On q-de Rham cohomology via Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document}-rings

We show that Aomoto’s q-deformation of de Rham cohomology arises as a natural cohomology theory for Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document}-rings. Moreover, Scholze’s (q-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(q-1)$$\end{document}-adic completion of q-de Rham cohomology depends only on the Adams operations at each residue characteristic. This gives a fully functorial cohomology theory, including a lift of the Cartier isomorphism, for smooth formal schemes in mixed characteristic equipped with a suitable lift of Frobenius. If we attach p-power roots of q, the resulting theory is independent even of these lifts of Frobenius, refining a comparison by Bhatt, Morrow and Scholze.


Introduction
The q-de Rham cohomology of a polynomial ring is a Z[q]-linear complex given by replacing the usual derivative with the Jackson q-derivative ∇ q (x n ) = [n] q x n−1 dx, where [n] q is Gauss' q-analogue q n −1 q−1 of the integer n. In [13], Scholze discussed the (q − 1)-adic completion of this theory for smooth rings, explaining relations to p-adic Hodge theory and singular cohomology, and conjecturing that it is independent of co-ordinates, so functorial for smooth algebras over a fixed base [13, Conjectures 1.1, 3.1 and 7.1].
We show that q-de Rham cohomology with q-connections naturally arises as a functorial invariant of -rings (Theorems 1.17, 1.23 and Proposition 1.25), and that its (q − 1)-adic completion depends only on a P -ring structure (Theorem 2.8), for P the set of residue characteristics; a P -ring has a lift of Frobenius for each p ∈ P. This recovers the known equivalence between de Rham cohomology and complete q-de Rham cohomology over the rationals, while giving no really new functoriality statements for smooth schemes over Z. However, in mixed characteristic, it means that

Comparisons for 3-rings
We will follow standard notational conventions for -rings. These are commutative rings equipped with operations λ i resembling alternating powers, in particular satisfying λ k (a + b) = k i=0 λ i (a)λ k−i (b), with λ 0 (a) = 1 and λ 1 (a) = a. For background, see [5] and references therein. The -rings we encounter are all torsion-free, in which case [16] shows the -ring structure is equivalent to giving ring endomorphisms n for n ∈ Z >0 with mn = m • n and p (x) ≡ x p mod p for all primes p. If we write λ t ( f ) := i≥0 λ i ( f )t i and t ( f ) := n≥1 n ( f )t n , then the families of operations are related by the formula t = −t d log λ −t dt . We refer to elements x with λ i (x) = 0 for all i > 1 (or equivalently n (x) = x n for all n) as elements of rank 1.

The 3-ring Z[q]
Definition 1.1 Define Z[q] to be the -ring with operations determined by setting q to be of rank 1.
We now consider the q-analogues [n] q := q n −1 q−1 ∈ Z[q] of the integers, with For any torsion-free -ring, localisation at a set of elements closed under the Adams operations always yields another -ring, since p (a −1 )−a − p = ( p (a)a p ) −1 (a p − p (a)) is divisible by p.

Lemma 1.3
For the -ring structure on Z[x, y] with x, y of rank 1, the elements λ n y−x q−1 ∈ Z[q, {(q n − 1) −1 } n≥1 , x, y] are given by Proof The second expression comes from multiplying out the Gaussian binomial expansions. The easiest way to prove the first is to observe that λ k ( y−x q−1 ) must be a homogeneous polynomial of degree k in x, y, with coefficients in the integral domain Z[q, {(q n − 1) −1 } n≥1 ], and to note that Thus λ k ( y−x q−1 ) agrees with the homogeneous polynomial above for infinitely many values of y x , so must be equal to it.

Remark 1.4 Note that as
Indeed, for any rank 1 element x in a -ring we have which is just the q-exponential e q (xt). Multiplicativity and universality then imply that λ (q−1)t ( a q−1 ) is a q-deformation of exp(at) for all a. Thus (q − 1) k λ k ( a q−1 ) is a q-analogue of the kth divided power (a k /k!). An explicit expression comes recursively from the formula , which arises because q is of rank 1 and qa q−1 = a + a q−1 .

Lemma 1.5 For elements x, y of rank
By induction on j, it thus follows that  10 The cochain complex qDR(A/R) naturally carries much more structure than these Adams operations. Whenever we can factor the functor O through a model category C equipped with a forgetful functor to Ch(R[q]) preserving weak equivalences and homotopy limits, we can regard qDR(A/R) as an object of the homotopy category of C by taking the defining homotopy limit in C.
The universal such example for C is given by the model category of cosimplicialrings over R [q], with weak equivalences being quasi-isomorphisms (i.e. cohomology isomorphisms) and fibrations being surjections; the underlying cochain complex has differential (−1) j ∂ j . That this determines a model structure follows from Kan's transfer theorem [9,Theorem 11.3.2] applied to the cosimplicial Dold-Kan normalisation functor taking values in unbounded chain complexes with the projective model structure; the conditions of that theorem are satisfied because the left adjoint functor sends acyclic cofibrant complexes to cosimplicial -rings which automatically have a contracting homotopy in the form of an extra codegeneracy map.
In particular, qDR(A/R) naturally underlies a quasi-isomorphism class of cosimplicial -rings over R [q]; forgetting the λ-operations gives a cosimplicial commutative R[q]-algebra, and stabilisation then gives an E ∞ -algebra over R[q], all with underlying cochain complex qDR(A/R).

Definition 1.11
Given a polynomial ring R[x], recall from [13] that the q-de Rham (or Aomoto-Jackson) cohomology q-• R[x]/R is given by the complex is then set to be

Definition 1.13
Given a cosimplicial abelian group V • , we write N V for the Dold-Kan normalisation of V ( [15,Lemma 8.3.7] applied the opposite category). This is a cochain complex with N r V = V r ∩ j<r ker σ j and differential d = r +1 j=0 (−1) j ∂ j : generated by q and the elements x i and

Proof
We verify the conditions of Lemma 1.14 by showing that each U n is a flat -ring over R [q] representing (X q strat ) n . Taking X = Spec R[x], observe that any element of (X q strat ) n (B) gives rise to a morphism f : R[q, x 0 , . . . , x n ] → B of -rings over R [q], with the image of x i − x j divisible by (q − 1). Flatness of B then gives a unique element f (x i − x j )/(q − 1) ∈ B, so we have a map f to B from the free -ring L over R[q, x 0 , . . . , x n ] generated by elements (the only hypothesis we really need) implying that the image of f factors through the image To see that (X q strat ) n is represented by U n , we only now need to check that U n is itself flat over R [q], which follows because the argument of Lemma 1.5 gives a basis In fact, the proofs of Lemma 1.14 and Proposition 1.15 show that the natural cosimplicial -ring structure on U • gives a model for the cosimplicial -ring structure on qDR(R[x]/R) coming from Remark 1.10.

Definition 1.16
Following [13, Proposition 5.4], we denote by Lη (q−1) the décalage functor with respect to the derived (q − 1)-adic filtration. This is given on complexes and is extended to the derived category of R[q]-modules by taking torsion-free resolutions.
Proof It suffices to prove the first statement, the second following immediately by , and similarly for the simplicial functor (Spec A ⊗ R A ) q strat of Definition 1.12. Since coproduct of flat -rings over R[q] is given by ⊗ R [q] , it follows from Lemma 1.14 and Proposition 1.15 that qDR(R[x 1 , . . . , x n ]/R) can be calculated as the Dold-Kan normalisation of (U • ) ⊗ R[q] n (given by the n-fold tensor product (U m ) ⊗ R[q] n in cosimplicial level m), for the cosimplicial module U • of Proposition 1.15.
The proof now proceeds in a similar fashion to the comparison between crystalline and de Rham cohomology in [3]. We consider the cochain complexes˜ • (U m ) given by In order to see that this differential takes values in the codomains given, observe that and similarly The first calculation also shows that the inclusion˜ for k ≥ 1, allowing us to define a contracting homotopy Since contracting homotopies interact well with tensor products, it also follows that the inclusion˜ n , and hence their retractions given by diagonals U m → U 0 , are quasi-isomorphisms. These combine to give a quasi-isomorphism on total complexes of normalisations. Now, the cosimplicial module˜ r (U • ) is given by the cosimplicial (i.e. levelwise) tensor product of U • with the cosimplicial Z-module with operations induced by those in Proposition 1.15. For r > 0, this cosimplicial Z-module is contractible, via the extra codegeneracy map given by The Eilenberg-Zilber theorem ([15, §8.5] applied to the opposite category) ensures that the normalisation of a cosimplicial tensor product is quasi-isomorphic to the tensor product of the normalisations. Tensoring with a complex which has an extra codegeneracy map always produces an acyclic complex, so˜ r (U • ) and its tensor powers are all acyclic for r > 0. The brutal truncation maps and we just observe that˜

Remark 1.18
Note that Theorem 1.17 and Remark 1.10 together imply that q-• R[x 1 ,...,x n ]/R naturally underlies the décalage of a cosimplicial -ring over R [q]. Even the underlying cosimplicial commutative ring structure carries more information than an E ∞ -structure when Q R.

Completed q-cohomology
Definition 1.19 Given a morphism R → A of -rings, we define the categorŷ Strat q A/R ⊂ Strat q A/R to consist of those objects which are (q − 1)-adically complete. Equivalently,Ŝtrat q A/R is the Grothendieck construction of the functor on the category of flat (q − 1)-adically complete -rings over R[q]. Definition 1.20 Given a flat morphism R → A of -rings, define qDR(A/R) to be the cochain complex of R q − 1 -modules given by taking the homotopy limit of the functorŜ The following is immediate: Given a flat morphism R → A of -rings, the complex qDR(A/R) is the derived (q − 1)-adic completion of qDR(A/R).

Definition 1.22
As in [13, §3], given a formally étale map : where ∇ q is defined as follows. First note that the R q −1 -linear ring endomorphisms The induced quasi-isomorphisms are independent of the choice of framing.
Proof Since the framing is formally étale, for any (q − 1)-adically complete commutative R[q]-algebra B, any commutative square of R-algebra homomorphisms admits a unique dashed arrow as shown.
For any (q − 1)-adically complete flat -ring B over R, we then have the same property for -ring homomorphisms over R instead of R-algebra homomorphisms: the diagram above gives a unique dashed R-algebra homomorphism, and uniqueness of lifts ensures that it commutes with Adams operations, so is a -ring homomorphism (R being flat over Z). Similarly (taking B = A q − 1 ) uniqueness of lifts ensures that the operations γ i are -ring endomorphisms of A q − 1 .
We can now proceed as in the proof of Theorem 1.17. The complex qDR(A/R) can be realised as the cochain complex underlying a cosimplicial -ringÛ (A), representing the functorX q strat of Definition 1.12 for X = Spec A, restricted to (q − 1)-adically complete -rings B. By the consequences of formal étaleness, we have x d ] and the fibre product is given via the projection of (Ỹ q strat ) n onto the first factor.
In particular, this means thatÛ (A) n is the (q − 1)-adic completion of is a copy of the cosimplicial ring U from Proposition 1.15. This isomorphism respects the cosimplicial operations; note that ∂ 0 is not linear for the left multiplication by A, but is still determined via formal étaleness of the framing. We now define a cosimplicial cochain complex˜ ) is a copy of the complex˜ • (U n ) from the proof of Theorem 1.17. Compatibility of this construction with the cosimplicial operations follows because the γ i are -ring homomorphisms.
The calculations contributing to the proof of Theorem 1.17 are still valid after base change, with contracting homotopies giving quasi-isomorphisms Reduction of this modulo (q −1) 2 , or of its décalage modulo (q −1) (cf. [4, Proposition 6.12]), replaces ∇ q with d throughout, removing any dependence on co-ordinates.
Definition 1.24 Given a flat morphism R → A of -rings with X := Spec A, denote the forgetful functor (B, f ) → B fromŜtrat q A/R to rings by OX q ,strat . There is then a notion of OX q ,strat -modules in the category of functors fromŜtrat q A/R to abelian groups; we will simply refer to these as OX q ,strat -modules. Given a property P of modules, we will say that an OX q ,strat -module F has the property P if for each In [13,Conjecture 7.5], Scholze predicted that the category of q-connections on finite projective A q − 1 -module is independent of co-ordinates on A. The following proposition gives the weaker statement that the category depends only on the -ring structure on A.
is an isomorphism.

Proof
The map is determined by its restriction to M, so using the basis for U 1 from Lemma 1.5, and taking v ∈ M, we have t (b), the cocycle condition becomes j+k = j • k , meaning is determined by the operators e i at the basis vectors, which must moreover commute. Linearity of with respect toÛ (A) 1 then reduces to the condition that (av) Finally, note that the condition that N /(q − 1) be the pullback of an A-module (necessarily (X q strat , N /(q −1))) is equivalent to saying that ∂ 0 N ≡ ∂ 1 N mod (q −1), or that (q − 1) divides k whenever k = 0. In particular, (q − 1) divides e i , and setting ∇ i,q := (q − 1) −1 e i gives a q-connection (∇ i,q ) 1≤i≤d on M = N 0 uniquely determining .
The inverse construction is given by

Comparisons for 3 P -rings
Since very few étale maps R[x 1 , . . . , x d ] → A give rise to -ring structures on A, Theorem 1.23 is fairly limited in its scope for applications. We now show how the construction of qDR and the comparison quasi-isomorphism survive when we weaken the -ring structure by discarding Adams operations at invertible primes.

q-cohomology for 3 P -rings
Our earlier constructions for -rings all carry over to P -rings, as follows.

Definition 2.1
Given a set P of primes, we define a P -ring A to be a Z,P -ring in the sense of [5]. This means that it is a coalgebra in commutative rings for the comonad given by the functor W (P) of P-typical Witt vectors. When a commutative ring A is flat over Z, giving a P -ring structure on A is equivalent to giving commuting Adams operations p for all p ∈ P, with p (a) ≡ a p mod p for all a.
Thus when P is the set of all primes, a P -ring is just a -ring; a ∅ -ring is just a commutative ring; for a single prime p, we write p := { p} , and note that a p -ring is a δ-ring in the sense of [10]. Definition 2.2 Given a P -ring R, say that A is a P -ring over R if it is a P -ring equipped with a morphism R → A of P -rings. We say that A is a flat P -ring over R if A is flat as a module over the commutative ring underlying R.

Lemma 2.5 For a set P of primes, the forgetful functor from -rings to P -rings has a right adjoint W (/ ∈P) . There is a canonical ghost component morphism
which is an isomorphism when P contains all the residue characteristics of B.
Proof Existence of a right adjoint follows from the comonadic definitions of -rings and P -rings. The ghost component morphism is given by taking the Adams operations n coming from the -ring structure on W (/ ∈P) (B), followed by projection to B. When P contains all the residue characteristics of B, a -ring structure is the same as a P -ring structure with compatible commuting Adams operations for all primes not in P, leading to the description above.
Note that the big Witt vector functor W on commutative rings thus factorises as W = W (/ ∈P) • W (P) , for W (P) the P-typical Witt vectors.

Proposition 2.6 Given a morphism R → A of -rings, and a set P of primes, there are natural maps
and the latter map is a quasi-isomorphism when P contains all the residue characteristics of A.
Proof We have functors We have (q n − 1) = (q − 1)[n] q , and [n] q is a unit in Z[ 1 n ] q − 1 , hence a unit in B when n is coprime to the residue characteristics. Thus the map (W (/ ∈P) B)/(q − 1) → W (/ ∈P) (B/(q − 1)) gives an isomorphism whenever B is (q − 1)-adically complete and admits a map from A, so the transformation (Spec A) is a natural isomorphism on the category of flat (q − 1)-adically complete P -rings over R[q], and hence qDR P (A/R) − → qDR(A/R). Remark 2.7 Remark 1.10 shows that qDR(A/R) can naturally be promoted to a cosimplicial -ring, and the same reasoning promotes qDR P (A/R) to a cosimplicial P -ring. The proof of Proposition 2.6 then ensures that the map qDR P (A/R) → qDR(A/R) is naturally a morphism of cosimplicial P -rings, Over Z[{ 1 p : p ∈ P}], every P -ring can be canonically made into a -ring, by setting all the additional Adams operations to be the identity. However, this observation is of limited use in establishing functoriality of q-de Rham cohomology, because the resulting -ring structure will not satisfy the conditions of Theorem 1.23. We now give a more general result which does allow for meaningful comparisons. : R[x 1 , . . . , x d ] → A is a formally étale map of P -rings, the elements x i having rank 1, then there are zigzags of R q − 1 -linear quasi-isomorphisms

Theorem 2.8 If R is a flat P -ring over Z and
whenever P contains all the residue characteristics of A.

Proof
The key observation to make is that formally étale maps have a unique lifting property with respect to nilpotent extensions of flat P -rings, because the Adams operations must also lift uniquely. In particular, this means that the operations γ i featuring in the definition of q-de Rham cohomology are necessarily endomorphisms of A as a P -ring. Similarly to Theorem 1.23, qDR P (A/R) is calculated using a cosimplicial Pring given in level n by the (q − 1)-adic completionÛ • P,A of the P -ring over

Cartier isomorphisms in mixed characteristic
In [13, Conjecture 7.1], Scholze predicted that q-• A/R, is a functorial invariant of the R-algebra A, independent of the choice of framing, so extends to all smooth schemes. Theorem 2.8 shows that q-• A/R, is functorial invariant of the P -ring A over R. The only setting in which Theorem 2.8 leads to results close to Scholze's conjecture is when R = W ( p) (k), the p-typical Witt vectors of a perfect field of characteristic p, and A = lim ← −n A n is a formal deformation of a smooth k-algebra A 0 . Then any formally étale morphism W ( p) (k)[x 1 , . . . , x d ] → A of topological rings gives rise to a unique compatible lift p of absolute Frobenius on A with p (x i ) = x p i , so gives A the structure of a topological p -ring. The framing still affects the choice of p -ring structure, but at least such a structure is guaranteed to exist, giving rise to a complex Our constructions now allow us to globalise the quasi-isomorphism q- [13,Proposition 3.4], where * A/R denotes the complex A

Lemma 2.9 Under the quasi-isomorphism qDR p (A/R) ( *
A/R q − 1 , (q − 1)∇ q ) from Theorem 2.8, the semilinear Adams operation p on qDR p (A/R) described in Definition 1.8 corresponds to the operation on * A/R q − 1 given by setting for a ∈ A q − 1 .
Proof Just observe that this expression defines a chain map on , and that the quasi-isomorphisms in the proof of Theorem 1.23 commute with these operations.
As in [13, §4], we refer to formal schemes over W ( p) (k) as smooth if they are flat deformations of smooth schemes over k. We refer to morphisms of such schemes as étale if they are flat deformations of étale morphisms over k.

Proposition 2.10 Take a smooth formal scheme X over R = W ( p) (k) equipped with a lift p of Frobenius which étale locally admits co-ordinates {x
in the derived category of étale sheaves on X.
Proof The unique lifting property of formally étale morphisms ensures that each affine formal scheme U étale over X has a unique lift p | U of Frobenius compatible with the given operation p on X. Functoriality of the construction qDR p for rings with Frobenius lifts thus gives us an étale presheaf qDR p (O X /R) ∧ p of complexes on X. As in Definition 1.8, the Adams operation p on O X then extends to (R q − 1 , p )semilinear maps and thus, denoting good truncation by τ , of the Cartier quasi-isomorphism. The local calculation of [13,Proposition 3.4] then ensures that C −1 q is a quasi-isomorphism.

Functoriality via analogues of de Rham-Witt cohomology
In order to obtain a cohomology theory for smooth commutative rings rather than for P -rings, we now consider q-analogues of de Rham-Witt cohomology. Our starting point is to observe that if we allow roots of q, we can extend the Jackson differential to fractional powers of x by the formula where d log x = x −1 dx, so terms such as [n] q 1/n x m/n have integral derivative, where [n] q 1/n = q−1 q 1/n −1 .

Motivation
Definition 3.1 Given a P -ring B, define 1/P ∞ B to be the smallest P -ring which is equipped with a morphism from B and for which the Adams operations are automorphisms.
In the case P = {p}, the p -ring 1/ p ∞ B is thus the colimit of the diagram By Remark 2.7, qDR p (A/R) naturally underlies a cosimplicial p -ring, so applying 1/ p ∞ levelwise gives another cosimplicial p -ring. For the Adams operation p of Definition 2.3, the underlying cochain complex is just 1/ p ∞ qDR p (A/R) := lim − → p qDR p (A/R). As an immediate consequence of Lemma 2.9, we have:

Lemma 3.2 If R is a flat p -ring over Z ( p) with p an isomorphism, then 1/ p ∞ q D R p (R[x]/R) is quasi-isomorphic to the complex
are quasi-isomorphic after (q − 1)-adic completion.
Thus in level 0 (resp. level 1), Lη (q−1) 1/ p ∞ qDR(R[x]/R) is spanned by elements of the form [ p n ] q 1/ p n x m/ p n (resp. x m/ p n d log x), so setting q 1/ p ∞ = 1 gives a complex whose p-adic completion is the p-typical de Rham-Witt complex.

Lemma 3.3 Let R and
A becomes an isomorphism on p-adic completion, because is flat and we have an isomorphism modulo p. Thus Combined with the calculation of Lemma 2.9, this gives us a quasi-isomorphism between ( 1/ p ∞ qDR p (A/R)) ∧ p and the ( p, q − 1)-adic completion of where I ranges over finite subsets of {1, . . . , d} and α ranges over elements of p −∞ Z d with 0 ≤ α i < 1 if i / ∈ I and −1 < α i ≤ 0 if i ∈ I . We then observe that the contributions to the décalage η (q−1) from terms with α = 0 must be acyclic, via a contracting homotopy defined by the restriction to η (q−1) of the q-integration map

Remark 3.4
The endomorphism given on 1/P ∞ qDR P (A/R) by a → 1/n ([n] q a) = [n] q 1/n 1/n a descends to an endomorphism of H 0 ( 1/P ∞ qDR P (A/R)/(q − 1)), which we may denote by V n because it mimics Verschiebung in the sense that n V n = n · id (since [n] q ≡ n mod (q − 1)). For A smooth over Z, we then have for ζ n a primitive nth root of unity.
By adjunction, this gives an injective map of P -rings, which becomes an isomorphism on completing 1/P ∞ qDR(A/Z) with respect to the system {([n] q 1/n )} n∈P ∞ of ideals, where we write P ∞ for the set of integers whose prime factors are all in P. This implies that the cokernel is annihilated by all elements of (q 1/P ∞ − 1), so leads us to consider almost mathematics as in [7].

Almost isomorphisms
From now on, we consider only the case P = {p}. Combined with Lemma 3.3, Remark 3.4 allows us to regard Lη (q−1) is equal to the p-adic completion of its square, since we may write it as the kernel . If we set h 1/ p n to be the Teichmüller element it is idempotent and flat, so gives a basic setup in the sense of [7, 2.1.1]. We thus regard the pair (Z[q 1/ p ∞ ] ∧ ( p,q−1) , W ( p)(m)) as an inverse system of basic setups for almost ring theory.
We then follow the terminology and notation of [7], studying p-adically complete given by localising at almost isomorphisms, the maps whose kernel and cokernel are W ( p) (m)-torsion. Since the counit (M * ) a → M of the adjunction is an (almost) isomorphism, we may also regard almost modules as a full subcategory of the category of modules, consisting of those M for which the natural map M → (M a ) * is an isomorphism. We can define p-adically complete (Z[q 1/ p ∞ ] ∧ ( p,q−1) ) a -algebras similarly, forming a full subcategory of Z[q 1/ p ∞ ] ∧ ( p,q−1) -algebras.

Perfectoid algebras
We now relate Scholze's perfectoid algebras to a class of p -rings, by factorising the tilting equivalence. For simplicity, we work over Z[ζ p ∞ ] ∧ p , although Lemma 3.8 has natural analogues over the ring K o ⊂ K of power-bounded elements of any perfectoid field K in the sense of [12]. Definition 3.6 Define Fontaine's period ring functor A inf from commutative rings to Definition 3.7 Define a perfectoid p -ring to be a flat p-adically complete p -algebra over Z p , on which the Adams operation p is an isomorphism. By analogy with [2, Notation 1.4], we say that a perfectoid p -ring over

Lemma 3.8 We have equivalences of categories
Proof A perfectoid p -ring B is a deformation of the perfect F p -algebra B/ p. As in [12,Proposition 5.13], a perfect F p -algebra C has a unique deformation W ( p) (C) over Z p , to which Frobenius must lift uniquely; this shows that W ( p) gives an equivalence between perfect F p -algebras and perfectoid p -rings. To obtain the bottom equivalence of the diagram, we will show that the functor W ( p) commutes with the respective functors C → C * of almost elements, then appeal to the tilting equivalence.
Because the idempotent ideals of the basic setups in each of our three categories are generated by the rank 1 elements h p −n constructed before Definition 3.5, we can write C * = n h − p −n C in each setting. For a Teichmüller element and taking inverse limits with respect to p gives A inf (C) * ∼ = A inf (C * ) as well.
Next, we observe that since B := A inf (C) is a perfectoid p -ring for any flat p-adically complete Z p -algebra C, we must have B ∼ = W ( p) (B/ p). Comparing rank 1 elements then gives a monoid isomorphism (B/ p) ∼ = lim ← −x →x p C, from which it follows that whenever C is perfectoid. Since tilting gives an equivalence of almost algebras by [12,Theorem 5.2], this completes the proof.

Functoriality of q-de Rham cohomology
Since for the ring C * of almost elements.
Proof By definition, X q, p Since right adjoints commute with limits and A inf = lim ← − p W ( p) , we may rewrite the first term as lim ← − p Hom p (A, W ( p) (C * )) = lim ← − p X (C * ). Setting B := lim ← − p W ( p) (C) * , observe that because [ p n ] q 1/ p n (q 1/ p n −1) = (q−1), we have n [ p n ] q 1/ p n B = (q −1)B, any element on the left defining an almost element of (q − 1)B, hence a genuine element since B = B * is flat. Then note that since the projection map θ : B → C * has kernel ([ p] q 1/ p ), the map θ • p n−1 has kernel ([ p] q 1/ p n ), and so B → W ( p) (C) * has kernel n [ p n ] q 1/ p n B. Thus In fact, the tilting equivalence gives lim ← − p X (C * ) ∼ = X (C * ), so the only dependence of X q, p strat (A inf (C) * ), and hence (( 1/ p ∞ qDR p (A/Z p )) ∧ p ) a , on the Frobenius lift p is in determining the image of X (C * ) → X (C * ) as C varies.
Although the map X (C * ) → X (C * ) is not surjective, it is almost so in a precise sense, which we now use to establish independence of p , showing that, up to faithfully flat descent, qDR p (A/Z p ) ∧ p /[ p] q 1/ p is the best possible perfectoid approximation to A[ζ p ∞ ] ∧ p . Definition 3.10 Given a functor X from (Z[ζ p ∞ ] ∧ p ) a -algebras to sets and a functor A from perfectoid (Z[ζ p ∞ ] ∧ p ) a -algebras to abelian groups, we write where Pfd(S a ) denotes the category of perfectoid almost S-algebras, and RHom [C,Set] (−, −) is as in Definition 1.9.
When X is representable by a (Z[ζ p ∞ ] ∧ p ) a -algebra C, we simply denote R Pfd (X , A ) by R Pfd (C, A ) -when C is perfectoid, this will just be A (C).
Thus R Pfd (C, A ) is the homotopy limit of the functor A (regarded as taking values in cochain complexes) on the category of perfectoid (Z[ζ p ∞ ] ∧ p ) a -algebras equipped with a map from C. This is closely related to the pushforward from the pro-étale site of the generic fibre, whose décalage for A = A inf is the complex A of [4, Definition 9.1].

Theorem 3.11 If R is a p-adically complete p -ring over Z p , and A a formal Rdeformation of a smooth ring over (R/ p), then the complex
of ( 1/ p ∞ R[q]) ∧ ( p,q−1) -modules is almost quasi-isomorphic to ( 1/ p ∞ qDR p (A/R)) ∧ p for any p -ring structure on A coming from a framing over R as in Theorem 2.8.
Proof Since passage to almost modules is an exact functor, it follows from the definition of qDR p that the cochain complex (( 1/ p ∞ qDR p (A/R)) ∧ p ) a is given by RHom [ fˆ p (R q−1 ),Set] (X q, p strat , (( 1/ p ∞ O) ∧ p ) a ) in the notation of Definition 1.9, where fˆ p (R q − 1 ) denotes the category of flat ( p, q − 1)-adically complete palgebras over R q − 1 .
Now note that It thus follows that the cochain complex (( 1/ p ∞ qDR p (A/R)) ∧ p ) a is the homotopy limit of the functor (B, x, y) → B a on the category of triples (B, x, y) for integral perfectoid p -rings B over Z[q 1/ p ∞ ] ∧ ( p,q −1) and where X = Spec A and Y = Spec R. By Lemma 3.8, such p -rings B are uniquely of the form A inf (C * ) for C ∈ Pfd((Z p [ζ p ∞ ] ∧ p ) a ), so this homotopy limit becomes Writing X ∞ (C) := Im (lim ← − p X (C * ) → X (C * )), Lemma 3.9 then combines with the description above to give We now introduce a Grothendieck topology on the category [Pfd (Z[ζ p ∞ ] ∧ p ) a , Set] by taking covering morphisms to be those maps C → C of perfectoid algebras which are almost faithfully flat modulo p. Since C = lim ← − (C/ p), the functor A inf satisfies descent with respect to these coverings, so the map is a quasi-isomorphism, where (−) denotes sheafification. In other words, the calculation of ( qDR p (A/R) ∧ p ) a is not affected if we tweak the definition of X ∞ by taking the image sheaf instead of the image presheaf. We then have where C → C runs over all covering morphisms. Now, lim ← − p X is represented by the perfectoid algebra ( 1/ p ∞ A) ∧ p , which is isomorphic to A[x 1/ p ∞ 1 , . . . , x 1/ p ∞ d ] ∧ p as in the proof of Lemma 3.3. This allows us to appeal to André's results [1, §2.5] as generalised in [2,Theorem 2.3]. For any morphism f : A → C, there exists a covering morphism C → C i such that f (x i ) has arbitrary p-power roots in C i . Setting C := C 1 ⊗ C . . . ⊗ C C d , this means that the composite A f − → C → C extends to a map ( 1/ p ∞ A) ∧ p → C , so f ∈ (X ∞ ) (C). We have thus shown that (X ∞ ) = X , giving the required equivalence Finally, compatibility of these equivalences with the ( 1/ p ∞ R[q]) ∧ ( p,q−1) -module structures is given by functoriality, multiplicativity and the identification

Remark 3.12
Corresponding to the cohomology theory (( 1/ p ∞ qDR p (A/R)) ∧ p ) a , it is natural to consider q-connections on finite projective modules M over η 0 It follows from the proof of Proposition 1.25 that these are equivalent, for X = Spec A, to finite projective almost ( 1/ p ∞ OX q ,strat ) ∧ p -modules N for which N /(q −1) is the pullback of the almost H 0 (( 1/ p ∞ qDR p (A/R)) ∧ p /(q −1))-module (X q strat , N /(q − 1)) =: M 0 . Up to almost isomorphism, these correspond via the proof of Theorem 3.11 to those finite projective A inf -modules N on the site of integral perfectoid algebras C over A[ζ p ∞ ] ∧ p ⊗ R 1/ p ∞ R for which there exists a W ( p) (A[ζ p ∞ ] ∧ p )-module M 0 with W ( p) (C)-linear isomorphisms This establishes a weakened form of [13, Conjecture 7.5] on co-ordinate independence of the category of q-connections, giving the statement for almost ( n [ p n ] q 1/ p n 1/ p n A[q 1/ p ∞ − 1]) ∧ p )-modules rather than A q − 1 -modules.
In particular, for A = R[x] these include all the elements (q − 1)[m] q p n x p n m−1 dx, since ε i (x m ) = 0 for all i > 0, x m having rank 1. This suggests that in general the image of H 1 (α * ) might be (q − 1)H 1 qDR p (A/R) ∧ p , tying in well with (q − 1)-adic décalage. Explicit descriptions for much of the functoriality from Corollary 3.13 can also be inferred from this analysis, since it implies that the transformations n−1 i=0 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.