality of D/C would give a point z:D\to\Omega extending C\to\kappa(\mathfrak m), for an algebraically closed field \Omega. Integrality follows directly from the orbit polynomials \prod_{f\in F}(T-f(d)), and the point follows by lying-over. In D_\Omega=D\otimes_C\Omega, consider the surjective \Omega-algebra evaluations

Geometric freeness says that these maps are distinct. Their maximal kernels are therefore distinct, and the Chinese remainder theorem gives b\in D_\Omega such that e_1(b)=1 and e_f(b)=0 for f\ne1. Hence e_1(\Tr_F(b))=1. On the other hand, writing b=\sum_i d_i\otimes\lambda_i gives

because all \Tr_F(d_i) lie in \mathfrak m. This contradiction shows that I=C. Thus a single element of D has trace one. The argument uses no assertion that invariants commute with residue-field base change. Apply it with D=B_U.

Finally, write a_F=e x^{-c}a in the whole finite model, with a\in A' and c finite. Multiplication is a finite matrix over B_m in the whole basis. Clearing its finitely many denominators makes it a continuous map from any compact lattice into a fixed enlarged lattice. The same statement holds for finite coefficient bundles, after choosing lattices. On an analytic bound where x is bounded away from zero, the same function and finite matrices are bounded. Multiplication is continuous on sections, preserves supports, and commutes with restriction; it therefore acts on the section and support complexes. Retaining a compact profinite parameter means using the uniform seminorm on that parameter, so the same bound applies. These are precisely the trace-one multiplication and semilinearity conditions in the cited cohomological propositions. The additional continuity of the later Laurent representatives under changes of coordinates is verified in Lemma 8.2.□

Proof. The quotient x^aL/x^bL has a finite filtration whose layers are x^jL/x^{j+1}L for a\leq j<b. Multiplication by x^j and Equation (3.1) identify the jth layer with

If L\in\mathscr C_m, this belongs to \mathscr C_{m+1} by Lemma 3.2. For a whole marking model it has a finite filtration with the same property by Proposition 3.4. The next-height hypothesis and the long exact cohomology sequences of these finite filtrations prove finiteness of every strip.

Put V=L[1/x]/x^aL. It is the countable increasing union of the finite-width strips x^{a-j}L/x^aL, j\geq0. With the stipulated bounded-denominator cochains, its cochain complex is their filtered colimit. Filtered colimits of vector spaces are exact, so its cohomology is the filtered colimit of their finite cohomology groups. It is countable, since \kk is finite. The short exact coefficient sequence

is exact on continuous cochains by Lemma 2.2; in concrete terms one lifts a cochain through a compact strip after enlarging its denominator bound. The long exact sequence identifies the localization kernel with an image of H^{i-1}_\cts(G,V) and its cokernel with a subspace of H^i_\cts(G,V). Both are countable, as claimed.□

Proof of Lemma 4.1. Work on an affinoid perfectoid test base T=\Spa(R,R^+) over \kk. The chosen inclusion \mathbb F_{p^j}\subset\kk induces the unramified field F_j in the relative period coefficients; Witt Frobenius \phi restricts to arithmetic Frobenius there. For a topologically nilpotent universal-cover coordinate X, the covariant Lubin–Tate section map is

Here brackets denote Teichmüller lifts. This is the natural additive isomorphism of Fargues and Scholze [14, Proposition II.2.2], applied with coefficient field F_j, uniformizer p, and residue-field cardinality p^j. Pushforward along the unramified degree-j map of coefficient-field curves gives \cO(1/j)=f_*\cO(1), as in the proof of Fargues and Scholze [14, Theorem II.2.14]. Thus its target is \mathcal H_j, and \phi^j\psi_j(X)=p\psi_j(X).

We record the continuity needed to use this comparison in families. On a closed period annulus, its Gauss seminorms can be normalized by |p|=p^{-1} and |[z]|_s=|z|^s, with s in a compact interval [a,b] and a>0. On an input chart \|X\|\leq\rho<1, the ith term of (4.3) has norm at most

For i\to+\infty the first factor forces convergence. For i=-h the bound is p^h\rho^{a p^{jh}}, which also tends to zero. Both tails converge uniformly on the input chart. The formula is therefore continuous, commutes with bounded maps of perfectoid algebras, and glues on test bases. Termwise calculation gives

Consequently the Honda Frobenius \Pi_j:X\mapsto X^p and the Teichmüller scalars satisfy, on both sides with the same composition,

These are the Honda algebra relations of Ravenel [28, A2.2.17–A2.2.18]. They identify the original integral endomorphism ring as a finite free \Zp-algebra generated by \Pi_j and a primitive unramified Teichmüller root. In particular finitely many monomials in these generators span it over \Zp. Compatibility with integers extends to \Zp by continuity: the discrepancy after an approximation modulo p^h has Honda norm at most \rho^{p^{jh}}, and its period-side norm tends to zero on each annulus. The displayed action identities therefore hold for the entire original maximal order, and then for its rational division algebra after inverting p.

For the required sheaf linearity, let \alpha\in C^0(|T|,\Qp) and fix an input section v. Quasicompactness supplies d\geq0 such that b=p^d\alpha is \Zp-valued. Choose its finite clopen-constant approximation b_h modulo p^h, and write b-b_h=p^hc_h, with c_h integral. Integral Honda scalar operations do not increase the norm, so

On each fixed period annulus, multiplication by an integral scalar has norm at most one, and

On each clopen piece constant scalar compatibility gives \psi_j([b_h]v)=b_h\psi_j(v). Taking the two limits and then commuting with p^{-d} proves the assertion for \alpha on the same test base.

Positive-slope vanishing identifies derived sections of \mathcal V_j with its ordinary section sheaf. Relative full faithfulness [2, Corollary 3.11], with the just-verified topological-field sheaf convention, turns the Honda action into a unital, composition-preserving map \End^0(\Gamma_j)\to\End(\mathcal V_j). Its source is a division algebra, so the map is injective. Over a geometric base both algebras have dimension j^2 over \Qp: for the target, the endomorphism bundle has slope zero and rank j^2, and geometric classification together with H^0(\cO)=\Qp computes its sections. Thus the map is an isomorphism. The unique maximal orders correspond, identifying the actual Honda automorphism group with the bundle-frame group. Uniqueness in full faithfulness makes this construction compatible with relative base change. On a general test base the corresponding action is by the sheaf \underline{D_j} of continuous D_j-valued functions; its global sections need not have finite dimension over \Qp. No opposite action is introduced.□

Proof. Adjoining the point coordinates to the powered étale coordinates is a pure-root extension by Proposition 3.5; it does not change a perfected analytic level. Thus these point coordinates compute the completed étale marking tower. Use first the reduced level-l closure in Proposition 3.5. Its coordinate ring has regular parameters t_{1,l},\ldots,t_{r,l}. Two consequences of that proposition are

with F_{i,j,l} integral convergent formal series, without a constant term. Formula (4.8) follows also by iterating the second assertion of (4.4) and expressing the base parameters in the powered level-l coordinates.

For l\geq1 and 0<\rho<1 in the divisible value group, take the closed chart

in the perfected generic fiber of this regular ring. If \lambda=\max |u_i| at a point of this chart, then (4.7) and finite telescoping of (4.4) give

If \lambda>\rho^{1/q_n} these inequalities would give \lambda\leq\lambda^{q_m}<\lambda. Therefore

For y_{i,l}=t_{i,l}^{q_n^l} this yields

The finite integral injective transition maps are surjective on these charts. To check the radius condition in this assertion, lift a point by lying over in the finite map. The bound (4.9) holds at the original point and therefore at its lift; (4.10) then forces the lifted normalized coordinate to have norm at most \rho. Consequently pullback on functions is isometric for spectral norms, also after adjoining all p-power roots.

Let L_\rho be the completed direct limit of these perfected affinoid algebras. The limits z_i=\lim_l y_{i,l} define a map from the perfected radius-\rho Gauss algebra to L_\rho. This map is isometric. To see this without inferring it from points, take a polynomial f in finitely many p-power roots of the variables. At every sufficiently late level, f(y_{1,l},\ldots,y_{r,l}) has exactly the radius-\rho Gauss norm of f: the t_{i,l} are free regular polydisc coordinates, and taking powers followed by perfection does not change the perfected Gauss algebra. These evaluations converge to f(z_1,\ldots,z_r) in the common spectral norm. The limit has the same norm, since the differences eventually have smaller norm than a given nonzero Gauss norm. Completion gives the asserted isometry; its image is therefore closed.

It is surjective as well. For fixed i,j, replace t_{a,l} in (4.8) by z_a^{1/q_n^l}. The substituted series converges on the radius-\rho polydisc and the error is bounded by

Thus t_{i,j} is in the closed image. Taking any fixed p-power root in this argument merely takes that root of the error bound, so all perfected finite-level generators are in the image. They are dense in L_\rho, proving surjectivity. This establishes inverse affinoid morphisms. Varying \rho gives the isomorphism of open balls.

All the preceding estimates are uniform norm estimates for integral series. Hence the constructed morphisms commute with bounded base change, glue on overlapping charts, and apply with any compact profinite parameter space \mathcal D by using the uniform norm on C(\mathcal D,C^\flat). In particular they establish a relative analytic comparison, not just a comparison of geometric points.

The normalized limits respect addition, integral scalar operations, and the full G-action. For example, replace the deformation addition series by its Honda reduction. Their difference on integral inputs has norm at most \lambda. Transporting to a fixed earlier division level contracts this error by q_m^{l-j}, exactly as in (4.11). The same argument applies to the integral coordinate change associated with g\in G and its Honda reduction; compact families of g have the same coefficient bounds. It applies also to the finite sums defining integral changes of an étale basis. Thus the comparison is equivariant for G and \GL_r(\Zp), continuously in both groups.

We can now identify the exact-height open. An integral relation among the normalized sections gives, by the inverse construction, the same relation among their compatible division sequences: if the normalized relation is zero, its point at a fixed earlier level is bounded by \lambda^{q_m^{l-j}} for every l, and is zero. The converse follows by applying the limit formula. Clearing denominators gives the same statement for rational relations. Over height exactly m, the marked étale Tate module has rank r and its marked basis is independent. Over a deeper height its rank is less than r, so the compatible tuple is dependent. It follows that restricting this analytic isomorphism to the exact-height open gives

Here the ambient analytic morphisms have already been constructed; the geometric-point criterion is used only to identify their open subfunctors. The original basis condition is open and closed at each finite level above x\ne0, so no additional basis component is lost in this restriction.□

Completion of the proof of Theorem 4.2. We translate (4.12) into bundle geometry. The geometric classification and the relative constant Harder–Narasimhan theorem used in this step are [14, Theorems II.2.14 and II.2.19].

Let s_1,\ldots,s_r be independent sections of \mathcal V_n over a geometric point, and let \mathcal W be the saturation of their generic span. It is generically generated by \cO^r, so it has no negative-slope quotient: a map from \cO to a negative-slope bundle is zero. If b=\rk\mathcal W\leq r<n, stability of \mathcal V_n gives \deg\mathcal W<b/n<1. Its degree is a nonnegative integer, so it is zero. Slope classification now gives \mathcal W\simeq\cO^b. The map from \cO^r to this bundle is a matrix over \Qp. Independence forces b=r and the matrix to be invertible onto its image. This proves the subbundle assertion over every geometric base curve.

For a general perfectoid test base T, write X_T for its relative Fargues–Fontaine curve and Y_T for the covering space with Frobenius quotient X_T. Let i_T:\cO^r\to\mathcal V_n be the actual map supplied by the sections. Its dual i_T^\vee:\mathcal V_n^\vee\to\cO^r is surjective on every geometric base curve. To see that these curves detect every point, take x\in X_T and lift it to y\in Y_T, whose Frobenius quotient is X_T. Choose a geometric point \Spa(L^\sharp,L^{\sharp+}) over y. The identification Y_T^\diamond=T\times\operatorname{Spd}(\Qp) makes its tilt a map \Spa(L,L^+)\to T. The untilt L^\sharp gives a point of Y_L mapping to y, hence a point of the geometric base curve X_L mapping to x [14, Definition II.1.15 and Propositions II.1.16–II.1.18]. Thus at every x some maximal minor of i_T^\vee is nonzero in the residue field: otherwise it would remain zero at the point of X_L, contradicting surjectivity there. That minor is a unit in the stalk and on a neighborhood, so i_T^\vee is surjective and splits locally. Dualizing makes i_T a locally split injection with locally free cokernel \mathcal Q. Its formation commutes with base change. We have therefore obtained the relative exact sequence

On each geometric fiber, every slope of \mathcal Q is positive: a nonpositive-slope quotient would give a forbidden map from the positive stable bundle \mathcal V_n. It has degree one and rank m. Each positive basic summand has positive integral degree, so there is exactly one such summand, of degree one; hence \mathcal Q\simeq\mathcal V_m. Thus the already constructed vector bundle \mathcal Q has constant basic type. The relative classification [14, Theorem II.2.19(ii), including the dual-surjection argument in its proof] now supplies its local frames.

Conversely, an extension

has no negative-slope quotient, by applying the same vanishing of maps separately to the two ends. A nonnegative bundle of degree one is positive basic unless it has a quotient \cO. The connecting map

sends (a_i) to \sum_i a_i\xi_i, where (\xi_i) is the extension class. Since \Hom(\mathcal V_m,\cO)=0, a quotient \mathcal E\to\cO exists exactly when this map has a nonzero kernel. Thus the middle term is \mathcal V_n exactly on (\mathcal N_m^r)_{\mathrm{ind}}. Moreover \Hom(\mathcal V_m,\cO^r)=0, so an extension with both ends fixed has no nonidentity automorphism inducing the identity on the ends. This identifies the extension moduli with a sheaf, with no residual automorphism group.

It remains to show that an actual connected formal marking supplies the claimed quotient frame in families. Its recursively monic equations in Proposition 3.3 bound all its coefficients by one on the charts above. Its series therefore converges on the open ball and induces a map on universal covers \mathcal H_n\to\mathcal H_m. This is a morphism of sheaves of \underline{\Qp}-modules, where \underline{\Qp}(T)=C^0(|T|,\Qp). Indeed it commutes with addition and the integral scalar operations, and multiplication by p is invertible on universal covers. It therefore commutes with every constant rational scalar. Here is the continuity argument for a varying scalar on the same affinoid perfectoid test object T. Write this section map as \psi, fix v\in\mathcal H_n(T), and let \alpha\in C^0(|T|,\Qp). Compactness gives an integer a\geq0 such that b=p^a\alpha takes values in \Zp. Let b_k be the finite-valued, clopen-constant representative of b modulo p^k, and put c_k=(b-b_k)/p^k. In the fixed Honda coordinates choose \|v\|\leq\rho<1 and \|\psi(v)\|\leq\rho'<1. Integral scalar operations do not increase these norms. Since [p^k]_{\Gamma_j}(T)=T^{p^{jk}}, the Honda differences satisfy

Both bounds tend to zero. On each clopen piece, \psi([b_k]v)=[b_k]\psi(v) by constant scalar compatibility. The continuity of \psi, established by the convergent analytic formulas, therefore gives \psi([b]v)=[b]\psi(v). Commuting with p^{-a} proves the same identity for \alpha on T. This proves linearity for the topological-field sheaf without changing the test object.

The marking kills the compatible integral division sections, since their images are Honda torsion on a perfect characteristic-p base, and hence kills their \Qp-span. The relative section sequence of (4.13) is exact, since H^1(\cO)=0 as a v-sheaf. Thus \mathcal H_n\to H^0(\mathcal Q) is an epimorphism of \underline{\Qp}-module sheaves. The map factors through that epimorphism, and its factor remains \underline{\Qp}-linear.

The relative section functor on perfect complexes is fully faithful by [2, Corollary 3.11, pp. 3680–3681], applied over the present small v-stack. The preceding construction gives a morphism in D(T_v,\underline{\Qp}), compatibly on all test bases, with the topological-field convention of that source. Positive bundles have no higher relative cohomology sheaves, so this statement applies to the ordinary section sheaves just constructed. It gives a unique bundle map

This map is nonzero on every geometric fiber. Indeed, the linear coefficient of the formal marking is nonzero on x\ne0, and on a sufficiently small ball it dominates the higher terms. Universal cover projection is locally surjective, so the induced section map is nonzero there. Both bundles on a geometric fiber have the same stable basic type, whose endomorphisms form a division algebra. Thus f is an isomorphism on every fiber, and hence is a relative bundle isomorphism by the same pointwise maximal-minor criterion. Full faithfulness and uniqueness make this construction compatible with every base change and with the three group actions. This is the needed relative frame construction.

4.3 The integral height of a frame

For a frame of \mathcal Q, use the ordered exact sequence (4.13) to identify \det\mathcal Q with \det\mathcal V_n, and fix the standard identifications \det\mathcal V_n\simeq\cO(1)\simeq\det\mathcal V_m. The determinant of the frame is then a section of the locally profinite sheaf \underline{\Qp^{\times}}. Its valuation is a locally constant integer. The reduced-norm valuation

is surjective and has kernel P_m. Therefore frames of any specified height form a P_m-torsor, locally nonempty.

The height of the frame obtained from a connected marking is unchanged by P_m. An integral étale basis change multiplies the ordered determinant by a unit, so it also preserves this height. The height therefore descends from the full marking tower through both marking torsors to X_m. The domain X_m is connected: it is exhausted by connected annulus-times-polydisc charts with nonempty successive intersections; perfection preserves their underlying spaces. Hence the descended height is one constant integer c.

Changing the fixed determinant normalization absorbs c. Over (\mathcal H_n^r)_{\mathrm{ind}}, the connected-marking tower is a P_m-torsor, and so is the space of quotient frames of this normalized height. The frame map is equivariant for that same group. An equivariant map between two torsors for the same group is an isomorphism: after a local section of the source, its image is a section of the target, and the map becomes the identity on the group coordinate. This proves surjectivity onto every frame of the specified height as well as injectivity.

Finally start with an arbitrary independent extension (4.14). Its middle term has type \mathcal V_n, so its rational frames form a D_n^{\times}-torsor. The determinant of the framed ends specifies a height for a middle frame. Surjectivity of the reduced-norm valuation makes the locus of that height locally nonempty, and it is a P_n-torsor. A frame in this locus produces (4.13) together with a quotient frame of the normalized height. By the preceding paragraph this is precisely a point of Z_\infty. Thus every independent extension occurs and its middle-frame fiber is exactly G=P_n. Taking the quotient gives (4.2). Integral changes at either end, and the ordinary stabilizer on the middle frame, change determinant only by a unit. They preserve the selected heights and act by the stated pushout and pullback operations. More explicitly, for an extension with maps (i,q), postcomposing the connected marking by a changes q to aq; the resulting class is the pullback along a^{-1}. Under the stated kernel-frame convention, b changes i to ib^{-1} and gives pushout by b. These use the covariant Honda action of Lemma 4.1 and commute with the middle-frame action. This completes the proof of Theorem 4.2.□

Proof. After pullback to a perfectoid test space, a torsor for a locally profinite group is represented by a pro-étale, universally open v-cover [32, Lemma 10.13]. For a profinite group H, that lemma presents it as \varprojlim_K P/K, where K ranges over open subgroups and P/K is finite étale over the test space. Over an affinoid perfectoid test space these finite covers are affinoid, and their limit is affinoid perfectoid, hence quasi-compact. Quasi-compactness descends from these test covers to the torsor morphism by [32, Proposition 10.11(o)]. Thus the same assertion holds for the locally spatial diamonds here; in particular a quasi-compact bound has quasi-compact inverse image. Conversely a quasi-compact bound upstairs has a quasi-compact image. The chosen increasing, slightly open box exhaustion covers the space; any such image has a finite subcover by these boxes and is contained in one larger box. The same argument after a finite refinement of charts works on finite covers. The actual relative isomorphism of Theorem 4.2 transports all quasi-compact bounds, so these observations prove the asserted cofinality on both sides. Compact parameter spaces do not change the argument: include the parameter in the quasi-compact bound, or use a finite clopen refinement and its uniform norm.

On an invariant bound, the Čech nerve of a compact-group torsor is its continuous bar construction. Supports pull back to the specified inverse-image bound in every degree. Every bound upstairs can be enlarged to a bound invariant under both groups: its saturation by the compact product G\times U is quasi-compact and is contained in a larger bound. Descent for the function sheaf and for the restriction fiber therefore gives the continuous group-cochain description on these cofinal invariant bounds. The two actions on Z_\infty commute, so taking first U-descent and then G-descent, or in the opposite order, gives the same double bar complex. The support and sheaf complexes are bounded below. In each total degree only finitely many bar and sheaf degrees contribute, and filtered colimits commute with these finite sums and restriction fibers. We may consequently pass to the support colimit in the double bar complex, obtaining (4.15). This also proves its assertion about all compact bar parameters. The noncompact additive cover in (4.1) is not used in this compact-torsor argument; supports for that cover are computed separately in Sections 5 and 6.□

Proof. The countable derived limit is represented by

We prove surjectivity. Given (a_j)_j, construct finite strings v^{(h)}_0,\ldots,v^{(h)}_h satisfying v^{(h)}_j-f_j(v^{(h)}_{j+1})=a_j for j<h. Having constructed such a string, use density to choose v^{(h+1)}_{h+1} so that

is as small as desired. Define the earlier terms by the required relations. Their changes are f_j\cdots f_{h-1}(e_h) for j<h and e_h for j=h. By continuity of these finitely many maps, choose e_h so small that all those changes have norm at most 2^{-h}. For each fixed j the sequence v^{(h)}_j is Cauchy. Completeness and continuity give limits v_j satisfying every required relation.

For the assertion with parameters, a continuous map from \mathcal D to E_j can be approximated uniformly by a map constant on a finite clopen partition. Approximate its finitely many values by images from E_{j+1}. Thus C(\mathcal D,E_{j+1})\to C(\mathcal D,E_j) has dense image, and the same proof applies. Finally the displayed product complex computes the derived inverse limit degreewise. Surjectivity in each degree identifies it with the kernel complex, which is the termwise inverse limit. No closed-image assumption is needed.□

Proof. Adjoin all p-power roots of T. The resulting perfectoid annulus is a pro-étale \Zp(1)-torsor over A_S[a,b]. Choose compatible roots of the two scalar radii. Tilting identifies its structural ring with Equation (5.3), with the same radii: the norm of a tilted coordinate equals the norm of its untilt. The hth term of the torsor nerve is the product of this perfectoid annulus with \Zp(1)^h. Its structural ring is the continuous function ring on that compact parameter space. All these objects are affinoid perfectoid and are v-acyclic for \cO^\flat. Descent consequently gives continuous group cochains. The two-term completed resolution for \Zp, justified for these complete coefficients by Lemma 2.3, gives Equation (5.4). Notice that the topology here is the Banach topology, not the discrete topology on the underlying ring.

For u\ne0, write u=p^v m, where v\in\Z and m is an integer prime to p. In characteristic p,

For v\geq0 this follows from Frobenius and from the unit linear term of (1+X)^m-1; for v<0 take unique p-power roots. Since p^v\leq |u|_{\R}, we obtain

The real absolute value of the exponent is essential in this inequality.

Let P_0 denote projection onto the coefficient of T^0. Define the degree-(-1) map from the source complex to the target by

on degree one, and set it equal to zero on degree zero. For u>0 the ratio of the target and source Gauss weights is (b'/b)^u\leq e^{-\kappa u}. For u<0 it is (a'/a)^u\leq e^{-\kappa|u|}. These inequalities and Equation (5.6) show that H has operator norm at most one. In particular its series converges, including when root denominators are unbounded. Direct calculation gives

This is the claimed chain homotopy. Its entries are fixed scalars, so it commutes with scalar extension and has the same bound for C(\mathcal D,R) coefficients. A translation merely replaces T by the translated coordinate before applying the same formula.□

Proof. We first prove the restriction assertion. By translation take c=0. Set \tau_p=|p|^{p/(p-1)}. Choose fixed constants B>e^{2\kappa} and A_p sufficiently large that, whenever r_{\mathrm{out}}/r\geq A_p, there exists s\in|C^\times| satisfying

Increase B slightly if needed to choose all radii strictly separated. Pick \alpha\in C with |\alpha|=s. Inside D_S(0,r_{\mathrm{out}}) there are annuli about \alpha, one contained in the other, satisfying Equation (5.5), such that both contain D_S(0,r). For example their inner and outer radii may be chosen on opposite sides of s, each pair separated by a factor exceeding e^\kappa, with outermost radius below Bs. The restriction between the discs factors through the restriction between these annuli. By Proposition 5.2 this factors through R[0]\oplus R[-1].

The degree-one constant summand is the image of the mod-p Kummer cocycle of T-\alpha, extended to R. On D_S(0,r),

Choose a pth root of -\alpha in C. The binomial series for (1-T/\alpha)^{1/p} converges because \|T/\alpha\|\leq r/s<\tau_p; this convergence bound is independent of the base. Let u(T) be this analytic pth root of T-\alpha, and let U denote the first root on the restricted Kummer tower. The ratio U/u(T) is of the form \zeta_p^{b} for a locally constant \Fp-valued function b on that tower. With the chosen generator, \gamma(U)=\zeta_pU, and hence (\gamma-1)b=1. Multiplication by this zero-cochain gives an R-linear nullhomotopy of the constant degree-one class. Thus the summand R[-1] maps to zero in the derived category. The remaining factorization through R[0] is evaluation: composing with the section of the smaller disc shows that it agrees with evaluation on the larger disc.

For completeness, degree-zero sections on a fixed disc already equal R. Cover D_S(0,r) by

These annuli cover because the two omitted small discs are disjoint. In the complex Equation (5.4) the kernel is exactly R: every nonzero diagonal multiplier is an invertible scalar. The intersection contains a constant section T=\beta with |\beta|=|\beta-\alpha|=r, chosen using the infinite residue field of C. Hence two annular constants that agree on the intersection are equal in R. Sheaf descent proves the assertion for the disc. The restriction factorization and exactness of filtered colimits now prove the shrinking-germ statement in (i).

Choose an expanding sequence of closed discs covering the line, with successive radius ratios at least A_p. Sections on this open exhaustion are the derived inverse limit of their section complexes; using slightly enlarged open discs gives the same limit. By (i), its degree-zero cohomology system is the constant system R, and every positive-degree system is pro-zero. Equation (5.2) proves the first assertion of Equation (5.8).

We next compute the germ of the exterior. Let E_S(r)=\{|T|>r\}. The Kummer tower identifies its section complex with

where the coefficient ring consists of Laurent series converging on every closed annulus contained in r<|T|<\infty. To justify this assertion, exhaust E_S(r) by a countable increasing sequence of closed annuli. Their coefficient restriction maps have dense image, since finite Laurent polynomials with bounded root denominator occur on every such annulus and are dense on each target. Apply Lemma 5.1 to the two terms of Equation (5.4). This gives exactly Equation (5.9), with its complete family of Gauss seminorms. The same argument works with compact parameters.

For r'>e^\kappa r, division by nonzero \epsilon^u-1 defines a homotopy from Equation (5.9) at r to that at r'. For negative exponents the increased inner radius absorbs Equation (5.6). For positive exponents, to bound a target seminorm with outer radius b, use the source seminorm with outer radius larger than e^\kappa b; such a seminorm is available because the outer radius is unbounded. Thus the homotopy is continuous and factors restriction through the constant copies in degrees zero and one. We conclude that

The map from ordinary sections of the line is the identity on the first summand. Its fiber is R[-2], proving (ii). Moreover the homotopy preserves the constant inclusion from ordinary sections. Taking its fiber proves the asserted pure image statement for maps between support bounds, before taking their final colimit. Explicitly, the boundary of the constant exterior cocycle gives a section s_r:R[-2]\to A_r of q_r. The fiber homotopy identifies a sufficiently separated transition A_r\to A_{r'} with s_{r'}q_r. With the splitting A_r\simeq R[-2]\oplus\operatorname{fib}(q_r), the transition is the identity on the first summand and zero on the second. This proves the assertion about the cones as well.

Here is an explicit normalization of the support generator. The constant degree-one Kummer cocycle of T-c on an exterior of c maps, under the boundary map in Equation (5.1), to the section-support class. For two bounded sections c,c' and a sufficiently large exterior, the quotient (T-c')/(T-c) has a pth root: its difference from 1 has norm below \tau_p. Their Kummer cocycles therefore agree there. This proves translation invariance and shows that every section-support class maps to the same generator in Equation (5.8). Excision for a finite disjoint union of discs gives a direct sum of support complexes. The map from that direct sum to a common enlarged bound consequently sums the section generators.

The arguments so far are relative: after a further affinoid perfectoid base change all formulas are unchanged. They therefore identify the relative section, germ, and support complexes as sheaves on the base v-site. Apply them successively in ordered coordinates. The finite box-support diagram is the iterated fiber of the restriction maps in those coordinates; finite fibers commute, and the one-coordinate constant inclusions and Kummer classes commute with every remaining coordinate restriction. The resulting complex is the tensor product of d copies of R[-2], namely R[-2d]. The same argument with evaluation gives ordinary and section-germ complexes R[0]. Taking a fixed sufficient radius ratio in each coordinate gives the uniform pure-image statement in (iii). This verifies the product assertion using the actual box fibers, without assuming a compact-support comparison for arbitrary analytic coefficient sheaves.

Finally use the absolute Artin–Schreier sequence on the v-site,

It is exact because its equation is solved by a finite étale cover on every characteristic-p perfectoid chart. This is also the sequence used in [30, proof of Theorem 5.1, p. 47]. The constant inclusions, evaluations, and chosen Kummer classes are compatible with this sequence. On the Laurent complexes this compatibility is explicit: Frobenius sends c_uT^u to c_u^pT^{pu}, projection onto the constant term commutes with it, and the homotopy does too, since

Taking the fiber of Frobenius minus the identity in the preceding relative computations therefore gives the corresponding base complex \RGamma(S,\Fp), with the same shifts. Finite fibers commute with the filtered colimits used here.

If R=R_{\mathcal D}, Frobenius minus the identity is surjective on R. Indeed the Artin–Schreier map on \Cflat has a continuous inverse branch on a sufficiently small ball about each point of its target. A finite clopen refinement of \mathcal D permits a continuous choice of these branches. The kernel is C(\mathcal D,\Fp), whose functions have finite image. This proves (iv). For a general affinoid perfectoid base we have kept the base complex, and have not asserted this global surjectivity. The finite-p affine-line calculation also agrees with the direct Kummer computation in [1, Example 2.19, proof of Lemma 2.20, Equation (2.22)].□

Proof. The map E_{j+N}\to E_j goes from a smaller support bound to a larger one. Excision inside the separated collars identifies it with a finite sum of maps from the support in child discs to the support in their parent discs. To see the excision used here directly, cover an ambient chart by a neighborhood of its support and the complement of a slightly smaller support bound. The Mayer–Vietoris square identifies the two corresponding fibers in Equation (5.1). Interleaving the inner and outer bounds yields the stated collar excision. It is compatible with restriction on the base and with a further finite cluster refinement.

By Proposition 5.4, after the fixed radius change each child-to-parent map has pure geometric support image in degree at least two. The estimate is uniform before the finite sum, by Remark 5.5. Thus it induces zero on geometric cohomology in degrees zero and one. For \Fp the base complex in Proposition 5.4(iv) is connective, so its shift by a support degree of at least two has the same low-degree vanishing. This proves the pro-zero assertion locally over the base.

We spell out its passage through descent. Use a qc covering diagram in each cosimplicial degree. Fix a total degree i. Because the geometric complexes are bounded below by zero, only cosimplicial degrees at most i+1 can affect the vanishing in degrees up to i and its comparison map. Make the finitely many required chart and collar refinements in these degrees simultaneously. The spectral sequence obtained by first taking geometric cohomology has zero transition on the relevant low-degree rows. If necessary repeat the fixed refinement once for each filtration piece. A map zero on associated graded pieces lowers the finite filtration, so this finite composite is zero on the required total cohomology. Hence the total degree-zero and degree-one systems are pro-zero as claimed. This argument uses no common refinement for all degrees of an unbounded descent diagram.

By (c), sections on X\setminus Z are the derived inverse limit of sections on the increasing complements X\setminus T_j. Finite fibers commute with this limit, so

All E_j are connective. In Equation (5.2), the limit and first derived limit of the pro-zero low-degree systems vanish. This proves H^0\RGamma_Z=H^1\RGamma_Z=0. No surjectivity assertion about the possibly large degree-two system is needed. Finally the long exact sequence for Equation (5.1), with Z in place of K, identifies degree-zero sections before and after deletion. The same proof applies to every retained compact parameter, since all radius bounds and finite sums were uniform in it.

For the last assertion use the interleaved open bounds of the exhaustion and put

These support complexes have canonical restriction maps as a varies, independently of the tube choices used to compute each one. The fixed-bound argument gives H^0(C_a)=H^1(C_a)=0, and each C_a is connective. Both X_a and X_a\setminus Z admissibly exhaust Y and Y\setminus Z, respectively. Taking their section limits and commuting finite fibers with the derived inverse limit gives

The Milnor sequence again gives zero in degrees zero and one: the relevant ordinary limits and \varprojlim^1 H^0(C_a) are all zero. This uses the already formed support complexes C_a; it requires no common tube depth for the different output bounds.□

Proof. A completed Koszul resolution on the e commuting generators of L computes its continuous cohomology. With trivial coefficients its differentials vanish, giving the asserted exterior algebra. One can compute the maps first over \Zp. If a basis of L' is v_1,\ldots,v_e, then a basis of L is pv_1,\ldots,pv_e. Restriction in degree i is multiplication by p^i on the corresponding exterior basis. The identity \operatorname{res}\operatorname{cor}=[L':L]=p^e and torsion-freeness then give multiplication by p^{e-i} for corestriction. Reducing modulo p proves both assertions. The same finite Koszul complexes, tensored with R_{\mathcal D} or with a base complex, compute the relative assertions. Thus no infinite product of coefficient spaces is introduced in this computation.□

Proof. Consider first s=1. Let D_a be the closed disc of radius R_a=R_0|p|^{-a} in the untilted affine line, and put

Choose R_0 between adjacent nonzero norm values of F_m and interleave these discs with slightly larger ones. Then L_{a+1}=p^{-1}L_a, the B_a exhaust the quotient, and these bounds are cofinal for Equation (6.1). Indeed every point has a representative of finite norm, and every quasicompact bound is contained in one of the increasing slightly-open images.

The inverse image of B_a in the affine line is

The displayed pieces are disjoint and locally finite: on any bounded disc only finitely many cosets occur. Consequently sections with this support form the product of the sections with the indicated disc supports. With the translation action this is the continuous coinduction from L_a. This assertion also applies to derived sections: on bounded charts it is finite disjoint excision, and an increasing exhaustion gives the displayed product. Descent through the translation torsor and Shapiro’s map therefore identify the quotient-support complex with

Here Shapiro’s map can be read directly on the bar complexes: choose the continuous coset section for the open subgroup L_a\subset F_m and insert the chosen representatives. The inverse inserts the same representatives in the coinduced function. Their homotopies are the usual insertion homotopies. Thus this use of Shapiro does not impose compact inverse image support under the noncompact translation cover. After Shapiro all bar parameters belong to compact powers of L_a.

Put A_a=\RGamma_{D_a}(\mathbb G_{a,C}^{\diamond},\mathscr E). The canonical maps

are the oriented comparison maps of Proposition 5.4. Their cones become cohomologically zero under a sufficiently large fixed ratio of support radii. This holds before group cohomology, with arbitrary compact parameters. To use it in Equation (6.4), first enlarge the geometric disc while retaining the smaller group L_a. The cone map is zero on the geometric cohomology rows with parameters L_a^j. Only afterwards group the translated copies in a larger disc. Each grouping is a finite sum, indexed by the appropriate lattice quotient, and hence preserves the vanishing of the cone map on those rows. This order avoids any requirement that a chosen representative of a Kummer class remain invariant when its center lies near the new boundary.

Under increasing a, the old disc supports inside a new disc are indexed by L_{a'}/L_a. The support map is the sum of their translated extension-of-support maps. On the oriented base classes it is precisely corestriction. We therefore obtain

To justify the comparison on total complexes, fix a total degree. The geometric and positive bar complexes are uniformly bounded below, so only finitely many bar degrees enter. Take a common radius enlargement in those degrees. The cone has zero colimit on every relevant geometric row, and the resulting spectral sequence gives zero colimit in that total degree. Filtered colimits are exact on the underlying vector spaces. Thus this reasoning proves Equation (6.5) without interchanging an uncontrolled limit and an unbounded totalization.

Lemma 6.1 leaves only the degree-m orientation line in Equation (6.5). This proves the case s=1. For s coordinates use product boxes in the affine space and the lattice L_a^s of rank ms. Product boxes with a common increasing bound are cofinal. The product Kummer calculation gives degree 2s, and the same lattice calculation leaves degree ms, proving both displayed formulas.

All these calculations are relative. On an arbitrary perfectoid base the phrase “base class” means its base complex, not that arbitrary perfectoid bases have acyclic \Fp-cohomology. For the external base \underline{\mathcal D}\times\Spa(\Cflat) these base complexes are E_{\mathcal D} and R_{\mathcal D}, respectively, by Proposition 5.4. The Kummer operators are linear over the retained coefficient ring and their bounds are independent of the compact parameter. This proves the parameter assertion and its compatibility with extension of constants. The constructions use canonical restriction and support maps, so their identifications are natural under compact continuous families of automorphisms.□

Proof. Let E=\Qp^t\oplus F_m, endowed with a fixed \Qp-norm. Independence says that \ell_z:E\to C' is injective at each geometric point with untilt field C'. Its unit sphere is compact. On a bounded base chart there is a uniform upper bound |\ell_z(w)|\leq M\|w\|. At a fixed geometric point there is also a positive lower bound on the unit sphere. Choose a sufficiently fine finite net in that sphere. The upper bound shows that the error in replacing a coefficient by a net point is smaller than the chosen lower bound. Thus finitely many strict analytic inequalities on those net points give the same lower bound on an open base neighborhood. Quasicompactness gives a finite refinement with uniform bounds

One may refine by rational charts after introducing the strict margins. A strict rank-one inequality persists at all the higher-rank points in such a chart, which is why interleaved radii are used.

At one geometric point choose a compact coefficient annulus large enough to contain all coefficients whose images could have norm near a proposed radius. This is possible by Equation (6.7). On this annulus the image avoids zero. The nonarchimedean norm is locally constant away from zero; hence it takes only finitely many values on the compact annulus. Choose \alpha<\beta in a gap between those values. The preimage of the disc is an open compact additive subgroup of E, and therefore a lattice L. A finite coefficient net on the annulus, with the upper bound in Equation (6.7), makes the inequalities defining this same L persist on an open base neighborhood. Outside the annulus the upper and lower bounds already determine membership. Taking a finite base refinement proves Equation (6.6) on the entire refined chart. Finally \ell_z(p^jw)=p^j\ell_z(w), proving persistence for every depth on the same chart.

Both L_a and L_b are lattices. In addition, the exact sequence

splits as a sequence of finite free \Zp-modules. Fix such a splitting once on the chart. Its F_m-component gives a continuous translation choice for every member of a slope cluster. In particular, if a'-a\in p^jL_a, there is b\in F_m with (a'-a,b)\in p^jL, so that

Thus the graph of that entire slope coset lies in the tube centered at \sum_i a_i z_i. The splitting ensures this assertion holds for the relative family, not only on individual fibers.

Conversely, two such quotient tubes of open radius \beta|p|^j can intersect only if, for some b\in F_m, |\ell_z(a-a',b)|<\beta|p|^j. By Equation (6.6) this forces a-a'\in p^jL_a. Distinct slope cosets therefore give disjoint tubes. The stabilizer of any one covering disc is \{b\in F_m:|b|<\beta|p|^j\}=p^jL_b, with the same group for the closed radius. Its full inverse image under the translation cover is the disjoint union indexed by F_m/p^jL_b.

Because the slope patch is compact and open, it is a finite union of p^jL_a-cosets for every sufficiently large j. Choose representatives at each depth. If a refined representative lies over a coarser one, the splitting just chosen supplies a constant translation making its disc lie in the coarser disc. Thus all maps between the tubes are maps between actual relative analytic discs. On passing to the quotient they are independent of that translation choice. For a child disc, the ultrametric inequality also permits its parent disc to be centered at the child’s section without changing the parent disc: the two centers differ by at most the parent radius. Thus the child-to-parent maps are the concentric support maps of Proposition 5.4. Taking a fixed number of further depths makes their radius ratio at least the constant required there.

We spell out neighborhood cofinality. A rational neighborhood of an analytic section contains a relative disc of some positive radius around that section after a quasicompact base refinement. Indeed write its finitely many defining inequalities using an analytic denominator. On a suitable base chart that denominator has a positive lower norm bound along the section. Taylor estimates on a bounded ambient disc bound all the changes in the numerators and denominator by a constant times the distance to the section. For a sufficiently small radius, the denominator keeps its norm and the defining inequalities remain true, by the ultrametric inequality. This argument also works when an inequality holds with equality at the section.

Apply this observation to an open neighborhood of the compact graph family. The product of a quasicompact base chart and a compact profinite coefficient patch is quasicompact; its topology has the usual basis of base opens and clopen parameter patches. A finite refinement therefore supplies finitely many positive disc radii. A sufficiently deep common p^jL-refinement, and the smaller of those finitely many radii, put all its tubes in the given neighborhood. Their intersection is exactly the graph: by Equation (6.7), a bounded convergent sequence of centers has a convergent coefficient subsequence, and the coefficient patch is closed. This proves both cofinality and the intersection assertion.

For support fibers, make this construction first on a slightly larger independent-base bound and on the finitely many rational pieces of its intervening collars. These bounds retain the strict lower bound in Equation (6.7). The first t tuple coordinates are unchanged by forming a normal tube, so the tubes remain in the rank-at-least-t locus. Bounded matrix slopes bound all remaining tuple coordinates. Consequently product bounds in the base and the compact slope patch are cofinal with the tuple bounds in this chart. Conversely, a tuple bound separated from lower rank has an independent-base bound of this form, by Equation (6.7). This proves the assertion about collars and supports.

Finally an external compact parameter may be included in the base chart and in its supremum norm, or treated by finite clopen refinement. All finite-net estimates retain the same bounds. Products use finitely many copies of E; taking common refinements and the maximum of finitely many required radius losses proves the last assertion.□

Proof. First fix a support bound in X_{s,\geq t} and a slightly larger bound. Restrict a section with that support to a neighborhood of the closed rank-t locus inside the larger bound. The fiber of this restriction is a section supported off that neighborhood; one can replace the neighborhood by a slightly smaller one to separate all boundary radii. This is the Mayer–Vietoris support fiber sequence on the bound and its collar. As the neighborhood shrinks, the remaining bounds are cofinal among compact bounds in X_{s,\geq t+1}. Conversely any such compact bound is disjoint from a sufficiently small neighborhood of the closed locus, by the strict separation on a slightly enlarged bound. Taking the exact filtered colimit of these support triangles and then increasing the original bound proves Equation (6.9). In particular its third term uses restriction to shrinking neighborhoods. It is not an inverse limit of cohomology supported in shrinking tubes.

To calculate that third term, use Proposition 6.3. On a fixed base chart, at depth j there are finitely many slope clusters and actual relative disc tubes. Upstairs under the translation cover, other discs are the disjoint translates of these discs. Shapiro reduces their cohomology to the compact stabilizers p^jL_b; this is the same explicit bar construction as in Equation (6.4).

The comparison is induced by actual base inclusions. On a refined base chart S, write C_S=\RGamma(S,\mathscr E), retaining any external parameter in S. Pullback from S to each covering disc is translation-equivariant. After Shapiro it gives the map

These maps commute with refinements and with the restriction maps on the base and its support collars.

Restriction through a sufficiently large fixed ratio of tube radii has pure base image on geometric cohomology, by Proposition 5.4. The radius ratio is independent of the center and of compact parameters. The finite refined cover of Proposition 6.3 works for every depth; hence one further depth works simultaneously for its charts, its support collars, and all the compact stabilizer-bar parameters in any fixed total degree. It follows that the geometric cone rows vanish before applying the group maps. The group maps here are restrictions to p^{j'}L_b\subset p^jL_b. By Lemma 6.1, they kill every positive lattice-cohomology degree and preserve the base degree zero. The resulting neighborhood-germ complex is therefore a copy of the base complex for each slope cluster. When a cluster splits, restriction replicates its class in each new cluster.

This argument is relative over the independent base, rather than just a calculation on its geometric fibers. More explicitly, the centers in Equation (6.8) are sections on entire refined charts, so the Kummer homotopies are linear over their base rings. Pullback of a base class realizes the comparison on each tube. On overlaps the comparisons are the restrictions of the same class; a further interleaving of tubes compares different choices. For descent over a support bound and its collar, choose a v-hypercover by quasicompact perfectoid charts in every degree needed. Fiber products are quasicompact after the bounded translation reduction. In a fixed total degree only finitely many of these degrees occur. One common further tube depth kills the cone rows there, so the bounded-below descent spectral sequence proves the relative comparison. Applying the support fiber on the base gives Equation (6.10).

There are only finite sums at each partition. Their exact filtered colimit is \operatorname{LC}(P,-), and an external compact parameter enters the coefficient ring of these finite sums. Thus the construction allows continuous coefficients in \mathcal D but does not allow a single germ to use unboundedly fine slope partitions with no common neighborhood. The comparisons agree under restriction to the actual graphs, which also proves their naturality under changes of graph chart and changes of lifts.□

Proof. Induct on s. The case s=0 is the coefficient calculation on the external parameter space. For the induction step start with the whole space X_{s,\geq0}=\mathcal N_m^s, whose comparison is Proposition 6.2. Delete the ranks t=0,\ldots,s-1 successively using Equation (6.9). For each such t, the germ term is locally Equation (6.10). By the inductive hypothesis it has the scalar comparison in question. The functor \operatorname{LC}(P,-) is an exact filtered colimit of finite direct sums and commutes with algebraic scalar extension. For its gluing, use integral frames of the coefficient quotients: the image of \Zp^s in a t-dimensional coefficient quotient is a free lattice, and its kernel in \Zp^s is a saturated direct summand. Dually, the coefficient plane meets the dual integral lattice in a saturated rank-t summand. Consequently M_s acts transitively on the Grassmannian, and integral graph charts give continuous local sections of M_s\to M_s/\Pi_t, where \Pi_t stabilizes the standard plane. The resulting transition functions on the independent base lie in \GL_t(\Zp). Their action on any finite set of cohomology classes is locally constant by inductive smoothness; compactness gives a common finite clopen refinement. A finite disjoint clopen trivializing cover computes global sections by finite sums of the exact functors \operatorname{LC}(P,-). Thus the germ comparison glues with its algebraic finite-span structure on the compact Grassmannian. Exactness of scalar extension over \Fp and the support triangles prove Equations (6.11) and (6.12) for the final complement Y_s.

The proof took place on uniform compact parameters before the support colimit and before each bounded-below totalization, as established in Lemma 6.4. It therefore gives the asserted comparisons on actual coefficient complexes. In particular it does not replace continuous families of unrestricted functions by an algebraic tensor after taking cohomology.

Smoothness, including the commuting P_m-action, now follows from the parameter comparison. For a compact acting group H and a class v, pullback by its action gives a family of classes with parameter \mathcal D=H. Supports remain uniform: a compact group translates a quasicompact bound into a quasicompact image contained in a larger bound. By Equation (6.11), this family lies in C(H,\Fp)\otimes_{\Fp}H_c^i(Y_s,\Fp). It is therefore a locally constant finite-valued orbit map. Thus the stabilizer of v is open. This proves smoothness for both commuting actions before we use the category of smooth representations below.

We next identify the representation in a germ term. Let \Pi_t\subset M_s be the stabilizer of the standard coefficient t-plane. Its action on the independent base is through its \GL_t(\Zp) Levi factor; the other Levi factor and the unipotent radical act trivially on the normal degree-zero germ. Hence

where induction is smooth induction and no support condition on M_s/\Pi_t is needed because that quotient is compact. To see the identification, use the integral trivializations just chosen on the graph charts. Sections are locally constant choices of an independent-base class, and transition functions are the changes of basis of that plane. These are exactly the equivariant locally constant functions defining the right side. The comparison is by restriction to the plane, so it is compatible with the group action and not just an isomorphism of vector spaces.

Smooth parabolic induction in Equation (6.13) preserves admissibility. Indeed an open subgroup H\subset M_s has finitely many orbits on M_s/\Pi_t. For each orbit, the stabilizer H\cap g\Pi_tg^{-1} maps onto an open subgroup of the \GL_t(\Zp) Levi factor. An H-fixed induced function is determined by a vector fixed under that open subgroup. There are finitely many such choices, each finite-dimensional by induction.

For completeness, admissible smooth representations of a compact p-adic analytic group over \Fp are closed under subobjects, quotients, and extensions. Take continuous duals into \Fp. A smooth module is admissible precisely when its compact dual is finitely generated over the completed group algebra: finite coinvariants for an open uniform pro-p subgroup give finite generation by compact Nakayama, and the converse follows by taking coinvariants at open subgroups. The completed group algebra is Noetherian, since the uniform subgroup algebra has its polynomial associated graded ring and the full algebra is finite over it. Finite generation is therefore closed under submodules, quotients, and extensions. Dualizing proves the assertion for smooth modules.

The whole-space cohomology in Proposition 6.2 is finite. The long exact sequences of Equation (6.9), Equation (6.13), and the preceding closure property now prove admissibility for Y_s. Countability follows in the same induction: the compact p-adic Grassmannian has a countable clopen basis, so locally constant functions on it with values in a countable \Fp-space form a countable space, and the support exact sequences preserve countability.□

Proof. Let V^\vee=\Hom(V,\Fp) be the compact dual. It is finitely generated over \Fp[[H]], as in the proof of Theorem 6.5. Noetherianity gives a resolution of V^\vee with finitely generated free terms, finite in each degree. Tensoring it with the trivial module \Fp shows that \Tor_a^{\Fp[[H]]}(\Fp,V^\vee) is finite in each degree. Continuous duality identifies this with the dual of H^a_{\cts}(H,V).

For scalar extension use the finite free resolution of the trivial module supplied by Lemma 2.3. Exhaust V by finite-dimensional H-stable subspaces. Such an exhaustion exists because V is countable and every smooth vector has finite orbit under the compact group. Give R_{\mathcal D}\otimes V the union of the complete finite-span steps so obtained. Its actual bounded-step bar cochains are

Indeed this is true on every finite span and then on their union. These are exactly the row families given by Theorem 6.5 with parameter \mathcal D\times H^a, compatibly with every bar face. The finite-type completed resolution and its comparison with continuous cochains apply by Lemma 2.3: each vector belongs to a finite continuous representation, and compact families are retained in one such step. In every degree the resolution calculation is a finite power of this coefficient module. A matrix entry in \Fp[[H]] acts on a finite H-stable span through a finite quotient of H, so its action after extension is R_{\mathcal D}-linear scalar extension of its action on V. The differential is therefore scalar extension of the differential over \Fp. Exactness of tensoring over \Fp gives the displayed isomorphism. These are the same finite-span families as in Theorem 6.5, not a newly imposed completion or topology on its cohomology.□

Proof. Write U=U_m\times U_r. The two compact frame torsors and Lemma 4.4 identify the complex in the assertion with

Indeed, on the completed full marked tower the actions commute. Descent first by U and then by G, or in the opposite order, gives the two displayed descriptions. Compact torsors preserve the cofinal systems of support bounds. The uniform-parameter comparisons of Theorem 6.5 apply to their bar diagrams. Bounded-below totalization in each degree therefore justifies this interchange of descents with the support exhaustion.

By Theorem 6.5 the geometric cohomology rows are scalar extensions of the smooth admissible modules H_c^b(Y_r,\Fp) for M_r. Their U_r-cohomology is finite in each degree by Lemma 6.6. The residual U_m-action on these finite \Fp-spaces is smooth, since the original commuting action is smooth and the finite-type cochain comparison retains its compact parameters. A finite-type resolution for U_m then has finite-dimensional terms with these finite coefficients; its cohomology is finite in each degree as well.

The successive first-quadrant spectral sequences have only finitely many terms in any prescribed total degree. Their entries, and hence their abutments, are finite over \Fp before scalar extension. The same finite-type comparisons and Equation (6.12) identify the \cO^\flat answers with extension to \Cflat, or to R_{\mathcal D} when the parameter is retained. This proves the assertion. No finite cohomological dimension of a torsion-containing full stabilizer is used here.□

Proof. For s\geq1, work locally over an affinoid perfectoid chart S of Y_{s-1} carrying untilted lifts z_1,\ldots,z_{s-1}. Such charts exist by the translation presentation. The inverse image of the last independent coordinate in the untilted affine line is the complement of the closed graph family

Refine the bounded independent-base chart using Proposition 6.3. Choose \rho strictly between its persistent radii \alpha<\beta and exhaust the output coordinate by discs

The centers in X_a correspond exactly to the compact open coefficient lattice p^{-a}L, and all have norm at most \alpha|p|^{-a}. They therefore have a separated outer collar inside X_a. This uses the persistence for negative as well as positive powers of p; the same linear identity proves both. At fixed a, and sufficiently large j, the closed tubes T_j before translation quotient are the finite disjoint discs indexed by p^{-a}L/p^jL, with radius \alpha|p|^j and centers \ell_z(w) at constant coefficient representatives w. Their open companions have radius \beta|p|^j and lie inside the same collar. By Proposition 6.3 these companions are a cofinal neighborhood system, and their complements admissibly exhaust X_a\setminus A_z. On the finite base refinement the exhaustions just chosen have common enlargements, so they compute the same support fibers on overlaps.

Let X denote this bounded ambient chart with its chosen collar, and put

The maps are C_{j'}\to C_j for j'\geq j: smaller support maps to larger support. A small cluster is included in its containing larger disc; the transition sums these maps when several clusters are included in one larger cluster. By Lemma 5.6, after a sufficiently large fixed ratio of these radii the maps on H^0 and H^1 are zero. This is vanishing of the entire cohomology-group map, not elementwise eventual vanishing. The uniform persistent lattice charts and support collars are precisely the hypotheses needed for that lemma. Finite sums of the cluster maps retain the vanishing. On a finite descent cover one may compose finitely many further refinements to kill the finite filtration in these two total degrees. Hence both low-degree inverse systems are pro-zero on the whole bounded chart.

Since complements exhaust X\setminus A_z, limits preserve the support fiber and give

The complexes C_j have no negative cohomology. The Milnor exact sequence for this countable tower therefore shows

both ordinary limits of H^0,H^1 vanish, and the possible \varprojlim^1 H^0 vanishes because that system is pro-zero. No condition on the surviving degree-two support system is required.

Now increase the output-coordinate bound. At every fixed bound the coefficient set p^{-a}L is compact, so the preceding argument applies. Use the interleaved open bounds and set

Although the tube refinements may depend on a, these support complexes restrict canonically. The bounds and their complements admissibly exhaust the relative affine line and its complement of A_z. Thus the exhaustion assertion of Lemma 5.6 gives

Each C_a is connective with H^0(C_a)=H^1(C_a)=0, so the Milnor sequence proves the same low-degree vanishing on the whole line. The two limits have been taken in order: first the tubes at a fixed output bound, then the canonical support complexes as that bound increases.

The support triangle and Proposition 5.4 now give an isomorphism from base functions to the ordinary functions on the affine line minus A_z. This is a relative statement on the perfectoid base charts: all maps and estimates are linear over the base and agree on overlaps. Degree-zero descent by the translation group F_m takes invariants of these base functions, on which the group acts trivially. Hence projection Y_s\to Y_{s-1} has the claimed relative degree-zero direct image. Induction, starting with Y_0, proves Equation (6.14). Retaining \mathcal D in the same relative calculation gives its parameter version.

For the last assertion, the finite model injects into its perfected analytic realization and then into the full marking tower: analytic evaluation on the exhausting bounds is injective, and the marking extension is faithful. A G-invariant function descends through the actual torsor of Theorem 4.2. By Equation (6.14), its image is a scalar \lambda\in\Cflat. Apply coefficient-only |\kk|-Frobenius to this equality back on the tower, whose finite model is defined over \kk. It fixes the original algebraic function, whereas it sends \lambda to \lambda^{|\kk|}. Injectivity gives \lambda^{|\kk|}=\lambda, so \lambda\in\kk. This argument does not require the chosen untilted quotient presentation itself to commute with coefficient Frobenius. If a preliminary finite constant extension was needed to define the model or its idempotent, the conclusion is in that finite field.□

Proof. The perfectness in Equation (7.1) makes the h-power map an isomorphism on the total lift torsor. Write \rho:\cO(\mathscr P)\to\cO(\mathscr P)\otimes_\kk\cO(T) for its coaction. Absolute powering satisfies

Here \Fr_h^* raises all coefficients, including the base parameters, to the hth power. Thus this is a map over the matching Frobenius base, not an endomorphism over the original marked base. The Honda relation identifies the translation Frobenius with [p^\ell]. In torsor-action notation the same identity is \Fr_h(t\cdot z)=[p^\ell]t\cdot\Fr_h(z). This endomorphism of T gives an isomorphism

onto a subgroup of T. It is not an automorphism onto all of T. For example the projection of [p^\ell]T to the level-\ell quotient is zero. The group T need not be reduced, although the total torsor in Equation (7.1) is perfect.

It follows from Equation (7.6) that, over B_\infty^{1/h}, the same total lift space with structure group [p^\ell]T is the root of the original T-torsor. Associated-bundle induction in stages now gives

where \pi:\Spec B_\infty^{1/h}\to\Spec B_\infty is the finite root map. The pushforward is the restriction of scalars specified above. One can verify the equality without a choice of a torsor point: after a faithfully flat trivialization it is the usual evaluation isomorphism for coinduced equivariant functions, and that isomorphism commutes with the two projections in the descent diagram. It therefore descends, is natural in E, and identifies each morphism of any finite diagram. This also proves exactness, since the trivialized functors are exact and the base changes are faithfully flat.

All the maps used here commute with G. They also commute with the changes of markings, since [p^\ell] is central and the chosen power fixes \kk. For two powers, the identities (z^{h_1})^{h_2}=z^{h_1h_2} and [p^{\ell_2}][p^{\ell_1}]=[p^{\ell_1+\ell_2}] show that the two resulting torsor maps are equal. The evaluation map for coinduction in stages has the same composition law. This proves the asserted coherence, including the maps of the diagrams, rather than only an identification of their terms.

Finally, the original finite diagram, its torsor quotient, and its coaction involve finitely many matrix coefficients and relations. They descend through the filtered union of finite étale marking algebras. Choose one stage containing this data and shrink U so that the original diagram has the required auxiliary action there. At every coinduction level the preceding identity is still the same intrinsic power identity, with its natural auxiliary action; it therefore descends to that stage. Equivariance can be checked after the one faithfully flat full-marking extension, since G preserves each finite marking algebra. This does not require intersecting depth conditions for individual elements of G, or making all high coinduced representations trivial under U.□

Proof. The intermediate order-p root cover is a torsor under the height-one Frobenius quotient of translations, of rank p^r. Over the chosen Honda constants this quotient is \alpha_p^r for m>1 and \mu_p^r for m=1. This follows by taking the first Frobenius kernel of the Honda group; its Verschiebung is zero in the first case and invertible in the second. These finite Hopf algebras and their Lie spaces are defined over a finite field. Their continuous marking actions factor through a finite group. Shrink U to fix the quotient and a Lie basis. For m=1 choose the standard toral basis of the fixed \mu_p^r. Finite presentation of the coaction descends its torsor action to a finite stage. Torsor freeness identifies the resulting invariant vector fields with a basis

For completeness this freeness can be checked after trivializing the torsor: the vectors are \partial/\partial z_i on \alpha_p^r and z_i\partial/\partial z_i on \mu_p^r. Their coefficient matrices are invertible, and invertibility descends faithfully flatly.

For b\in R, define

This formula is additive, satisfies the ordinary Leibniz rule, and is \kk-linear because \kk is perfect. To see that it is a frame, let (t_1,\ldots,t_r)=(x,y_1,\ldots,y_{r-1}), a p-basis after the finite étale base change. Write

Then \partial_i=\sum_j a_{ij}^p\partial/\partial t_j and the new determinant is \det(a_{ij})^p\in R^\times. The derivations commute with G, because the full-marking action commutes with translations and with powers. The identities descend to this same finite stage: they hold after the faithful full-marking extension, and G preserves R. The dual frame and its top wedge give \eta. All differentials in this construction were taken at a finite root stage.

We describe the Cartier operator used in Equation (7.7). If dt=dt_1\wedge\cdots\wedge dt_r and h=p^\nu, the unique p-basis expansion of a form is

The operator is

For power series the expansion groups the exponents modulo h; localization and finite étale base change preserve it. First take h=p. In one variable the de Rham differential on a residue exponent other than p-1 gives an exact term, while the last exponent gives the class of t^{p-1}dt. Tensoring these one-variable complexes proves the Cartier isomorphism in r variables, and iterating it gives Equation (7.9) for h=p^\nu. The inverse of the one-step isomorphism is characterized on one-forms by db\mapsto[b^{p-1}db] and by multiplication on exterior powers. This characterization shows that the formula commutes with coordinate changes and étale base change. For the customary inverse-Frobenius convention and its duality interpretation, see Blickle and Böckle [5, Definition 2.1 and Example 2.23]. It is continuous on the indicated formal coefficient modules. In particular it commutes with G and satisfies \Cartier^\nu(b^h\omega)=b\Cartier^\nu(\omega).

Equation (7.9) proves the asserted linearity and dual-generator property. More explicitly, for \eta=dt the pairing

on R^{1/h} pairs the R-basis t^{\alpha/h} with its complementary basis t^{((h-1)\mathbf1-\alpha)/h}, and has invertible matrix. Replacing dt by a volume generator multiplies by units on the source and target and preserves perfectness. This establishes the claim without applying a finite-type smoothness theorem to the completed power-series ring.

The torsor augmentation has the same dual-generator property. Here is a finite-level verification. The coordinate algebra of the translation quotient is a tensor product of truncated polynomial algebras \kk[z_i]/(z_i^h), with its Honda Hopf structure. A nonzero translation-invariant functional \lambda gives a pairing \lambda(ab). Its radical is a translation-stable ideal: invariance of \lambda and the antipode identity show that the map to the dual regular representation is equivariant. If the radical is proper, it is contained in the augmentation ideal, since this coordinate algebra is local. For a translation-stable ideal I contained there, apply \id\otimes\epsilon to \Delta(I)\subset\cO(H)\otimes I; the counit identity gives I=0. The radical cannot be the whole algebra because \lambda\ne0. The pairing is therefore perfect. After a faithfully flat trivialization this is precisely the pairing of the torsor functional, so its perfectness descends.

There is a useful precision about the comparison. The dual module is rank one over the source R^{1/h}. Thus the two functionals initially differ by

not by an a priori constant multiplying the output. Both maps are G-equivariant, and the generator property makes u unique, hence G-invariant. By Theorem 4.2 and Corollary 6.8, its image after perfected geometric extension and full markings is a constant on (\mathcal N_m^r)_{\mathrm{ind}}. This extension is injective on the finite algebraic root ring: power-series expansion on the analytic annuli is injective in each entry of the whole finite free basis, remains so after the idempotent summand, and full marking is faithfully flat. The coefficient-only |\kk|-Frobenius fixes its finite model, and therefore fixes this constant. Consequently u\in\kk^\times, with the understood finite enlargement if the model was defined there. Since h fixes \kk, Equation (7.10) is the required scalar comparison.□

Proof. The formula and Proposition 7.2 give equivariance and h_0^{-1}-linearity. This is \kk-linearity because h_0 fixes the finite field. Continuity follows by grouping finitely many residue exponents in the h_0-basis expansion; multiplication by the fixed volume and its inverse is continuous on bounded-pole steps. The intermediate volume factors cancel under composition, so S^a(f)=\Cartier^{a\nu_0}(f\eta)/\eta for every a\geq1.

Suppose first that m>1. Put s_q=\prod_iD_i^{q-1}\in\Lambda_q. A \Lambda-linear functional \mathsf N_q\to\kk is evaluation at a socle element of \Lambda_q. Its space is the one-dimensional line generated by evaluation at s_q. Make U pro-p. The action on this line is a continuous character to \kk^\times, a group of order prime to p, so it is trivial. The maps therefore descend with their natural actions. Successive coinduction composes the basic socle evaluations into evaluation at

which is nonzero in \Lambda_{q_0^a}. Thus these composites do not vanish. The coherence of Proposition 7.1 identifies their coefficient maps with the root-twisted composites of T_{\eta,h_0}. In selfmap notation these are S^a. Proposition 7.2 identifies any other nonzero augmentation with a nonzero constant multiple; the constants introduce no semilinear twist because all chosen powers fix \kk.

Suppose now that m=1. We use the order-p torsor C=R^{1/p} under the fixed group \mu_p^r from Proposition 7.2. Its character decomposition is strongly graded by (\Z/p\Z)^r, with neutral component R and invertible character components [27, Theorem 12.12 and Remark 12.13]. The strong grading uses the torsor hypothesis: after a faithfully flat trivialization, each character component is free of rank one and multiplication between components is an isomorphism. These properties descend to R. Zariski locally on \Spec R, choose generators z_i of the fundamental character components. Strong grading makes each z_i a unit, with b_i=z_i^p\in R^\times. The monomials z^\alpha, 0\leq\alpha_i<p, form an R-basis of C. Taking pth powers shows that the b^\alpha form an R^p-basis of R, so the b_i are a p-basis.

For the standard toral Lie basis, \delta_i(z_j)=\delta_{ij}z_j. Equation (7.8) gives \partial_i(b_j)=\delta_{ij}b_j, so the already constructed volume is locally

On overlaps a different choice has z_i'=c_i z_i with c_i\in R^\times, hence b_i'=c_i^p b_i and d\log b_i'=d\log b_i. These local descriptions therefore agree with the global volume. We retain this standard toral normalization; the comparison constants multiply the torsor functionals. The logarithmic Cartier formula gives \Cartier(\eta)=\eta. Consequently, for every chosen h_0=p^{\nu_0} fixing \kk,

This proves Equation (7.12) at the existing finite frame stage. The natural auxiliary actions at higher levels are retained; no splitting of a higher torsion quotient or further shrink of U is required. No positive-dimensional local distribution ring is used in this case.

We next prove the pole bound in the whole model. Fix an A-basis b_1,\ldots,b_v of A'. The finite étale base change makes B'^{1/h_0} free over B on

The B-linear map T_{\eta,h_0} after multiplication by e, with output included in B', is a fixed finite matrix in these bases. Its entries belong to B and so have a common finite x-pole bound. For f\in x^{-D}A', its root is a sum of x^{-D/h_0}a_i^{1/h_0}b_i^{1/h_0}, with a_i\in A. Group the exponents in each a_i^{1/h_0} modulo the integral lattice. The resulting coefficients lie in A. Reducing -D modulo h_0 loses at most \lceil D/h_0\rceil integral powers of x. The fixed matrix contributes at most a further c, proving Equation (7.13) for all D, including negative D. This also proves continuity on compact coefficient lattices.

If D_{a+1}\leq D_a/h_0+c+1, then

Choose D_0>(c+1)/(1-h_0^{-1}). The same inequality implies \lceil D_0/h_0\rceil+c\leq D_0 and, for each fixed D, eventually D_a\leq D_0. The estimates apply to the whole lattice, uniformly over any compact parameter set. A compact bounded-pole cochain family uses one such D, so the assertion about cochains follows as well.

For the trace and volume assertion, set

Write \eta=a_R\theta with a_R\in R^\times. Extend its coefficient to the unit a=a_R+(1-e)\in B'^\times and use a\theta on the whole algebra. The trace form of the finite étale algebra is perfect; therefore the matrices

and their inverses have entries in B. The multiplication matrices of e,a,a^{-1} do also. Taking a maximum bounds all their poles by one c'. Transport by the root isomorphism replaces every matrix entry by its hth root. Its pole bound in the original x-valuation is consequently c'/h. The volume at this level is

This is transport along an isomorphism, not differential pullback under an inseparable map.

Finally take a finite Laurent principal-part representative w on one root grid and a bounded-pole coefficient on the next grid, and pair them by the coefficient residue of their product times \eta_h, after the finite étale trace. In the logarithmic x-coordinate the residue selects exponent zero; in each ordinary y-coordinate it selects exponent -1. Equation (7.9) therefore gives

where T_{h,hh_0} is the h-root of T_{\eta,h_0} and \iota is inclusion of the same Laurent expression on the finer grid. For instance the ordinary one-variable test is h_0i+j=-1 on both sides; the logarithmic test is h_0i+j=0. The coefficient field is fixed by h_0, so taking the selected coefficient introduces no field twist. Finite étale trace commutes with this calculation, since its root matrix is the transported trace matrix. Multiplication by the volume coefficient and its inverse cancels as in Equation (7.7). Thus the scalar monomial check proves Equation (7.14) in the whole basis and then after e. Continuity extends the identity whenever a convergent Laurent expression defines a residue functional. This is the adjoint identity used below; no trace for the purely inseparable root extension is involved.□

Proof. Every term and image bundle is finite locally free on the affine scheme \Spec R. Each surjection onto such an image has an underlying R-linear section. Choose these finitely many sections once. Embed their finite projective modules in finite free modules and choose finitely generated whole A-lattices; each section is then a finite matrix over the localization and has a fixed pole loss. The group action on a fixed lattice has a uniform pole bound, by the finite torsor realization and the bounded-step continuity of Lemma 2.2 and Proposition 3.4. Since g(x)/x is an integral unit, the same bound applies to every shift of that lattice. The finitely many group differentials and chosen sections computing the composite boundary thus have a common total pole loss. This constructs the required operator on continuous cocycles, with their bounded-pole convention.

The natural identity Equation (7.5) carries the entire diagram, including its arrows, to its root diagram. Transport each of the already chosen sections by the same root isomorphism. Computing the boundary with those sections is exactly the root of the original computation. Hence the increasing ranks of the coinduced representations require no new choices of matrices or new pole estimate. This statement concerns realization of a fixed diagram; comparisons of extension classes after one common auxiliary shrink are supplied in Section 10.□

Proof. We first calculate on X with scalar coefficients. Compact bounds are cofinal among boxes

In fact a quasicompact bound has its x-norm bounded away from zero and all coordinate norms bounded strictly below one; one can then choose \rho larger than its x-bound and every y-bound. Interleaving a slightly smaller closed box and a slightly larger open box gives the same support colimit. This permits strict gaps in every radius used below and avoids assertions about a particular higher-rank boundary point.

The complement of a box is the union of the two x-exteriors and the s individual y-exteriors. Its finite Čech restriction diagram, augmented by sections on X, calculates the support fiber. Every intersection is a product of open discs, punctured discs, and annuli. Exhaust it by a countable sequence of closed polyannuli and polydiscs, after adjoining all coordinate roots. The root polydisc, rational-localization and product constructions make these closed domains affinoid perfectoid [29, Proposition 5.20, Theorem 6.3(i)–(ii), and Proposition 6.18]. Their structural sheaf is v-acyclic by [32, Proposition 8.8]. Their function algebras are the corresponding completed Laurent algebras. The restriction maps have dense image: finite Laurent polynomials belong to both domains and are dense in the smaller-domain algebra. With parameters, finite sums whose coefficients are in R_{\mathcal D} have the same property. Equivalently, first approximate a continuous parameter family by a finite clopen partition and then approximate its finitely many values by Laurent polynomials. The complete normed-space inverse-limit lemma, Lemma 5.1, eliminates the first derived inverse limit. There are no higher derived limits for a countable sequence. Thus these open intersections have their ordinary Laurent functions in degree zero, with no additional cohomology.

Here is the resulting splitting of the support diagram. In an unpunctured coordinate y_j, a function on the exterior annulus is the sum of its nonnegative and strictly negative Laurent parts. The nonnegative part extends to the whole disc, since it converges at every radius less than one. Its summand in the restriction complex is contractible. The negative part is the quotient summand, in degree one. These projections have norm at most one on each closed annulus; on open domains one first inserts a radius gap. They are R_{\mathcal D}-linear and act coefficientwise in the other coordinates.

For the punctured coordinate write an exterior pair as (f_{\rm in},f_{\rm out}). The negative part of f_{\rm in} extends to the whole punctured disc: convergence towards zero is the only extra condition for a negative-power series. Likewise the nonnegative part of f_{\rm out} extends to the whole punctured disc. Subtracting these global functions diagonally leaves the unique pair

Encode it by the single Laurent series

Thus its normal representative is (f_+,-f_-). The minus sign ensures that multiplication or substitution crossing the exponent I=0 is represented by the usual Laurent operation: eliminating an outer nonnegative part by a diagonal section introduces its negative at the inner end. This quotient is again in degree one, and it retains every exponent I\in\Z[1/p], including zero exactly once.

Apply these splittings successively to the finite product restriction diagram. A summand which extends in any one coordinate is contractible by the corresponding one-coordinate splitting. The only remaining summand has every y-exponent negative and the indicated two-ended x-representative. It lies in degree s+1=r. The splittings commute in distinct coordinates, so this is a calculation of the entire complex, not only of its highest quotient. It also proves the assertion for s=0.

For the surviving inner positive x-powers choose a radius \exp(-L) slightly below the inner support radius. For the surviving negative x-powers and negative y-powers choose a common outer radius \exp(-\delta) slightly above every outer support radius. Their Gauss norms are exactly

Convergence on a closed polyannulus is precisely decay outside finite subsets for these norms. Every support class has these bounds after inserting the radius gaps. Conversely a series in one such step converges on the smaller inner domain and the larger outer collars, and hence supplies a class for a box with separated radii. Enlarging supports decreases the inner radius and increases the outer radii; on these representatives the map retains the same coefficients. The support colimit is therefore exactly \mathscr L(\mathcal D). This calculation uses the finite restriction diagram followed by a filtered colimit of supports. It does not interchange that colimit with an inverse limit of shrinking supports.

We next pass to the actual finite marked algebra. On a closed base chart with perfectoid algebra R, the whole algebra

is already complete as a finite free R-module. It is finite étale and affinoid perfectoid by [32, Theorem 6.1(i)]. Relative Frobenius for an étale algebra is an isomorphism [37, Lemma 41.14.3, Tag 0EBS]; consequently this algebra is also the perfected marked extension, rather than only an unperfected approximation. Restrictions use the same whole basis, so the function complex is the finite sum of the scalar complexes. The matrix e commutes with every restriction and is an actual chain idempotent. Taking its image commutes with the finite fibers and filtered support colimit. For the intermediate inverse limits its image remains complete and the transition maps remain dense: apply the target idempotent to ambient approximants. Hence the preceding inverse-limit argument applies there as well.

This computation uses all compact supports on Z_U. Pullbacks of quasicompact bounds under a finite map are quasicompact, and the image of a quasicompact bound is a quasicompact bounded subset of X. Thus pulled-back boxes form a cofinal support system upstairs. This establishes the first quasi-isomorphism in Equation (8.3) on the actual cover.

Coefficient Frobenius and its inverse are continuous between steps. For example, raising a coefficient to its Q_kth power changes a weight w to Q_kw; this is the step with parameters (Q_kL,Q_k\delta), followed if needed by a cofinal enlargement. It remains to check surjectivity of \varphi-1, including parameters. For b\in F, the equation

has a solution with |a|\leq|b|. If |b|<1, take a=-\sum_{j\geq0}b^{Q_k^j}. If |b|=1, any root is integral. If |b|>1, a root has norm |b|^{1/Q_k}. These choices have continuous local branches: around a root a_0 for b_0, add the small solution for b-b_0. Near a point with |b_0|\geq1 take |b-b_0|<1, so the same bound is retained; on |b|<1 use the displayed series itself. For a continuous function on a compact profinite \mathcal D, a finite clopen refinement of these local charts therefore gives a continuous solution with \|a\|_{\sup}\leq\|b\|_{\sup}.

Apply this construction separately to every Laurent coefficient. The weighted norms do not increase, so all the chosen coefficients assemble to a family in the original complete step. Uniformly small tails make that assembled family continuous even though the finite clopen partitions used for different coefficients may differ. Solving on the whole finite sum and then applying e proves surjectivity on \mathscr W_{\mathcal D}. Its kernel is exactly W_{\mathcal D}, since the fixed elements of F are \kk and e is fixed. This proves the second quasi-isomorphism.

Finally, relative Frobenius identifies each fixed root version of A'[1/x] with its base change to B^{1/h}. Thus b_\nu^{1/h} is a basis over the perfected base. Any two fixed such bases are related by invertible matrices over a finite root version of B, with bounded x-poles and no negative y-powers. The bound Equation (8.2), rescaled to that root grid, makes both basis changes continuous. If a fixed matrix before taking roots has pole bound c, its h-root has bound c/h. Moreover, when h\mid h', each b_\nu^{1/h} expands integrally in the basis b_\mu^{1/h'}: it is the (h'/h)th power of a basis element, and the multiplication laws of the root model are integral. These observations prove both the basis assertion and compatibility with the stated operations.□

Proof. We check the scalar coordinate changes first. An integral formal automorphism preserving (x) factors into three types:

where u is an integral unit, c(x)\in x\kk[[x]]^s, and the last map preserves (y) and has invertible linear Jacobian in y over \kk[[x]]. To obtain the factorization, first remove the x-unit substitution by its formal inverse. For the residual map fixing x, subtract the value of its y-coordinates at y=0; the remaining map fixes (y) and its y-Jacobian is invertible. All inverse maps are integral. These operations are continuous on formal jets, uniformly in a compact family.

Fix a root grid 1/h and a monomial x^Iy^{-J}. For each of the three types every surviving output monomial x^{I'}y^{-J'} satisfies

For the x-unit substitution, the expansion of u^I on the root grid has only nonnegative additional x- and y-orders, giving the stronger bound |J'|\leq|J|. For translation, write X=x^{1/h} and Y_j=y_j^{1/h}. Expanding (Y_j+c_j(x)^{1/h})^{-hJ_j}, an increase a_j/h in its pole requires the factor c_j(x)^{a_j/h}, whose x-order is at least a_j/h. Summing over j gives Equation (8.4).

For the third type use finite principal-parts duality, which also avoids choosing a preferred expansion of a general linear combination of the y_j. On the grid just fixed the negative Y-powers form

with coefficients extended in X as needed. A principal part Y^{-j} pairs with Y^{j-1}\,dY_1\cdots dY_s by coefficient extraction. This is a perfect pairing between every finite principal-part truncation and the corresponding quotient of the power-series ring. If v preserves (Y), then its inverse and the determinant of its inverse Jacobian are integral, and

has total Y-degree at least |j'|-s. It cannot pair nontrivially with Y^{-j} when |j'|>|j|. Every coefficient remains in \kk[[X]]. Thus here I'\geq I and |J'|\leq|J|. The pairing description is the usual change-of-variables map on the finite Čech complex: it is first checked on monomial fractions by coefficient extraction and then on their finite principal-part span. Hence it represents the geometric support pullback. The case s=0 requires no such step.

The two inequalities in Equation (8.4) are stable under composition. They imply

When both x-exponents are negative this uses the slope \delta; when both are nonnegative it uses L\geq\delta; when they cross zero the excess is at least (L-\delta)I'. The output of a fixed monomial is convergent. For translations, after I' becomes nonnegative the weight increases by at least L-\delta per additional unit of pole, and at a fixed grid only finitely many terms precede that crossing. For a map preserving (y), the total y-pole order is bounded and its remaining x-series converges. The x-unit case has the same property.

These expansions also occur on actual analytic collars. For example

converges on |x|<|y|, and its term weights are (L-\delta)j-\delta. Any compact support bound can be enlarged so that its y-radius exceeds its x-radius. The same enlargement works for every integral c(x) because |c(x)|\leq|x|. For maps preserving (y) the ideal and its inverse have the same total-radius bounds; finite Čech refinements give the principal-parts pullback above. Thus the algebraic formulas agree with the canonical support action used in Proposition 8.1.

All coefficients of these integral maps have norm at most one. The weight estimate therefore gives a uniform operator bound on each complete scalar step. For a fixed monomial, every bounded-weight output truncation involves finitely many monomials on its fixed root grid and depends on finitely many formal jets. It consequently varies continuously with the group element. For an arbitrary convergent sum, discard a uniformly small tail and apply this assertion to its finite remainder. This proves joint continuity and equicontinuity, including all compact parameters. Operator-norm continuity in the group element is neither asserted nor needed.

On the whole finite model, the group action is the scalar pullback just treated followed by a finite basis matrix. The integral model of Proposition 3.4 and compactness give a uniform pole bound for those matrices and their inverses. Their entries vary continuously in the bounded-lattice topology. Apply Equation (8.2); then pass to the invariant image of e. This proves the same assertions for the marked module. That bound also proves the claims about multiplication and root bases.

For completeness, these estimates allow integration against compact free resolution terms as in Lemma 2.3. If a compact family has image K_0 in a complete linearly topologized \kk-step, the closed \kk-linear span of K_0 is compact. Modulo any open linear subgroup its image is the span of a finite set over the finite field, hence finite; this proves total boundedness, and completeness gives compactness. Enlarge the step first to contain the compact group translates if necessary. Summation against compact distributions, and the resolution comparison homotopies, therefore remain inside a complete step. No boundary image is replaced by its closure.

Finally, absolute Frobenius raises coefficients and basis elements to their pth powers and scales every exponent by p. The relative-Frobenius basis identifications in Proposition 8.1 and the rescaled weight condition show that this operation and its inverse act continuously between steps. It commutes with the geometric action. A power fixing \kk is \kk-linear, so it commutes with the differentials and homotopies of resolutions over \kk. This operation is distinct from the coefficient-only \varphi in Equation (8.3).□

Proof. Put K=\RGamma(G,\RGamma_c(Z_U,\cO^\flat)), with no external parameter. By Corollary 6.7, every H^j(K) is finite-dimensional over F. By Lemma 8.2 the complete-step comparisons in Lemma 2.3 apply. The finite-type resolution in each degree commutes with taking the fiber of \varphi-1. Thus

One can equally form this fiber on bounded-step bar cochains: the parameter-uniform surjectivity proved above applies to the compact bar parameters, and the two descriptions agree.

We recall explicitly the finite-dimensional semilinear argument. If V is an F-vector space of dimension d and \psi is an invertible Q_k-semilinear map, choose coordinates \psi(z)=Az^{[Q_k]} with A invertible. For any target t, the equations Az^{[Q_k]}-z=t are, after multiplication by A^{-1}, monic equations with pairwise coprime leading monomials z_1^{Q_k},\ldots,z_d^{Q_k}. Monomial reduction gives a quotient algebra of dimension Q_k^d, with basis \prod z_i^{a_i} for 0\leq a_i<Q_k. Their Jacobian is the invertible matrix -A^{-1}, so that algebra is finite étale. As F is algebraically closed, every fiber has Q_k^d points. Consequently \psi-1 is surjective and its kernel is a d-dimensional \kk-vector space.

Apply this to the bijective semilinear action of \varphi on each H^j(K). The long exact sequence of its fiber and the surjectivity in the preceding degree give

which is finite over \kk. The shift in Equation (8.5) gives finiteness of H^a_{\cts}(G,W).

The marked algebra acts continuously on W by Equation (8.2), and its finite subgroups have the trace-one functions supplied by Proposition 3.7. All the hypotheses of Proposition 2.7 are therefore satisfied. Its orientation for G=P_n is trivial, and it identifies

The right side is finite (and zero in negative degree), proving the assertion. The argument uses parameter families to justify the complexes; it makes no claim of finite dimension over F or \kk with an infinite external parameter retained. In that case the respective scalar rings are R_{\mathcal D} and C(\mathcal D,\kk).□

Proof. First work at a bounded root level h. In the normalized whole basis, multiplication by Equation (9.4) converts Equation (9.8) into the scalar residue test on each basis coefficient. Testing against all elements of x_h^{-D_0}A^{1/h} detects every strictly negative-y monomial of this product whose x-order is at most D_0/h. There is no undetected smaller y-pole at this level: every positive J_j in the root grid is at least 1/h. Thus

Here one multiplies the detected product by \mathcal T_h^{-1}, which loses at most c/h in x-order. These are invertible matrices on the negative-y quotient: their entries and those of their inverses have no negative y-powers. Consequently projection to negative y-powers does not invalidate their inverse identity. This justifies using the inverse matrix in Equation (9.11).

Conversely, suppose that w has x-order at least \epsilon>0 in the normalized basis at level h. At a later level h' the bound is still at least \epsilon. The trace/volume product then has x-order at least \epsilon-c/h'. If h'>(D_0+c)/\epsilon, it pairs to zero with \mathcal P^{1/h'}. In particular every bounded-root element of W_+ maps to zero in Q. If a bounded-root element maps to zero in Q, its functional is zero at some later level of the direct system; Equations (9.11) and (9.5) then put it in W_+. We have proved

To prove surjectivity, take a functional at any stage of Equation (9.6). On the whole free lattice its finite coefficient expression is represented by finite scalar principal parts. Applying the inverse trace/volume matrix gives a bounded-root representative v for Equation (9.8); it lies in the whole Laurent module because that matrix has a fixed x-pole bound and no negative y-powers. Replace v by ev. These two functionals agree after one transpose transition: the image of the corresponding transfer lies in eB', where multiplication by e is the identity. Thus ev represents the same element of the colimit and belongs to W_{\mathrm{fin}}. This argument uses the whole trace matrix and only then projects by e; it does not specialize an idempotent to the boundary or presume a free model for its component there.

We next extend the map to W. Express w\in W on one fixed whole normalized basis. Truncate to the finitely many terms of x-order I\leq M, obtaining v_0. Let c_e be a fixed x-pole bound for the matrix of e in that basis. Taking M>c_e+1 gives

The finitely many terms of v_0 have a common root denominator, and e has a bounded root denominator at this fixed basis level, so the first inclusion holds. Define \pi(w)=\pi_{\mathrm{fin}}(v). Any two such choices differ by a bounded-root element of W_+; Equation (9.12) proves independence of the choice. If \pi(w)=0, the bounded-root representative v belongs to W_+, so w also belongs to W_+. This proves exactness.

For continuity, fix a complete weighted step and a normalized basis. A sufficiently small norm ball has all its nonzero monomials of weight greater than some a>0. Such monomials necessarily have I>a/L, because w_{L,\delta}(I,J)\leq LI when I\geq0 and is nonpositive when I<0. The ball is therefore contained in W_+. The kernel of the linear map to discrete Q is open in this step. The same argument in supremum norms works for a compact parameter space: only finitely many bounded-x monomials remain uniformly in the parameter, and each retained \kk-coefficient is locally constant. Its image in Q is finite.

It remains to identify the group action, rather than just the kernel as a vector space. Work first at finite root level, and put A_h=\kk[[x_h,y_h]], \mathfrak m_h=(x_h,y_h), and \Omega_h=\widehat\Omega^r_{A_h/\kk}. Write \alpha=\Tr(wp_h\eta_h) as an ordinary top form, including the factor x_h^{-1} from d\log x_h, with its negative-y Čech convention and its ordered-coordinate signs. Choose N so that x_h^N\alpha has nonnegative x-orders. Its truncation modulo x_h^N is a finite principal-part class

The equality holds because x_h is nilpotent on the quotient. The connecting map for multiplication by x_h^N on \Omega_h sends this class to the all-negative ordinary principal part of \alpha in H^r_{\mathfrak m_h}(\Omega_h), with the coordinate order fixing the Čech sign. This identifies the finite truncation with an intrinsic local-cohomology class. If g(x_h)=u x_h, the naturality square has g on the source copy of \Omega_h and u^{-N}g on the middle copy. Consequently the connecting map sends u^{-N}g\xi_N to the g-translate of the class of \alpha, as required.

These maps are the maps on the finite support Čech model of Proposition 8.1. In particular a translation y_h\mapsto y_h+c(x_h) gives finite generalized-fraction expansions modulo x_h^N, since c(x_h) is nilpotent there. The two-ended x-convention (f_+,-f_-) ensures that multiplication crossing x-order zero is ordinary Laurent multiplication. For a convergent bounded-root expression, truncation above all fixed x-pole bounds, including the group basis matrix bounds, changes neither residue before or after the given action. Thus the same naturality applies to every expression being paired.

The generalized-fraction transformation law and the independence of regular parameters for formal residue on H^r_{\mathfrak m_h}(\Omega_h) hold also in characteristic p [23, Lemmas (7.2)(b) and (7.3), pp. 60–67]. Here A_h is a complete regular local algebra over the perfect field \kk with residue field \kk, so all hypotheses of that result hold. The finite étale trace is natural, the action preserves (x) and the whole lattice, and the transported volume is invariant on the e-component. These facts give

This proves equivariance on bounded-root expressions and is compatible with Equation (9.9). Finite-root expressions are dense after a complete-step enlargement. Both the group action and \pi are continuous into enlarged steps and the discrete quotient by Lemma 8.2 and the preceding paragraph. Passing to the limit proves equivariance on all of W. In particular W_+ is G-stable. Its positive complete steps also satisfy the uniform action condition of Definition 2.1. Indeed, for a step with x-gap \epsilon, pass to a later normalized basis h'. The scalar substitutions preserve x-order by Equation (8.4); the remaining group basis matrices have one uniform pole bound c_G/h', because they are root transports of the matrices on the original whole model. Choose h' with c_G/h'<\epsilon/2. The entire compact group family then has x-gap at least \epsilon/2, and Lemma 8.2 puts it continuously into one common complete positive step. The same argument applies to compact parameter families. Compact-envelope integration from Lemma 2.3 therefore defines the completed group-ring action on W_+. The map \pi commutes with that action: approximate a distribution by finite group-ring sums in a compact envelope and use continuity of \pi into Q.

This proof uses finite principal-part residue invariance; it does not presume a separate duality theorem for completed functions at infinite root level.□

Proof. The empty tuple is immediate. Otherwise, since the tuple is finite, choose one normalized basis at level h_0^{a_*} on which all components have a common positive x-gap. Applying \Phi^{a_*} clears this basis denominator. It need not clear the root denominators in the series. In the original basis the resulting tuple v still has a positive x-gap, since all multiplication laws in A' are integral.

Choose a common weighted step (L_1,\delta) for the finitely many series of v, with I\geq\epsilon>0. Their original weights are bounded below. Increasing L_1 to L adds (L-L_1)I to every weight. Choose L large enough that all nonzero terms now have weight at least \alpha>0. They still form null series in this new step. Under h_0^a-powering the coefficient exponents are multiplied by h_0^a, so these weights are at least \alpha h_0^a. Each b_i^{h_0^a} expands on the original basis with coefficients in A. These coefficients add nonnegative x- and y-powers; projection to the strictly negative-y part can only discard terms or improve the weight. Their coefficients have norm at most one. This proves Equation (9.14). The x-gap is multiplied by h_0^a, hence remains at least \epsilon in the same complete step.□

Proof. An analytic subset of a Polish space has the Baire property [19, Theorem 21.6]. If B were meagre, its countably many cosets would make Z meagre in itself, contrary to the Baire theorem. Thus B is nonmeagre. Pettis’s theorem [19, Theorem 9.9] says that the product of a nonmeagre set with the Baire property and its inverse contains a neighborhood of the identity. For a subgroup this product is B itself, proving the assertion.□

Proof. Use the finite free resolution and the countability statement Equation (9.13) above. We first construct the complete cycle spaces to which Lemma 9.4 applies. Take a normalized basis level, rational weights L>\delta>0, and a rational positive cutoff \epsilon>0. In the whole weighted null-series space require x-order at least \epsilon and the equation ew=w. The first condition is a closed coordinate condition; the second is closed because e and the identity are continuous into a common enlarged complete step. This gives a complete separable metrizable \kk-vector space E\subset W_+. Separability follows from the countable exponent index and finite coefficient field. These spaces form a countable collection covering W_+: the basis levels are countable, and rational weights and positive cutoffs can be chosen cofinally. Each inclusion between two specified coefficient steps is continuous into some common enlarged step.

Fix i and one such E. The cycles

form a closed subspace of a Polish group. In fact the differential is continuous from this finite product into a common complete step by Lemmas 8.2 and 2.3. Let

Its index in Z_E is countable by Equation (9.13). It is analytic for the following explicit reason. For each complete positive primitive step E', the set

is closed: compare the two sides in a common enlarged complete step. Projecting this closed subset of a Polish product onto Z_E gives an analytic set. There are countably many choices of E', and their projected union is exactly B_E. Thus B_E is an analytic subgroup, and Lemma 9.4 makes it open. This argument does not require the global boundary image in W_+ to be closed or the inductive topology to be strict.

Now let z be any cycle in W_+^{a_i}. Absolute Frobenius is \kk-linear and G-equivariant. It commutes with the completed group-ring differential: this holds for finite group-ring sums and then for their convergent actions on the complete steps. Apply Lemma 9.3 and choose the rational weights and cutoff within its bounds. The forward iterates after the initial basis clearing lie in one Z_E and tend to zero there. Since B_E is open, some iterate \Phi^a z is du with u\in W_+^{a_{i+1}}. Both \Phi and its inverse preserve W_+ and commute with d. Hence z=d(\Phi^{-a}u). Every cycle bounds, as required.□

Proof. The preceding acyclicity and Equation (9.10) identify H_i(G,Q) with H_i(G,W), hence make it finite. Compact/discrete duality and the finite free resolution give

Filtered colimits commute with kernels and cokernels of vector-space maps, so no derived interchange is involved here. Dualizing gives

This inverse limit is finite. Its projection to the first term has image I_i. To check surjectivity onto that image description, note that every S^aM_i is compact. For x\in I_i, the nonempty compact sets

are nested. Their intersection gives a preimage in I_i. Thus S:I_i\to I_i is surjective, and successive choices of preimages in I_i lift x to the inverse limit. It follows that I_i is finite. We have used a surjection from the inverse limit onto the stable image; we have not assumed them equal before proving finiteness.□

Proof. The compact preimage argument in the preceding proof gives S(I)=I, and finiteness makes this restriction invertible. In V=M/I the nested compact subspaces S^aV have zero intersection. They shrink uniformly to zero: for every neighborhood O of zero there is an a_0 such that S^aV\subset O for every a\geq a_0. Otherwise the compact sets S^aV\cap(V\setminus O) would have nonempty intersection.

For f=\sum_{j\geq0}c_jz^j define

Every sufficiently late finite tail belongs to an arbitrarily specified open linear subspace, uniformly for v\in V. Completeness therefore supplies the limit. Polynomial module identities pass to these uniform limits and give a continuous \kk[[z]]-action.

Choose lifts v_1,\ldots,v_s of a basis of the finite space V/SV=M/SM. Given v=v^{(0)}, recursively choose coefficients c_{ij}\in\kk and v^{(j+1)}\in V such that

After t steps the error is S^tv^{(t)}, which tends uniformly to zero. Thus v=\sum_i(\sum_jc_{ij}z^j)v_i; the continuous map \kk[[z]]^s\to V is surjective. Each successive quotient S^jV/S^{j+1}V is an image of V/SV. Hence every V/S^aV is finite, and its compact kernel S^aV is open. Uniform shrinking makes these kernels a neighborhood basis, proving the topology assertion.

The Noetherian module V has a stabilizing ascending sequence \ker z\subseteq\ker z^2\subseteq\cdots. Its union is a finitely generated submodule killed by some z^t, hence finite over the finite ring \kk[[z]]/(z^t). The map from T_S(M) to this union is injective, since a locally nilpotent element of I is zero. Therefore T_S(M) is finite and closed. In fact this map is surjective: if S^a x\in I, subtract the unique element of I with the same S^a-image to obtain a nilpotent lift. The contraction and the power-series action in this argument concern M/I; they are not asserted on the finite summand on which S is invertible.□

Proof. Put A_N=S^NM_i, Z_N=S^NX_i, and f=f_i|_{A_N}. The induced map

has countable kernel. Indeed, the subgroup represented by \{a\in A_N:f(a)\in SZ_N\} maps to SZ_N/f(SA_N), which is countable by Equation (9.16) at N+1; the kernel of this map is an image of \ker f, countable by the comparison at N. Its image in Z_N/SZ_N is countable by Equation (9.17). Thus A_N/SA_N is countable. It is a compact Hausdorff group, because SA_N is a compact image and hence closed. A countable compact Hausdorff group is finite: its countable cover by closed singletons, together with Baire, forces a singleton to be open; the group is then discrete and compact. This proves finiteness of A_N/SA_N.

The stable intersection for S acting on A_N is I_i. Apply Theorem 9.6 and Lemma 9.7. This gives the asserted finite generation and makes both \ker(S|_{A_N}) and T_S(A_N) finite.

We first obtain countability of the kernel on Z_N without using closedness of \ker f. Write Z_0=f(A_N) and C_N=Z_N/Z_0, which is countable. An element of \ker(S|_{Z_0}) lifts to an a\in A_N with Sa\in\ker f. The latter kernel is countable and each nonempty fiber of S:A_N\to A_N is finite. Consequently S^{-1}(\ker f), and hence \ker(S|_{Z_0}), is countable. Mapping \ker(S|_{Z_N}) to C_N proves its countability. The cokernel is countable already by Equation (9.17).

It remains to prove the stronger assertion about \ker f, for which countability alone would not suffice. Let \alpha\in\ker f\subseteq S^NM_i. Choose a \mathcal P-valued cocycle c such that S^Nc represents \alpha. Since N\geq1, this representative is already e-supported after localization. If i>0, vanishing in X_i supplies a bounded-pole primitive b with

in the cochain complex for B_U, viewed inside that for B'. The convention of Lemmas 2.2 and 2.3 means that this one primitive belongs to some whole lattice x^{-D}A' uniformly on its compact cochain parameters. By Equation (9.3), all sufficiently high S^ab are \mathcal P-valued. The chain-map identity gives d(S^ab)=S^a(S^Nc) there. Thus S^a\alpha=0 in M_i. When i=0, vanishing of the already e-supported localized cocycle means that the cocycle itself is zero, since \mathcal P\hookrightarrow B' is injective. This proves local S-nilpotence of \ker f in every degree. The exponent may depend on its chosen bounded-pole primitive; no uniform exponent was assumed.

We now have \ker f\subseteq T_S(A_N), so \ker f is finite and closed. Its intersection with I_i is zero, because S acts invertibly on I_i. For an element \xi belonging to all the images in Equation (9.18), discard the first N terms and consider

These sets are nonempty and nested. Each is a coset of \ker f\cap S^aA_N, hence finite and closed in the compact space A_N. Their intersection is nonempty. Any point in it belongs to I_i and maps to \xi. This proves the nontrivial inclusion in Equation (9.18); the reverse inclusion is immediate. It also explains why no Hausdorff topology on localized cohomology was used.□

Proof. The embedding by constant objects is fully faithful and exact, which gives the kernel and cokernel statements. For extensions it is enough to give the dual argument in \operatorname{Ind}(\mathcal B). Suppose

is exact with A and B constant. Write E as a filtered system of objects E_j of \mathcal B. The images of the maps E_j\to B are objects of \mathcal B, and their filtered colimit is B. The defining Hom formula for ind-objects says that a map from a constant object to a filtered colimit factors through one term. Applied to \operatorname{id}_B, it shows that E_j\to B is an epimorphism for some j. Let K_j=\ker(E_j\to B). The induced map K_j\to A is a map in \mathcal B. The pushout of K_j\hookrightarrow E_j along K_j\to A is E: the natural map from that pushout has identity maps on the kernel A and quotient B, hence is an isomorphism. This pushout is a cokernel between objects of \mathcal B, so is constant. Dualizing proves extension closure in the pro-category.

For the second assertion, the morphism from the constant object I to \{J_a\} is a compatible family of maps I\to J_a. Since its composite with J_a\hookrightarrow X is the fixed inclusion, these are inclusions identifying I as a subobject of every J_a. The system \{J_a/I\} is pro-zero. Thus for any chosen a some later transition J_b/I\to J_a/I is zero. This transition is monic, so J_b/I=0. All still later subobjects contain I and are contained in J_b=I, proving the assertion.□

Proof. We descend on i, starting above the bound in Equation (10.3). Assume the assertion of constancy has been proved for \overline T^{\,j}(\Lambda) with j>i. The ideal \mathfrak m has a finite resolution by finite free \Lambda-modules, given by the positive part of the Koszul resolution of \kk. Induction on resolution length shows that \overline T^{\,j}(L) is constant for every j>i and every L with a finite free resolution. The case of a free L is the descending induction hypothesis and finite additivity. For 0\to L'\to F\to L\to0 with F finite free and L' of shorter resolution length, the long exact sequence gives

For j>i the outer terms are constant by the induction on length and Lemma 10.2; extension closure gives the middle term. Taking L=\mathfrak m and j=i+1 proves the required constancy of \overline T^{\,i+1}(\mathfrak m).

The exact sequence 0\to\mathfrak m\to\Lambda\to\kk\to0 now identifies the image of

with the kernel of a map between the two constant objects T^i(\kk) and \overline T^{\,i+1}(\mathfrak m). Call this constant subobject I^i. Under Equation (10.4), the image system is

The second part of Lemma 10.2 supplies one finite N_i with J_a=I^i for all a\geq N_i. Increase N_i to at least one. Equality of these subobjects in \mathcal A says exactly that Equation (10.1) holds for this N_i at every sufficiently deep actual level. Its depth is uniform because only this one consecutive-image comparison is involved.

By Proposition 9.8, the endomorphism induced by S on J_{N_i}=I^i has countable kernel and cokernel at these levels. It is therefore an automorphism in \mathcal A. Replacing the stationary inverse system by its N_ith images gives an isomorphic pro-object. This replacement has underlying object I^i and invertible transition S|_{I^i}, so is constant. Explicitly, its map from the constant object I^i to level a is (S|_{I^i})^{-a} followed by inclusion. At the initial level the projection is the inclusion I^i\hookrightarrow T^i(\kk). This proves the claimed monomorphism and completes the descending induction.

Finally, multiplication by any element of \mathfrak m followed by \Lambda\to\kk is zero. The resulting endomorphism of \overline T^{\,i}(\Lambda) has zero composite with the monic projection just proved. It is zero. Thus \mathfrak m annihilates the pro-object.□

Proof. Let \mathcal D be the category of \mathfrak m-power-torsion \Lambda-modules. Every finitely generated submodule of an object of \mathcal D has finite length: finitely many generators are killed by one power of \mathfrak m, and the corresponding quotient ring is Artinian. Thus \mathcal D=\operatorname{Ind}(\mathcal C).

Finite endpoints have the same Yoneda extensions in these two categories. One can see the finite realization directly. Starting at the right endpoint of an extension in \mathcal D, choose finitely many lifts of its generators, then take the finite-length submodule that they generate. Its kernel is finite-length. Lift generators of that kernel into the preceding middle term and continue. At the last step include the whole left endpoint. This constructs a finite-length subextension representing the given class. The same construction can be made with a prescribed finite subcomplex: include its terms before choosing the next lifts. To check injectivity, take a finite Yoneda zigzag witnessing equality. Its directed underlying path has no directed cycles. Process its vertices in an order compatible with the arrows, choosing a finite subextension that contains the images of every already chosen predecessor; at a finite original vertex retain the whole extension. All arrows then restrict to a finite zigzag. Hence the map on Ext is both surjective and injective. Equivalently, this is extension fullness of an abelian category in its ind-completion [9, Theorem 2.2.1].

For comparison with all \Lambda-modules, use that the \mathfrak m-power-torsion submodule of an injective module is itself injective over the Noetherian ring \Lambda [37, Lemma 47.3.9(2), Tag 08XW]. If M\in\mathcal D embeds in an ordinary injective I, that embedding factors through I[\mathfrak m^\infty]. Its cokernel is still \mathfrak m-power torsion. Iterating this construction gives an ordinary injective resolution entirely in \mathcal D. Each term is also injective in the full abelian subcategory \mathcal D, so the resolution computes both Ext groups, proving \operatorname{Ext}_{\mathcal D}^j(E,F) =\operatorname{Ext}_\Lambda^j(E,F) and the lemma.□

Proof. A composition series shows that T^j(E)=0 for every finite-length E and j>b. In particular \overline T^{\,b+1}(\mathfrak m)=0. The exact sequence for 0\to\mathfrak m\to\Lambda\to\kk\to0, together with Proposition 10.3, gives an isomorphism

For an allowed index s, identify \mathsf N_s with \Lambda_s as a plain \Lambda-module. Let K_s^\bullet be the Koszul resolution on D_1^s,\ldots,D_d^s, in degrees [-d,0]. Apply exact completion and then the exact cochain-term functors \mathcal C^j to this bounded complex. The augmented horizontal rows remain exact, so the total complex computes the constant object \mathcal C^\bullet(\Lambda_s). Its vertical-cohomology spectral sequence has

The finite column interval and Equation (10.3) give a finite filtration in every total degree. By Proposition 10.3, \mathfrak m annihilates every entry on this first page, so all first differentials vanish.

We compare these operations in the forward direction. Under the pairings \alpha_s above, the natural inclusion \mathsf N_s\to\mathsf N_{s'}, dual to the regular quotient, is

Its Koszul comparison preserves the top exterior term while making every other column map vanish on vertical cohomology. To see this, lift the displayed injection to a map K_s^\bullet\to K_{s'}^\bullet. On the exterior-basis summand indexed by I\subset\{1,\ldots,d\}, that map is multiplication by

The Koszul differential removes one index, and this formula commutes with it because the missing factor changes D_j^s to D_j^{s'}. On the first spectral-sequence page the comparison is the identity in column -d and zero in every other column. The latter maps remain zero on every subsequent page.

There is no incoming differential at (-d,b). Starting with E_2^{-d,b}=Y, increase s once for each possible page r=2,\ldots,d. Inductively suppose the whole object Y survives to page r at the current index. At a larger index the comparison remains the identity on this whole source. Naturality of d_r and the zero comparison on its target column show that the outgoing d_r at the larger index is zero on all of Y. The earlier differentials are still zero for the same reason. After these finitely many increases, E_\infty^{-d,b}=Y. The leftmost-column edge of the finite filtration is therefore an epimorphism

When d=1 there are no pages to eliminate, and the same conclusion holds directly.

The edge followed by Equation (10.5) is the operation of the chain map

which projects the top exterior term onto \Lambda and then augments. Applying \operatorname{Hom}_\Lambda(-,\kk) to K_s^\bullet gives zero differentials; the displayed map is therefore a generator of \operatorname{Ext}^d_\Lambda(\Lambda_s,\kk)\simeq\kk. By Lemma 10.4 it has a finite-length Yoneda representative. To verify that this representative gives the same operation, compare the two representing morphisms in the bounded derived category of finitely generated \Lambda-modules, using their finite free resolutions. They agree there because they represent the same Ext class. Exact completion and the exact cochain-term functors preserve quasi-isomorphisms of these bounded complexes, as is seen by taking the exact horizontal rows of their cones. Their induced operations thus agree after totalization. This identifies the edge map with the operation of the finite diagram and proves the proposition.□

Proof. There is an extension-descent spectral sequence

Here is a construction verifying the coefficient-category requirements. Regard U as the proconstant affine group scheme whose coordinate algebra is the algebra of locally constant \kk-valued functions on U. The underlying vector space of the coordinate coalgebra of T\rtimes U is \mathcal O(T)\otimes C_{\mathrm{lc}}(U,\kk). A cofree representation restricts to a cofree T-representation: the semidirect action is changed to ordinary translation on the T coordinate by conjugating that coordinate by the retained U coordinate. Its T-invariants are a cofree discrete U-representation. Such U-representations are acyclic for invariants, since the locally constant coinduction functor is exact; lifts through a vector-space surjection are chosen on the finite image of a locally constant function. Cofree resolutions therefore give the spectral sequence for the composite of the two invariant functors. Tensoring with the dual of the finite representation E identifies that spectral sequence with Equation (10.6).

By Lemma 10.4, the translation Ext groups are the ordinary \Lambda-Ext groups of finite-length modules. The Koszul resolution makes \operatorname{Ext}^b_\Lambda(\kk,\kk) finite-dimensional and zero for b>d. Composition series and the long exact sequences in each variable give the same two conclusions for all finite-length endpoints. Their induced U-actions are discrete continuous. Indeed a finite Yoneda representative of a translation Ext class becomes equivariant after shrinking U to act trivially on its finite endpoints and on the finite translation quotient through which its middle terms act; give those middle terms trivial auxiliary action. This open subgroup fixes the class. Finite-type resolutions for the analytic group U then make each group on the left of Equation (10.6) finite-dimensional. Only finitely many terms contribute in a fixed total degree. The abutment is therefore finite-dimensional over the finite field \kk, hence is a finite set.

Every representation of the indicated coalgebra is a union of finite-dimensional subrepresentations. As in the finite subdiagram construction in Lemma 10.4, an extension with finite endpoints, and any Yoneda comparison between such extensions, can be realized by finite diagrams. Fix a class in the displayed forgetting kernel. Choose a finite equivariant diagram representing it. Its underlying T-extension is zero, so there is a finite T-linear Yoneda comparison with a representative of zero. Shrink U so that it acts trivially on all the finite modules of the original diagram. Shrink further so that its action on the finite translation quotients through which the comparison modules act is trivial. Give those comparison modules trivial auxiliary action. Every comparison arrow is then equivariant, and the class has become zero. Finally the forgetting kernel is a finite set. Intersect the finitely many open subgroups just obtained for its elements. This one intersection kills the whole kernel. A sufficiently deep normal product congruence subgroup is contained in it.□

Proof. Choose finite Yoneda diagrams for e_s and e'_0. After one initial shrink U_0, their plain module diagrams are equivariant with trivial auxiliary actions, agreeing with the natural action at the fixed endpoint \mathsf N_s. The power-subalgebra identification commutes with marking changes by Proposition 7.1, so coinduction gives equivariant diagrams for every q without making the action on \mathsf N_q trivial. The natural inclusion \mathsf N_s\to\mathsf N_q is equivariant. The space of translation maps \mathsf N_q\to\kk is one-dimensional; the pro-p group U_0 acts trivially on this line because \kk^\times has order prime to p. Choose a nonzero map on that line for the last arrow.

After forgetting U_0, the composite in Equation (10.7) is a generator. This can be checked directly on the power-subalgebra free pairing. With \Lambda^{(q)}=\kk[[D_1^q,\ldots,D_d^q]], the coefficient of \prod D_i^{q-1} gives the perfect pairing identifying coinduction from \Lambda^{(q)} with extension of scalars. Extension of scalars carries the Koszul representative of e'_0 to the top projection in the Koszul resolution on D_1^q,\ldots,D_d^q. Under the regular-module identifications, the inclusion from the smaller block is multiplication by \prod D_i^{q-s} up to a unit. The Koszul comparison written in the proof of Proposition 10.5 has top coefficient one. Following with augmentation therefore leaves a nonzero top coefficient in \kk. Changes in the perfect-pairing generators only multiply that coefficient by a unit, whose augmentation is nonzero. Thus the ordinary Ext class is c_qe_s for some c_q\in\kk^\times.

Subtract c_qe_s from each composite, now in the fixed group \operatorname{Ext}^d_{\mathcal R_{U_0}}(\mathsf N_s,\kk). All differences lie in its forgetting kernel. Apply Lemma 10.6 to this fixed pair of endpoints. One subgroup U_* kills the entire kernel, and hence every difference, independently of q.□

Proof of Theorem 10.1 for m>1. If all T^i(\kk) vanish, their finitely many potentially nonzero degrees in Equation (10.3) have a common vanishing tail, which is the desired conclusion. Otherwise choose the largest nonzero degree b and the epimorphism e_s from Proposition 10.5. If b-d<0, its source is zero and already contradicts T^b(\kk)\ne0. We may therefore assume j=b-d\geq0.

Shrink the marking tail so that Lemma 10.7 holds and the consecutive-image conclusion of Proposition 10.3 holds in degree b. Fix any one level U=U_\nu in this tail. The set F^b_\nu of Equation (10.2) is finite. Take a class in the positive subspace Equation (10.9) and a cocycle representative w as described there. If its positive order is at least \epsilon>0, its hth powers have whole-basis order tending to positive infinity as h=q^{m-1} increases. Indeed first power out the fixed root denominator h_s. The resulting whole finite model has integral multiplication laws, so further Frobenius powers multiply the lower order bound; the fixed projection and basis changes contribute only their fixed pole costs. The estimate for \mathcal B therefore puts \mathcal B(w^h) in \mathcal P_\nu for all sufficiently large a. Regard this cochain in the whole localized model through B_U=eB'\subset B'. Its differential vanishes there because it vanishes in B_U. Since \mathcal P_\nu is G-stable and \mathcal P_\nu\hookrightarrow B' is injective, it is a \mathcal P_\nu-valued cocycle.

By Equation (10.8), the fixed class e_s[w] then belongs to

for every sufficiently large a. Multiplication by the nonzero constant in that equation does not change this subspace. These image subspaces are decreasing, so membership in every sufficiently late one implies membership in their entire intersection F^b_\nu. The threshold may depend on w; no uniform threshold over all cocycles is needed.

Thus e_s maps Z_U^j into a finite-dimensional space. The quotient of its source by Z_U^j is countable-dimensional, so the whole image of e_s at this fixed level is countable-dimensional. This holds at every level in the chosen tail. Its morphism in \mathcal A is therefore zero. It was an epimorphism onto the nonzero object T^b(\kk), a contradiction. Hence all T^i(\kk) vanish, and the uniform degree bound gives one common tail for all degrees.□

Proof of Theorem 10.1 for m=1. By Proposition 7.3, the normalized transfer satisfies

on the open coefficient algebra and on its group cochains. It is therefore surjective on every X^i_\nu. In particular Equation (10.1) holds, and Equation (10.2) gives a finite intersection F^i_\nu at each sufficiently deep marking level.

Choose a sufficiently positive whole-lattice shift x^LA', with L larger than the fixed projection pole cost. The image of its cohomology in X^i_\nu has countable cokernel, by Lemma 3.8. If a class z is represented by a projected cocycle from this lattice, its iterated Frobenius powers are integral in the whole model after projection: their order grows, while the idempotent and basis costs are fixed. They consequently lie in \mathcal P_\nu for every sufficiently large power. The identity splitting Frobenius gives

in the actual open cohomology. As before, the image subspaces are decreasing, so this class belongs to F^i_\nu. The image of the positive-lattice cohomology is therefore finite; its cokernel is countable. Hence X^i_\nu is countable. The uniform cohomological bound and the finitely many required choices give a common sufficiently deep tail for all degrees.□

Proof. By Theorem 10.1, whose boundary hypothesis is precisely Lemma 3.8 at height m+1, all groups H^i_{\cts}(G,B_U) are countable on sufficiently deep fixed levels. We show that the indicated coefficient extensions have the same property.

Apply Lemma 3.6 to the basic coefficient M. Its whole division-tuple factors, including their powered coordinates, give a finite-dimensional translation representation E; the linear connected marking trivializes the chosen integer periodicity power. At every sufficiently deep level this gives a G-equivariant identification

Here G acts on the generator-lift torsor and commutes with translation.

We will use a finite filtration or a finite direct-summand diagram of E. Its finitely many coaction matrices and arrows factor through one finite translation quotient. By the same finite-presentation descent as in Lemma 3.6 and Section 7, the entire diagram and its torsor descend to one further finite marking level. The auxiliary actions on its finite-dimensional \kk-spaces factor through finite groups, so a common shrink makes them trivial on the diagram. Naturality under formal-group isomorphisms retains G-equivariance: the identities hold at full markings and descend faithfully to the fixed finite stage. These choices concern finitely many modules and maps, not individual elements of G.

The associated-bundle functor is exact on these finite diagrams. After base change to the finite lift torsor it is tensoring with E; faithful flatness detects exactness. Its terms are finite locally free bundles on the affine open. Surjections between them have underlying module sections, so after choosing lattices their sections have bounded denominators. Consequently the associated exact sequences give exact continuous cochain sequences in the convention of Lemma 2.2.

Suppose first that m>1. The distribution algebra is

with maximal ideal \mathfrak m=(D_1,\ldots,D_d), as in Proposition 7.1. A finite translation representation is a finite-length continuous \Lambda-module. Its finite filtration

has quotients that are finite direct sums of the trivial representation \kk. Having retained this finite diagram at the chosen marking level, its associated filtration has quotients B_U^{\oplus a} with their ordinary G-actions. Countability of their cohomology and the long exact sequences of the filtration give countability of H^i_{\cts}(G,V_E) in every degree.

For m=1, the translation group is of multiplicative type. Choose H from the full cofinal sequence of finite generator-lift quotients, deep enough that E factors through H. Thus its torsor is the full root algebra of Proposition 3.5, rather than an intermediate subalgebra of that root extension. Its distribution algebra

is finite étale over \kk, because the Cartier dual H^D of a finite multiplicative-type group is étale. Every finite R_H-module is therefore a direct summand of R_H^{\oplus a} for some a. The perfect trace pairing of the finite étale algebra identifies R_H with R_H^\vee=\mathcal O(H) as R_H-modules. Thus E is a direct summand of a finite sum of regular function representations. This argument works over \kk itself and includes quotients whose order is divisible by p. For the general semisimplicity statement over the ground field, see Milne [27, Theorem 12.30, pp. 241–242]. Shrink U so that the splitting maps of this finite diagram descend as well.

The associated bundle of such a regular representation is the function algebra of the corresponding finite lift torsor. By Proposition 3.5 and (7.1), after enlarging the level it is

The exponent may be chosen to fix \kk. Powering gives an additive, G-equivariant homeomorphism

For the bounded-pole topology, this follows either from the finite root coordinates or by multiplying every root pole bound and parameter order by h; the inverse rescales those bounds. Hence powering identifies the continuous cochain complexes as additive complexes. It is also \kk-linear for the chosen exponents. It follows that each regular block, and then its finite sums and summands, has countable cohomology.

All choices involve finitely many levels and finite diagrams. They persist on a cofinal tail, where Theorem 10.1 also applies. This proves the assertion in both cases.□

Proof. Put H=J/U and N=B_U\otimes_B M_B. The G- and H-actions on N commute. Finite étale torsor descent identifies the augmented Amitsur complex with the finite-group cochain resolution

where the displayed N is in group-cochain degree zero. This augmented complex is exact. For completeness, after faithfully flat base change to B_U the torsor becomes the trivial H-torsor and its augmented complex has the usual extra-degeneracy contraction. Exactness therefore holds before base change as well.

It also holds with the required continuous G-cochain parameters. To make the topology explicit, choose \tau\in B_U with \sum_{h\in H}h(\tau)=1. Such an element exists: the torsor trace is surjective after the same faithfully flat base change, so its cokernel vanishes over B. The B-linear map

satisfies \lambda(1)=1. Applying \lambda to the first algebra factor contracts the augmented Amitsur complex; the insert-unit differential gives the contraction identity directly. These are finite matrices over B on the fixed finite locally free terms, so they send each lattice to a bounded pole shift and are continuous there. The contraction need not commute with G: exactness of each row after applying continuous cochains is what is needed. Thus applying C^a_{\cts}(G,-) to (11.2) remains exact for every a.

The first-quadrant double complex

therefore computes H^*_{\cts}(G,M_B), and its other spectral sequence is

Finite H-cochains are finite products. Their cohomology on a countable module is countable, regardless of divisibility of |H| by p. Every total-degree diagonal of (11.3) is finite, so its abutment is countable. This proves the claim.□

Proof of Theorem 11.1. Tensoring finite free modules preserves exact filtrations and retracts, and a tensor product of basic coefficients is basic. Induction on the finite constructions in Definition 3.1 therefore reduces the asserted closure properties to finite filtrations and retracts. We induct downwards on m. At m=n, every basic coefficient is a finite-dimensional vector space over the finite field \kk. A finite-type projective resolution for G computes its cohomology by a complex whose term in each degree is a summand of a finite power of this finite coefficient module; see Lemma 2.3. Each cohomology group is therefore finite. This uses finiteness of the resolution in each degree, not finite cohomological dimension of G. Finite sums, summands, and finite filtrations preserve the conclusion.

Now let m<n and assume the assertion at m+1. It suffices first to treat a basic tensor coefficient M: the long exact cohomology sequence and retracts then give all the closures allowed in Definition 3.1. By Lemmas 11.2 and 11.3,

We spell out the boundary comparison. The submodules x^aM are G-stable as submodules with their induced actions, since G preserves the ideal (x). The successive quotients of every fixed pole strip are restrictions of the original coefficients to A_{m+1}, with the appropriate conormal twists. These are still allowed coefficients: the conormal line is an integer periodicity twist, and a powered tuple with b\leq m\ell also satisfies b\leq(m+1)\ell. The whole-model filtrations in Proposition 3.4 and Lemma 3.8 consequently give finite cohomology for every fixed strip. In particular, for

each H^i_{\cts}(G,Q_a) is finite.

By the bounded-denominator convention, cochains on Q are precisely the filtered union of the cochains on the Q_a. Filtered colimits of vector spaces are exact, so

is countable. The exact cochain sequence supplied by Lemma 2.2 for 0\to M\to M[1/x]\to Q\to0 now shows that the kernel and cokernel of

are countable. Together with (11.4), this proves that H^i_{\cts}(G,M) is countable.

Finally, M is compact. The finite-type completed resolution in Lemma 2.3 computes H^i_{\cts}(G,M) as a quotient of a closed subspace of a compact module by the compact image of a continuous map. Its cohomology is thus compact and Hausdorff. A countable compact Hausdorff group is finite: by the Baire theorem one of its singleton closed subsets has interior; translations make the group discrete, and compactness makes that discrete group finite. Hence H^i_{\cts}(G,M) is finite. This closes the induction.□

Proof. The coefficient ring is

Multiplication by p is injective, and the ring is even. Therefore the odd homotopy groups of E_n/p vanish, while each even homotopy group is the special-fiber ring with an integer power of its intrinsic periodicity line. Extending \mathbb F_{p^n} to the finite field \kk identifies this coefficient with one of the m=1 twists in Theorem 11.1. Finite scalar extension commutes with the compact continuous-cochain complex and with its cohomology: after choosing a field basis it is a finite direct sum, with the product compact topology. Faithfulness of the extension therefore proves the asserted finiteness for P_n over \mathbb F_{p^n}.

For the extended group use the continuous group-extension spectral sequence

Its coefficients on the second page are finite, and a finite group has finite cohomology in each degree on a finite module. Each total-degree diagonal has finitely many entries. The abutment is therefore finite. This reasoning also applies when p divides the degree n of the constant-field extension; no averaging by the order of the Galois group is used.□

Proof. Every entry on the second page of (12.1) is finite by Corollary 11.4. The entries on all later pages are subquotients of those entries. Choose R,N as in the descent theorem. Only 0\leq s\leq N can contribute to the limiting page, and the entries contributing to the stem j have t=j+s. There are thus only finitely many finite limiting entries in that stem. Strong convergence supplies a separated exhaustive filtration of the abutment. Above filtration N, all successive quotients vanish, so that tail is constant; separatedness makes the constant tail zero. The remaining filtration is finite, with finite successive quotients, so the abutment is finite. The same argument covers negative j, and no evaluation or splitting of additive extensions is needed.□

Proof. The cofiber exact sequence gives, for every j,

Both end terms are killed by p. Thus every finite middle term is a finite p-group; we do not require it to be killed by p itself. Factoring p^{a+1} as p\circ p^a gives exact triangles

Their homotopy exact sequences inductively show that \pi_j(X/p^a) is a finite p-group for every a\geq1 and every j.

For fixed j, the inverse system of these finite groups is Mittag–Leffler: its decreasing images in any one finite target eventually stabilize. Its first derived inverse limit therefore vanishes. Applying the Milnor exact sequence, in the adjacent degree as well, and derived completeness gives

For clarity, vanishing of the derived limit requires no surjectivity of the transition maps. The map 1-\mathrm{shift} on the product of the finite groups is surjective: every finite set of its defining equations can be solved backwards, and compactness of the product supplies a simultaneous solution. This is also a direct proof of the required lim-one vanishing.

Put M=\pi_jX. The presentation (12.3) makes M a compact Hausdorff pro-p group with its canonical continuous \Zp-module structure. On each finite target \pi_j(X/p^a), this scalar structure is defined by reducing a p-adic integer modulo a power of p that kills that target; the definitions are compatible with the transition maps. Multiplication by p on M is continuous, so pM is compact and closed. By (12.2), the quotient M/pM is finite.

Choose a_1,\ldots,a_d\in M lifting generators of M/pM over \mathbb F_p. For z\in M, successively express

The scalars c_i=\sum_{\ell\geq0}p^\ell b_{i,\ell} converge in \Zp. The residual term after L steps is p^Lz_L. It tends to zero in M: any fixed finite target in (12.3) is killed by a fixed power of p, which kills the image of p^Lz_L for all sufficiently large L, independently of z_L. Therefore z=\sum_i c_i a_i. The continuous homomorphism

is surjective. This proves finite generation as an abstract \Zp-module, and not merely density of a finitely generated subgroup.

Conversely, if every \pi_jX is finitely generated over \Zp, then both end terms of (12.2) are finite. For the p-torsion term, use that a submodule of a finite module over the Noetherian ring \Zp is finite as a module, and here it is killed by p. The middle term is consequently finite as well.□

Proof of Theorem 1.1. Let X=L_{K(n)}S_p^\wedge. A K(n)-local spectrum is derived p-complete when n>0. To see this directly, the fiber of X\to\operatorname*{holim}_aX/p^a is

The spectrum S[1/p] is K(n)-acyclic, so this mapping spectrum vanishes by locality. This proves the required completeness without a connectivity assumption.

The map S\to S_p^\wedge is a mod-p equivalence, hence a K(n)-equivalence: its cofiber has invertible multiplication by p, whereas multiplication by p is zero on Morava K-theory. Exactness of localization therefore identifies

The latter spectrum has finite homotopy groups in every degree by Proposition 12.1. Applying Lemma 12.2 proves the sphere assertion.

For the finite-spectrum formulation, consider the class of finite p-local spectra F for which every \pi_j L_{K(n)}(F\wedge S/p) is finite. It contains the sphere by Proposition 12.1 and is closed under suspensions, finite cofiber sequences, and retracts, by exactness and the homotopy long exact sequence. It therefore contains every finite p-local spectrum. The spectrum L_{K(n)}F is derived p-complete by the same locality argument, and its cofiber of p is L_{K(n)}(F\wedge S/p). Applying Lemma 12.2 objectwise proves the full finite-spectrum conclusion of Theorem 1.1.□

Proof. By Barthel et al. [3, Theorem A], the rationalized homotopy ring is the indicated exterior algebra. For a finitely generated \Zp-module its \Qp-dimension after inverting p is its free rank. The exterior monomials are indexed by subsets I and have the degrees in (12.4). This proves the formula.□

Proof. At height one, Morava E-theory is p-completed periodic complex K-theory, with

The stabilizer is \Zp^\times=\mu_{p-1}\times(1+p\Zp), and the Adams operation \psi^a acts on \beta^t by a^t. Taking homotopy fixed points by the finite group \mu_{p-1} is exact on these coefficient modules, because p-1 is a unit in \Zp. The resulting Adams summand, denoted B, has

The element 1+p topologically generates 1+p\Zp. Its two-term continuous resolution, or the standard Adams-operation fiber sequence [16, §§1–2], gives

On a retained coefficient in degree 2t, the map is multiplication by (1+p)^t-1. For t=0 it is zero, giving \Zp in degrees 0 and -1. For t\neq0 it is injective, and its cokernel is cyclic of order determined by

Indeed, if p\nmid a, binomial expansion gives (1+p)^a-1\equiv ap\pmod{p^2}. If v_p(z)\geq1 and p is odd, the term pz in (1+z)^p-1 has valuation 1+v_p(z) and every other term has strictly greater valuation. Iterating this observation proves (12.5) for positive t. For negative t, the identity

shows that the valuation is unchanged by replacing t with -t.

The cokernel for a nonzero retained t lies in degree 2t-1; there is no kernel in degree 2t. Coefficients in other degrees vanish. The fiber exact sequence has at most one contributing kernel or cokernel in each stem, so the stated groups involve no unresolved additive extension.□