## Abstract

We show that Aomoto’s *q*-deformation of de Rham cohomology arises as a natural cohomology theory for \(\Lambda \)-rings. Moreover, Scholze’s \((q-1)\)-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.

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## 1 Introduction

The *q*-de Rham cohomology of a polynomial ring is a \(\mathbb {Z}[q]\)-linear complex given by replacing the usual derivative with the Jackson *q*-derivative \(\nabla _q(x^n)= [n]_qx^{n-1}dx\), where \([n]_q\) is Gauss’ *q*-analogue \(\frac{q^n-1}{q-1}\) 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 \(\Lambda \)-rings (Theorems 1.17, 1.23 and Proposition 1.25), and that its \((q-1)\)-adic completion depends only on a \(\Lambda _P\)-ring structure (Theorem 2.8), for *P* the set of residue characteristics; a \(\Lambda _P\)-ring has a lift of Frobenius for each \(p \in P\). 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 \(\mathbb {Z}\). However, in mixed characteristic, it means that complete *q*-de Rham cohomology depends only on a lift \(\Psi ^p\) of absolute Frobenius locally generated by co-ordinates with \(\Psi ^p(x_i)=x_i^p\). Given such data, we construct (Proposition 2.10) a quasi-isomorphism between Hodge cohomology and *q*-de Rham cohomology modulo \([p]_q\), extending the local lift of the Cartier isomorphism in [13, Proposition 3.4].

Taking the Frobenius stabilisation of the complete *q*-de Rham complex of *A* yields a complex resembling the de Rham–Witt complex. We show (Theorem 3.11) that up to \((q^{1/p^{\infty }}-1)\)-torsion, the *p*-adic completion of this complex depends only on the *p*-adic completion of \(A[\zeta _{p^{\infty }}]\) (where \(\zeta _n\) denotes a primitive *n*th root of unity), with no requirement for a lift of Frobenius or a choice of co-ordinates. The main idea is to show that the stabilised *q*-de Rham complex is in a sense given by applying Fontaine’s period ring construction \(A_{\inf }\) to the best possible perfectoid approximation to \(A[\zeta _{p^{\infty }}]\). As a consequence, this shows (Corollary 3.13) that after attaching all *p*-power roots of *q*, *q*-de Rham cohomology in mixed characteristic is independent of choices, which was already known after base change to a period ring, via the comparisons of [4] between *q*-de Rham cohomology and their theory \(A\Omega \).

The cohomology theories we construct thus depend either on Adams operations at the residue characteristics (for de Rham) or on *p*-power roots of *q* (for variants of de Rham–Witt), establishing correspondingly weakened versions of the conjectures of [13]; in Remark 3.15, we suggest a possible candidate for a theory without those restrictions. The essence of our construction of *q*-de Rham cohomology of *A* over *R* is to set *q* to be an element of rank 1 for the \(\Lambda \)-ring structure, and to look at flat \(\Lambda \)-rings *B* over *R*[*q*] equipped with morphisms \(A \rightarrow B/(q-1)\) of \(\Lambda \)-rings over *R*. If these seem unfamiliar, reassurance should be provided by the observation that \((q-1)B\) carries *q*-analogues of divided power operations (Remark 1.4). For the variants of de Rham–Witt cohomology in Sect. 3, the key to giving a characterisation independent of lifts of Frobenius is the factorisation of the tilting equivalence for perfectoid algebras via a category of \(\Lambda _p\)-rings, leading to constructions similar to [4].

I would like to thank Peter Scholze for many helpful comments, in particular about the possibility of a *q*-analogue of de Rham–Witt cohomology, and Michel Gros for spotting a missing hypothesis. I would also like to thank the anonymous referee for suggesting many improvements.

## 2 Comparisons for \(\Lambda \)-rings

We will follow standard notational conventions for \(\Lambda \)-rings. These are commutative rings equipped with operations \(\lambda ^i\) resembling alternating powers, in particular satisfying \(\lambda ^k(a+b)= \sum _{i=0}^{k} \lambda ^i(a)\lambda ^{k-i}(b)\), with \(\lambda ^0(a)= 1\) and \(\lambda ^1(a)=a\). For background, see [5] and references therein. The \(\Lambda \)-rings we encounter are all torsion-free, in which case [16] shows the \(\Lambda \)-ring structure is equivalent to giving ring endomorphisms \(\Psi ^n\) for \(n \in \mathbb {Z}_{>0}\) with \(\Psi ^{mn}=\Psi ^m \circ \Psi ^n\) and \(\Psi ^p(x) \equiv x^p \mod p\) for all primes *p*. If we write \(\lambda _t(f):=\sum _{i \ge 0} \lambda ^i(f)t^i\) and \(\Psi _t(f):= \sum _{n \ge 1} \Psi ^n(f)t^n\), then the families of operations are related by the formula \(\Psi _t= -t\frac{d\log \lambda _{-t} }{d t}\).

We refer to elements *x* with \(\lambda ^i(x)=0\) for all \(i>1\) (or equivalently \(\Psi ^n(x)=x^n\) for all *n*) as elements of rank 1.

### 2.1 The \(\Lambda \)-ring \(\mathbb {Z}[q]\)

### Definition 1.1

Define \(\mathbb {Z}[q]\) to be the \(\Lambda \)-ring with operations determined by setting *q* to be of rank 1.

We now consider the *q*-analogues \([n]_q:= \frac{q^n-1}{q-1} \in \mathbb {Z}[q]\) of the integers, with \([n]_q!=[n]_q[n-1]_q\ldots [1]_q\), and \(\left( {\begin{array}{c}n\\ k\end{array}}\right) _q= \frac{[n]_q!}{[n-k]_q![k]_q!}\).

### Remark 1.2

To see the importance of regarding \(\mathbb {Z}[q]\) as a \(\Lambda \)-ring observe that the binomial expressions

have as *q*-analogues the Gaussian binomial theorems

as well as Adams operations

For any torsion-free \(\Lambda \)-ring, localisation at a set of elements closed under the Adams operations always yields another \(\Lambda \)-ring, since \(\Psi ^p(a^{-1})-a^{-p} = (\Psi ^p(a)a^p)^{-1}(a^p- \Psi ^p(a))\) is divisible by *p*.

### Lemma 1.3

For the \(\Lambda \)-ring structure on \(\mathbb {Z}[x,y]\) with *x*, *y* of rank 1, the elements

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 \(\lambda ^k(\tfrac{y-x}{q-1})\) must be a homogeneous polynomial of degree *k* in *x*, *y*, with coefficients in the integral domain \(\mathbb {Z}[q, \{(q^n-1)^{-1}\}_{n \ge 1}]\), and to note that

Thus \(\lambda ^k(\frac{y-x}{q-1})\) agrees with the homogeneous polynomial above for infinitely many values of \(\frac{y}{x}\), so must be equal to it. \(\square \)

### Remark 1.4

Note that as \(q\rightarrow 1\), Lemma 1.3 gives \( (q-1)^k\lambda ^k(\frac{y-x}{q-1})\rightarrow \frac{(x-y)^k}{k!}\). Indeed, for any rank 1 element *x* in a \(\Lambda \)-ring we have

which is just the *q*-exponential \(e_q(xt)\). Multiplicativity and universality then imply that \(\lambda _{(q-1)t} (\frac{a}{q-1})\) is a *q*-deformation of \(\exp (at)\) for all *a*. Thus \((q-1)^k\lambda ^k(\frac{a}{q-1})\) is a *q*-analogue of the *k*th divided power \((a^k/k!)\). An explicit expression comes recursively from the formula

obtained by subtracting \(\lambda _t(\frac{a}{q-1})\) from each side of the expression \( \lambda _{qt}(\frac{a}{q-1})=\lambda _t(a)\lambda _t(\frac{a}{q-1})\), which arises because *q* is of rank 1 and \(\frac{qa}{q-1}= a+\frac{a}{q-1}\).

### Lemma 1.5

For elements *x*, *y* of rank 1, the \(\Lambda \)-subring of \(\mathbb {Z}[q, \{(q^n-1)^{-1}\}_{n \ge 1},x,y]\) generated by \(q,x,y, \frac{y-x}{q-1}\) has basis \(\lambda ^k(\frac{y-x}{q-1})\) as a \(\mathbb {Z}[q,x]\)-module.

### Proof

The \(\Lambda \)-subring clearly contains the \(\mathbb {Z}[q,x]\)-module *M* generated by the elements \(\lambda ^k(\frac{y-x}{q-1})\), which are also clearly \(\mathbb {Z}[q,x]\)-linearly independent. Since \(\mathbb {Z}[q,x]\) is a \(\Lambda \)-ring, it suffices to show that *M* is closed under multiplication.

By Lemma 1.3, we know that

We can rewrite \(\frac{y-q^ix}{q-1}= \frac{y-x}{q-1}- [i]_qx\), so \(\lambda ^j(\frac{y-q^ix}{q-1})-\lambda ^j(\frac{y-x}{q-1})\) lies in the \(\mathbb {Z}[q,x]\)-module spanned by \(\lambda ^m(\frac{y-x}{q-1}) \) for \(m<j\). By induction on *j*, it thus follows that

so the binomial expression above implies \( \lambda ^i(\frac{y-x}{q-1})\lambda ^j(\frac{y-x}{q-1}) \in M \). \(\square \)

### 2.2
*q*-cohomology of \(\Lambda \)-rings

### Definition 1.6

Given a \(\Lambda \)-ring *R*, say that *A* is a \(\Lambda \)-ring over *R* if it is a \(\Lambda \)-ring equipped with a morphism \(R \rightarrow A\) of \(\Lambda \)-rings. We say that *A* is a flat \(\Lambda \)-ring over *R* if *A* is flat as a module over the commutative ring underlying *R*.

### Definition 1.7

Given a morphism \(R \rightarrow A\) of \(\Lambda \)-rings, we define the category \(\mathrm {Strat}^q_{A/R}\) to consist of flat \(\Lambda \)-rings *B* over *R*[*q*] equipped with a compatible morphism \(f : A \rightarrow B/(q-1)\), such that *f* admits a lift to *B*; a choice of lift is not taken to be part of the data, so need not be preserved by morphisms.

More concisely, \(\mathrm {Strat}^q_{A/R}\) is the Grothendieck construction of the set-valued functor

on the category \(f\Lambda (R[q])\) of flat \(\Lambda \)-rings over *R*[*q*].

### Definition 1.8

Given a flat morphism \(R \rightarrow A\) of \(\Lambda \)-rings, define \(\mathrm {qDR}(A/R)\) to be the cochain complex of *R*[*q*]-modules given by taking the homotopy limit (in the sense of [6]) of the functor

The cochain complex \(\mathrm {qDR}(A/R)\) naturally carries \((R[q], \Psi ^n)\)-semilinear operations \(\Psi ^n\) coming from the morphisms \(\Psi ^n : B\otimes _{R[q], \Psi ^n}R[q] \rightarrow B\) of *R*[*q*]-modules, for \(B \in \mathrm {Strat}^q_{A/R}\).

Equivalently, can we follow the approach of [8, 14] towards the stratified site and de Rham stack by regarding \(\mathrm {qDR}(A/R)\) as the quasi-coherent cohomology complex of \((\mathrm {Spec}\,A)_{\mathrm {strat}}^q\), as follows.

### Definition 1.9

Given a category \(\mathcal {C}\), write \([\mathcal {C},\mathrm {Set}]\) and \([\mathcal {C},\mathrm {Ab}]\) for the categories of functors on \(\mathcal {C}\) taking values in sets and abelian groups, respectively. For any functor \(X : \mathcal {C}\rightarrow \mathrm {Set}\), we then denote by \(\mathbf R \mathrm {Hom}_{[\mathcal {C},\mathrm {Set}]}(X,-)\) the functor from \([\mathcal {C},\mathrm {Ab}]\) to cochain complexes given by taking the right-derived functor of the functor

of natural transformations with source *X*.

For the forgetful functor \(\mathscr {O}: f\Lambda (R[q])\rightarrow \mathrm {Mod}(R[q])\) to the category of *R*[*q*]-modules, we then have

with Adams operations \(\Psi ^n : \mathscr {O}\otimes _{R[q], \Psi ^n}R[q] \rightarrow \mathscr {O}\) giving the \((R[q], \Psi ^n)\)-semilinear operations \(\Psi ^n\) on \(\mathrm {qDR}(A/R)\).

### Remark 1.10

The cochain complex \(\mathrm {qDR}(A/R)\) naturally carries much more structure than these Adams operations. Whenever we can factor the functor \(\mathscr {O}\) through a model category \(\mathcal {C}\) equipped with a forgetful functor to \(\mathrm {Ch}(R[q])\) preserving weak equivalences and homotopy limits, we can regard \(\mathrm {qDR}(A/R)\) as an object of the homotopy category of \(\mathcal {C}\) by taking the defining homotopy limit in \(\mathcal {C}\).

The universal such example for \(\mathcal {C}\) is given by the model category of cosimplicial \(\Lambda \)-rings over *R*[*q*], with weak equivalences being quasi-isomorphisms (i.e. cohomology isomorphisms) and fibrations being surjections; the underlying cochain complex has differential \(\sum (-1)^j \partial ^j\). 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 \(\Lambda \)-rings which automatically have a contracting homotopy in the form of an extra codegeneracy map.

In particular, \(\mathrm {qDR}(A/R) \) naturally underlies a quasi-isomorphism class of cosimplicial \(\Lambda \)-rings over *R*[*q*]; forgetting the \(\lambda \)-operations gives a cosimplicial commutative *R*[*q*]-algebra, and stabilisation then gives an \(E_{\infty }\)-algebra over *R*[*q*], all with underlying cochain complex \(\mathrm {qDR}(A/R) \).

### Definition 1.11

Given a polynomial ring *R*[*x*], recall from [13] that the *q*-de Rham (or Aomoto–Jackson) cohomology \(q\text{- }\Omega ^{\bullet }_{ R[x]/R}\) is given by the complex

so \(\nabla _q(x^n)= [n]_qx^{n-1}dx\).

Given a polynomial ring \(R[x_1, \ldots ,x_d]\), the *q*-de Rham complex \(q\text{- }\Omega ^{\bullet }_{ R[x_1, \ldots ,x_d]/R}\) is then set to be

so takes the form

### Definition 1.12

Given a flat morphism \(R \rightarrow A\) of \(\Lambda \)-rings with \(X=\mathrm {Spec}\,A\), define the functor \({\tilde{X}}_{\mathrm {strat}}^q\) from flat \(\Lambda \)-rings over *R*[*q*] to simplicial sets by taking the Čech nerve of \(\mathrm {Hom}_{\Lambda ,R}(A,B)\rightarrow \mathrm {Hom}_{\Lambda ,R}(A,B/(q-1))\), so

with simplicial operations

### Definition 1.13

Given a cosimplicial abelian group \(V^{\bullet }\), we write *NV* for the Dold–Kan normalisation of *V* ( [15, Lemma 8.3.7] applied the opposite category). This is a cochain complex with \(N^rV= V^r \cap _{j <r} \ker \sigma ^j\) and differential \(d= \sum _{j=0}^{r+1} (-1)^j\partial ^j : N^rV \rightarrow N^{r+1}V\).

### Lemma 1.14

If, for \(X=\mathrm {Spec}\,A\), the functors \(({\tilde{X}}_{\mathrm {strat}}^q)_n\) are represented by flat \(\Lambda \)-rings \(\Gamma ( ({\tilde{X}}_{\mathrm {strat}}^q)_n,\mathscr {O})\) over *R*[*q*], then a model for \(\mathrm {qDR}(A/R)\) is given by the Dold–Kan normalisation of the cosimplicial module \(n \mapsto \Gamma ( ({\tilde{X}}_{\mathrm {strat}}^q)_n,\mathscr {O})\).

### Proof

The set-valued functor \(X_{\mathrm {strat}}^q=(\mathrm {Spec}\,A)_{\mathrm {strat}}^q\) of Definition 1.7 is resolved by the simplicial functor \({\tilde{X}}_{\mathrm {strat}}^q\) of Definition 1.12. In the notation of Definition 1.9, this implies that the functor \(\mathrm {Hom}_{[f\Lambda (R[q]),\mathrm {Set}]}(X_{\mathrm {strat}}^q,-)\) on \([f\Lambda (R[q]),\mathrm {Ab}]\) is resolved by the cochain complex

Although \(X_{\mathrm {strat}}^q\) is not representable on the category of flat \(\Lambda \)-rings over *R*[*q*], our hypotheses ensure that each functor \(({\tilde{X}}_{\mathrm {strat}}^q)_n\) is so. Thus the functors \(\mathrm {Hom}_{[f\Lambda (R[q]),\mathrm {Set}]}(({\tilde{X}}_{\mathrm {strat}}^q)_n,-)\) and their direct summands \(N^n \mathrm {Hom}_{[f\Lambda (R[q]),\mathrm {Set}]}(({\tilde{X}}_{\mathrm {strat}}^q)_{\bullet },-)\) are exact, and are their own right-derived functors. This implies that the cochain complex of functors above models \(\mathbf R \mathrm {Hom}_{[f\Lambda (R[q]),\mathrm {Set}]}(X_{\mathrm {strat}}^q,-)\), and the result follows by evaluation at \(\mathscr {O}\). \(\square \)

### Proposition 1.15

If *R* is a \(\Lambda \)-ring and *x* of rank 1, then \(\mathrm {qDR}(R[x]/R)\) can be calculated by Dold–Kan normalisation of the cosimplicial *R*[*q*]-module \(U^{\bullet }\) given by setting \(U^n\) to be the \(\Lambda \)-subring

generated by *q* and the elements \(x_i\) and \(\frac{x_i-x_j}{q-1}\), with cosimplicial operations

### Proof

We verify the conditions of Lemma 1.14 by showing that each \(U^n\) is a flat \(\Lambda \)-ring over *R*[*q*] representing \(({\tilde{X}}_{\mathrm {strat}}^q)_n\). Taking \(X=\mathrm {Spec}\,R[x]\), observe that any element of \(({\tilde{X}}_{\mathrm {strat}}^q)_n(B)\) gives rise to a morphism \(f : R[q,x_0, \ldots , x_n] \rightarrow B\) of \(\Lambda \)-rings over *R*[*q*], with the image of \(x_i-x_j\) divisible by \((q-1)\). Flatness of *B* then gives a unique element \(f(x_i-x_j)/(q-1) \in B\), so we have a map *f* to *B* from the free \(\Lambda \)-ring *L* over \(R[q,x_0, \ldots , x_n]\) generated by elements \(z_{ij}\) with \((q-1)z_{ij} =x_i-x_j\).

Since *B* is flat, it embeds in \(B[\{(q^m-1)^{-1}\}_{m \ge 1} ]\) (the only hypothesis we really need) implying that the image of *f* factors through the image \(U^n\) of *L* in \(R[q, \{(q^m-1)^{-1}\}_{m \ge 1},x_0, \ldots , x_n]\). To see that \(({\tilde{X}}_{\mathrm {strat}}^q)_n\) 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

for \(U^n\) over *R*[*q*]. We therefore have \(\mathrm {qDR}( R[x]/R)\simeq NU^{\bullet }\). \(\square \)

In fact, the proofs of Lemma 1.14 and Proposition 1.15 show that the natural cosimplicial \(\Lambda \)-ring structure on \(U^{\bullet }\) gives a model for the cosimplicial \(\Lambda \)-ring structure on \(\mathrm {qDR}(R[x]/R)\) coming from Remark 1.10.

### Definition 1.16

Following [13, Proposition 5.4], we denote by \(\mathbf L \eta _{(q-1)}\) the décalage functor with respect to the derived \((q-1)\)-adic filtration. This is given on complexes \(C^{\bullet }\) of \((q-1)\)-torsion-free *R*[*q*]-modules by

and is extended to the derived category of *R*[*q*]-modules by taking torsion-free resolutions.

### Theorem 1.17

If *R* is a \(\Lambda \)-ring and if the polynomial ring \(R[x_1, \ldots ,x_n]\) is given the \(\Lambda \)-ring structure for which the elements \(x_i\) are of rank 1, then there are *R*[*q*]-linear zigzags of quasi-isomorphisms

### Proof

It suffices to prove the first statement, the second following immediately by décalage. We have \((\mathrm {Spec}\,A\otimes _RA')_{\mathrm {strat}}^q(B)= (\mathrm {Spec}\,A)_{\mathrm {strat}}^q(B)\times (\mathrm {Spec}\,A')_{\mathrm {strat}}^q(B)\), and similarly for the simplicial functor \(\widetilde{(\mathrm {Spec}\,A\otimes _RA')}_{\mathrm {strat}}^q\) of Definition 1.12. Since coproduct of flat \(\Lambda \)-rings over *R*[*q*] is given by \(\otimes _{R[q]}\), it follows from Lemma 1.14 and Proposition 1.15 that \(\mathrm {qDR}(R[x_1, \ldots ,x_n]/R)\) can be calculated as the Dold–Kan normalisation of \((U^{\bullet })^{\otimes _{R[q]}n}\) (given by the *n*-fold tensor product \( (U^m)^{\otimes _{R[q]}n}\) in cosimplicial level *m*), for the cosimplicial module \(U^{\bullet }\) 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 \({\tilde{\Omega }}^{\bullet }(U^m)\) 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 \({\tilde{\Omega }}^{\bullet }(U^{m-1})\hookrightarrow {\tilde{\Omega }}^{\bullet }(U^m)\) is a quasi-isomorphism, since for \(\omega \in {\tilde{\Omega }}^{\bullet }(U^{m-1})\), we have

for \(k \ge 1\), allowing us to define a contracting homotopy

Since contracting homotopies interact well with tensor products, it also follows that the inclusion \({\tilde{\Omega }}^{\bullet }(U^{m-1})^{\otimes _{R[q]}n}\hookrightarrow {\tilde{\Omega }}^{\bullet }(U^m)^{\otimes _{R[q]}n}\) is a quasi-isomorphism. By induction on *m* we deduce that the inclusions \({\tilde{\Omega }}^{\bullet }(U^0)^{\otimes _{R[q]}n}\hookrightarrow {\tilde{\Omega }}^{\bullet }(U^m)^{\otimes _{R[q]}n}\), and hence their retractions given by diagonals \(U^m \rightarrow U^0\), are quasi-isomorphisms. These combine to give a quasi-isomorphism

on total complexes of normalisations.

Now, the cosimplicial module \({\tilde{\Omega }}^r(U^{\bullet })\) is given by the cosimplicial (i.e. levelwise) tensor product of \(U^{\bullet }\) with the cosimplicial \(\mathbb {Z}\)-module

with operations induced by those in Proposition 1.15. For \(r>0\), this cosimplicial \(\mathbb {Z}\)-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 \({\tilde{\Omega }}^r(U^{\bullet })\) and its tensor powers are all acyclic for \(r>0\).

The brutal truncation maps

are therefore quasi-isomorphisms of flat cochain complexes over *R*[*q*], so

and we just observe that \({\tilde{\Omega }}^{\bullet }(R[x]) =(\Omega ^*_{R[x]/R}[q], (q-1)\nabla _q)\). \(\square \)

### Remark 1.18

Note that Theorem 1.17 and Remark 1.10 together imply that \(q\text{- }\Omega ^{\bullet }_{R[x_1, \ldots ,x_n]/R}\) naturally underlies the décalage of a cosimplicial \(\Lambda \)-ring over *R*[*q*]. Even the underlying cosimplicial commutative ring structure carries more information than an \(E_{\infty }\)-structure when \(\mathbb {Q}\nsubseteq R\).

### 2.3 Completed *q*-cohomology

### Definition 1.19

Given a morphism \(R \rightarrow A\) of \(\Lambda \)-rings, we define the category \({\hat{\mathrm {Strat}}}^q_{A/R} \subset \mathrm {Strat}^q_{A/R}\) to consist of those objects which are \((q-1)\)-adically complete.

Equivalently, \({\hat{\mathrm {Strat}}}^q_{A/R}\) is the Grothendieck construction of the functor

on the category of flat \((q-1)\)-adically complete \(\Lambda \)-rings over *R*[*q*].

### Definition 1.20

Given a flat morphism \(R \rightarrow A\) of \(\Lambda \)-rings, define \({\widehat{\mathrm {qDR}}}(A/R)\) to be the cochain complex of \(R\llbracket q-1 \rrbracket \)-modules given by taking the homotopy limit of the functor

The following is immediate:

### Lemma 1.21

Given a flat morphism \(R \rightarrow A\) of \(\Lambda \)-rings, the complex \({\widehat{\mathrm {qDR}}}(A/R)\) is the derived \((q-1)\)-adic completion of \(\mathrm {qDR}(A/R)\).

### Definition 1.22

As in [13, §3], given a formally étale map , define to be the complex

where \(\nabla _q\) is defined as follows. First note that the \(R\llbracket q-1\rrbracket \)-linear ring endomorphisms \(\gamma _i\) of \(R[x_1, \ldots ,x_d]\llbracket q-1\rrbracket \) given by \(\gamma _i(x_j)= q^{\delta _{ij}}x_j\) extend uniquely to endomorphisms of \(A\llbracket q-1\rrbracket \) which are the identity modulo \((q-1)\), then set

Note that \(\widehat{q\text{- }\Omega }^{\bullet }_{ R[x_1, \ldots ,x_d]/R}\) is just the \((q-1)\)-adic completion of \( q\text{- }\Omega ^{\bullet }_{ R[x_1, \ldots ,x_d]/R}\).

### Theorem 1.23

If *R* is a flat \(\Lambda \)-ring over \(\mathbb {Z}\) and is a formally étale map of \(\Lambda \)-rings, the elements \(x_i\) having rank 1, then there are zigzags of \(R\llbracket q\rrbracket \)-linear quasi-isomorphisms

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 \(\Lambda \)-ring *B* over *R*, we then have the same property for \(\Lambda \)-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 \(\Lambda \)-ring homomorphism (*R* being flat over \(\mathbb {Z}\)). Similarly (taking \(B= A\llbracket q-1 \rrbracket \)) uniqueness of lifts ensures that the operations \(\gamma _i\) are \(\Lambda \)-ring endomorphisms of \(A\llbracket q-1 \rrbracket \).

We can now proceed as in the proof of Theorem 1.17. The complex \({\widehat{\mathrm {qDR}}}( A/R)\) can be realised as the cochain complex underlying a cosimplicial \(\Lambda \)-ring \({\hat{U}}(A)\), representing the functor \({\tilde{X}}_{\mathrm {strat}}^q\) of Definition 1.12 for \(X = \mathrm {Spec}\,A\), restricted to \((q-1)\)-adically complete \(\Lambda \)-rings *B*. By the consequences of formal étaleness, we have

giving \(({\tilde{X}}_{\mathrm {strat}}^q)_n \cong \mathrm {Hom}_{\Lambda ,R}(A,B)\times _{\mathrm {Hom}_{\Lambda ,R}( R[x_1, \ldots ,x_d],B)} ({\tilde{Y}}_{\mathrm {strat}}^q)_n \) for each *n*, where \(Y= \mathrm {Spec}\,R[x_1, \ldots ,x_d]\) and the fibre product is given via the projection of \(({\tilde{Y}}_{\mathrm {strat}}^q)_n\) onto the first factor.

In particular, this means that \({\hat{U}}(A)^n\) is the \((q-1)\)-adic completion of

where each \(U(R[x_i])\) is a copy of the cosimplicial ring *U* from Proposition 1.15. This isomorphism respects the cosimplicial operations; note that \(\partial ^0\) 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 \({\tilde{\Omega }}^{\bullet }({\hat{U}}(A))\) by setting \({\tilde{\Omega }}^{\bullet }({\hat{U}}(A)^n) \) to be the \((q-1)\)-adic completion of

where each \({\tilde{\Omega }}^{\bullet }(U(R[x_i]))\) is a copy of the complex \({\tilde{\Omega }}^{\bullet }(U^n)\) from the proof of Theorem 1.17. Compatibility of this construction with the cosimplicial operations follows because the \(\gamma _i\) are \(\Lambda \)-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 \(\nabla _q\) with *d* throughout, removing any dependence on co-ordinates. \(\square \)

As in [13, Definition 7.3], there is a notion of *q*-connection \(\nabla _q= (\nabla _{1,q}, \ldots , \nabla _{d,q})\) on a finite projective \(A\llbracket q-1 \rrbracket \)-module *M*, in the form of commuting \(R\llbracket q-1 \rrbracket \)-linear operators \(\nabla _{i,q}\) on *M*, with each \(\nabla _{i,q}\) satisfying \(\nabla _{i,q}(av)= \nabla _{q,x_i}(a)v + \gamma _i(a) \nabla _{i,q}(v)\) for \(a \in A, v \in M\).

### Definition 1.24

Given a flat morphism \(R \rightarrow A\) of \(\Lambda \)-rings with \(X:=\mathrm {Spec}\,A\), denote the forgetful functor \((B,f) \mapsto B\) from \({\hat{\mathrm {Strat}}}^q_{A/R}\) to rings by \(\mathscr {O}_{{\hat{X}}^q,\mathrm {strat}}\).

There is then a notion of \(\mathscr {O}_{{\hat{X}}^q,\mathrm {strat}}\)-modules in the category of functors from \({\hat{\mathrm {Strat}}}^q_{A/R}\) to abelian groups; we will simply refer to these as \(\mathscr {O}_{{\hat{X}}^q,\mathrm {strat}}\)-modules. Given a property *P* of modules, we will say that an \(\mathscr {O}_{{\hat{X}}^q,\mathrm {strat}}\)-module \(\mathscr {F}\) has the property *P* if for each \((B,f) \in {\hat{\mathrm {Strat}}}^q_{A/R}\), the *B*-module \(\mathscr {F}(B,f)\) has property *P*.

We say that an \(\mathscr {O}_{{\hat{X}}^q,\mathrm {strat}}\)-module \(\mathscr {F}\) is Cartesian if for each morphism \((B,f) \rightarrow (B',f')\) in \({\hat{\mathrm {Strat}}}^q_{A/R}\), the map \(\mathscr {F}(B,f)\otimes _BB' \rightarrow \mathscr {F}(B',f')\) is an isomorphism.

Given an \(\mathscr {O}_{{\hat{X}}^q,\mathrm {strat}}\)-module \(\mathscr {F}\), we define \(\Gamma ({\hat{X}}^q_{\mathrm {strat}},\mathscr {F}):= \varprojlim _{{\hat{\mathrm {Strat}}}^q_{A/R} }\mathscr {F}\).

In [13, Conjecture 7.5], Scholze predicted that the category of *q*-connections on finite projective \(A\llbracket q-1 \rrbracket \)-module is independent of co-ordinates on *A*. The following proposition gives the weaker statement that the category depends only on the \(\Lambda \)-ring structure on *A*.

### Proposition 1.25

Under the conditions of Theorem 1.23, with \(X:=\mathrm {Spec}\,A\), the category of finite projective \(A\llbracket q-1 \rrbracket \)-modules \((M,\nabla )\) with *q*-connection is equivalent to the category of those finite projective \(\mathscr {O}_{{\hat{X}}^q,\mathrm {strat}}\)-modules \(\mathscr {N}\) for which the map

is an isomorphism.

### Proof

The restriction on \(\mathscr {N}/(q-1)\) ensures that it is Cartesian; this also implies that \(\mathscr {N}\) is Cartesian, because finite projective modules are flat and \((q-1)\)-adically complete.

Now, the cosimplicial \(\Lambda \)-ring \({\hat{U}}(A)\) realising \({\widehat{\mathrm {qDR}}}( A/R)\) in the proof of Theorem 1.23 admits a natural map \(A \rightarrow {\hat{U}}(A)/(q-1)\) from the constant cosimplicial diagram. Thus \({\hat{U}}(A)\) defines a cosimplicial diagram in \({\hat{\mathrm {Strat}}}^q_{A/R}\). Since the functor \({\tilde{X}}_{\mathrm {strat}}^q\) of Definition 1.12 resolves \(X_{\mathrm {strat}}^q\), it follows that the functor \({\hat{U}}(A) : \Delta \rightarrow {\hat{\mathrm {Strat}}}^q_{A/R}\) from the simplex category is initial in the sense of [11, §IX.3].

In particular, this means that the category of Cartesian \(\mathscr {O}_{{\hat{X}}^q,\mathrm {strat}}\)-modules \(\mathscr {N}\) is equivalent to the category of Cartesian cosimplicial \({\hat{U}}(A)\)-modules *N*, where the Cartesian condition amounts to saying that the maps \(N^m\otimes _{{\hat{U}}(A)^m, \partial ^i}{\hat{U}}(A)^{m+1} \rightarrow N^{m+1}\) are all isomorphisms. Setting \(M=N^0\), Cartesian \({\hat{U}}(A)\)-modules are equivalent to \({\hat{U}}(A)^0=A\llbracket q-1 \rrbracket \)-modules *M* with isomorphisms \(\Delta : (\partial ^1)^*M \cong (\partial ^0)^*M\) satisfying the cocycle condition \(\partial ^1\Delta = (\partial ^0\Delta ) \circ (\partial ^2\Delta ) : (\partial ^2\partial ^0)^*M \rightarrow (\partial ^0\partial ^0)^*M\).

The map \(\Delta \) is determined by its restriction to *M*, so using the basis for \(U^1\) from Lemma 1.5, and taking \(v \in M\), we have

for \(R\llbracket q-1 \rrbracket \)-linear endomorphisms \(\Delta _{\underline{k}}\) of *M*. Since \(\lambda _t(a+b)= \lambda _t(a)\lambda _t(b)\), the cocycle condition becomes \(\Delta _{\underline{j}+\underline{k}}= \Delta _{\underline{j}}\circ \Delta _{\underline{k}}\), meaning \(\Delta \) is determined by the operators \(\Delta _{e_i}\) at the basis vectors, which must moreover commute.

Linearity of \(\Delta \) with respect to \({\hat{U}}(A)^1\) then reduces to the condition that \(\Delta ( av) = \partial ^1(a)\Delta (v)\) for \(a \in A\), \(v \in M\). Writing *A* for \(\partial ^0A\) and \(h_i^{[k]}:= \lambda ^{k}(\tfrac{\partial ^1x_i-\partial ^0x_i}{q-1})\), the ideal \(J:=(h_i^{[\ge 2]},h_ih_j)_{i \ne j}\) satisfies \(U^1 = A \oplus \bigoplus _i A h_i \oplus J\). The proof of Theorem 1.23 gives \(\partial ^1(a) \equiv a+ (q-1)\sum _i \nabla _{q,x_i}(a)h_i \mod J\), and in \(U^1/J\) we have \([h_i]^2 \equiv x_i[h_i]\). Comparing coefficients of \(h_i\) in the equation \(\Delta ( av) \equiv \partial ^1(a)\Delta (v) \mod J\) then gives

Finally, note that the condition that \(\mathscr {N}/(q-1)\) be the pullback of an *A*-module (necessarily \(\Gamma ({\hat{X}}^q_{\mathrm {strat}},\mathscr {N}/(q-1)) \)) is equivalent to saying that \(\partial ^0_N \equiv \partial ^1_N \mod (q-1)\), or that \((q-1)\) divides \( \Delta _{\underline{k}}\) whenever \(\underline{k}\ne 0\). In particular, \((q-1)\) divides \(\Delta _{e_i}\), and setting \(\nabla _{i,q}:= (q-1)^{-1}\Delta _{e_i}\) gives a *q*-connection \((\nabla _{i,q})_{1\le i\le d}\) on \(M=N^0\) uniquely determining \(\Delta \).

The inverse construction is given by \( \Delta _{\underline{k}}= (q-1)^{\sum k_i} \nabla _{1,q}^{k_1}\circ \cdots \circ \nabla _{d,q}^{k_d}. \)\(\square \)

## 3 Comparisons for \(\Lambda _P\)-rings

Since very few étale maps \(R[x_1, \ldots ,x_d]\rightarrow A\) give rise to \(\Lambda \)-ring structures on *A*, Theorem 1.23 is fairly limited in its scope for applications. We now show how the construction of \({\widehat{\mathrm {qDR}}}\) and the comparison quasi-isomorphism survive when we weaken the \(\Lambda \)-ring structure by discarding Adams operations at invertible primes.

### 3.1
*q*-cohomology for \(\Lambda _P\)-rings

Our earlier constructions for \(\Lambda \)-rings all carry over to \(\Lambda _P\)-rings, as follows.

### Definition 2.1

Given a set *P* of primes, we define a \(\Lambda _P\)-ring *A* to be a \(\Lambda _{\mathbb {Z},P}\)-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 \(\mathbb {Z}\), giving a \(\Lambda _P\)-ring structure on *A* is equivalent to giving commuting Adams operations \(\Psi ^p\) for all \(p \in P\), with \(\Psi ^p(a) \equiv a^p \mod p\) for all *a*.

Thus when *P* is the set of all primes, a \(\Lambda _P\)-ring is just a \(\Lambda \)-ring; a \(\Lambda _{\emptyset }\)-ring is just a commutative ring; for a single prime *p*, we write \(\Lambda _p:= \Lambda _{\{p\}}\), and note that a \(\Lambda _p\)-ring is a \(\delta \)-ring in the sense of [10].

### Definition 2.2

Given a \(\Lambda _P\)-ring *R*, say that *A* is a \(\Lambda _P\)-ring over *R* if it is a \(\Lambda _P\)-ring equipped with a morphism \(R \rightarrow A\) of \(\Lambda _P\)-rings. We say that *A* is a flat \(\Lambda _P\)-ring over *R* if *A* is flat as a module over the commutative ring underlying *R*.

### Definition 2.3

Given a morphism \(R \rightarrow A\) of \(\Lambda _P\)-rings, we define the category \(\mathrm {Strat}^{q,P}_{A/R}\) to consist of flat \(\Lambda _P\)-rings *B* over *R*[*q*] equipped with a compatible morphism \(A \rightarrow B/(q-1)\), such that the map \(A \rightarrow B/(q-1)\) admits a lift to *B*. We define the category \({\hat{\mathrm {Strat}}}^{q,P}_{A/R} \subset \mathrm {Strat}^q_{A/R}\) to consist of those objects which are \((q-1)\)-adically complete.

More concisely, \(\mathrm {Strat}^{q,P}_{A/R}\) (resp. \({\hat{\mathrm {Strat}}}^{q,P}_{A/R}\)) is the Grothendieck construction of the functor \((\mathrm {Spec}\,A)_{\mathrm {strat}}^{q,P}\) (resp. \(\widehat{(\mathrm {Spec}\,A)}_{\mathrm {strat}}^{q,P}\)) given by

on the category of flat \(\Lambda _P\)-rings (resp. \((q-1)\)-adically complete flat \(\Lambda _P\)-rings) over *R*[*q*].

### Definition 2.4

Given a flat morphism \(R \rightarrow A\) of \(\Lambda _P\)-rings, define \(\mathrm {qDR}_P(A/R)\) to be the cochain complex of *R*[*q*]-modules given by taking the homotopy limit of the functor

Define \({\widehat{\mathrm {qDR}}}_P(A/R)\) to be the cochain complex of \(R\llbracket q-1 \rrbracket \)-modules given by the corresponding homotopy limit over \({\hat{\mathrm {Strat}}}^{q,P}_{A/R}\).

For \(p \in P\), the cochain complex \(\mathrm {qDR}_P(A/R)\) naturally carries \((R[q], \Psi ^p)\)-semilinear operations \(\Psi ^p\) coming from the morphisms \(\Psi ^p : B\otimes _{R[q], \Psi ^p}R[q] \rightarrow B\) of *R*[*q*]-modules, for \(B \in \mathrm {Strat}^{q,P}_{A/R}\).

Thus when *P* is the set of all primes, we have \(\mathrm {qDR}_P(A/R)= \mathrm {qDR}(A/R)\). At the other extreme, for *A* smooth, \({\widehat{\mathrm {qDR}}}_{\emptyset }(A/R)\) is the Rees construction of the Hodge filtration on the infinitesimal cohomology complex [8] of *A* over *R*, with formal variable \((q-1)\). In more detail, there is a decreasing filtration *F* of \(\mathscr {O}_{\inf }\) given by powers of the augmentation ideal of \(\mathscr {O}_{\inf } \rightarrow \mathscr {O}_{\mathrm {Zar}}\) (with \(F^{\nu }\mathscr {O}_{\inf }=\mathscr {O}_{\inf }\) for \(\nu \le 0\)), and then

### Lemma 2.5

For a set *P* of primes, the forgetful functor from \(\Lambda \)-rings to \(\Lambda _P\)-rings has a right adjoint \(W^{(\notin P)}\). 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 \(\Lambda \)-rings and \(\Lambda _P\)-rings. The ghost component morphism is given by taking the Adams operations \(\Psi ^n\) coming from the \(\Lambda \)-ring structure on \(W^{(\notin P)}(B)\), followed by projection to *B*. When *P* contains all the residue characteristics of *B*, a \(\Lambda \)-ring structure is the same as a \(\Lambda _P\)-ring structure with compatible commuting Adams operations for all primes not in *P*, leading to the description above. \(\square \)

Note that the big Witt vector functor *W* on commutative rings thus factorises as \(W= W^{(\notin P)} \circ W^{(P)}\), for \(W^{(P)}\) the *P*-typical Witt vectors.

### Proposition 2.6

Given a morphism \(R \rightarrow A\) of \(\Lambda \)-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

on the category of flat \(\Lambda _P\)-rings over *R*[*q*]. There is an obvious map

and hence a natural transformation \((\mathrm {Spec}\,A)_{\mathrm {strat}}^q\circ W^{(\notin P)} \rightarrow (\mathrm {Spec}\,A)_{\mathrm {strat}}^{q,P}\), which induces the morphism \(\mathrm {qDR}_P(A/R) \rightarrow \mathrm {qDR}(A/R)\) on cohomology.

When *P* contains all the residue characteristics of *A*, the map \( (W^{(\notin P)} B)/(q-1) \rightarrow W^{(\notin P)}(B/(q-1))\) is just

since the morphism \(R[q] \rightarrow W^{(\notin P)} B\) is given by Adams operations, with \(\Psi ^n(q-1)= q^n-1\).

We have \((q^n-1)=(q-1)[n]_q\), and \([n]_q\) is a unit in \(\mathbb {Z}[\tfrac{1}{n}]\llbracket q-1 \rrbracket \), hence a unit in *B* when *n* is coprime to the residue characteristics. Thus the map \( (W^{(\notin P)} B)/(q-1) \rightarrow W^{(\notin P)}(B/(q-1))\) gives an isomorphism whenever *B* is \((q-1)\)-adically complete and admits a map from *A*, so the transformation \((\mathrm {Spec}\,A)_{\mathrm {strat}}^q\circ W^{(\notin P)} \rightarrow (\mathrm {Spec}\,A)_{\mathrm {strat}}^{q,P}\) is a natural isomorphism on the category of flat \((q-1)\)-adically complete \(\Lambda _P\)-rings over *R*[*q*], and hence \( {\widehat{\mathrm {qDR}}}_P(A/R) \xrightarrow {\simeq } {\widehat{\mathrm {qDR}}}(A/R) \). \(\square \)

### Remark 2.7

Remark 1.10 shows that \(\mathrm {qDR}(A/R)\) can naturally be promoted to a cosimplicial \(\Lambda \)-ring, and the same reasoning promotes \(\mathrm {qDR}_P(A/R)\) to a cosimplicial \(\Lambda _P\)-ring. The proof of Proposition 2.6 then ensures that the map \(\mathrm {qDR}_P(A/R) \rightarrow \mathrm {qDR}(A/R)\) is naturally a morphism of cosimplicial \(\Lambda _P\)-rings,

Over \(\mathbb {Z}[\{\frac{1}{p}: p \in P\}]\), every \(\Lambda _P\)-ring can be canonically made into a \(\Lambda \)-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 \(\Lambda \)-ring structure will not satisfy the conditions of Theorem 1.23. We now give a more general result which does allow for meaningful comparisons.

### Theorem 2.8

If *R* is a flat \(\Lambda _P\)-ring over \(\mathbb {Z}\) and is a formally étale map of \(\Lambda _P\)-rings, the elements \(x_i\) having rank 1, then there are zigzags of \(R\llbracket q-1\rrbracket \)-linear quasi-isomorphisms

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 \(\Lambda _P\)-rings, because the Adams operations must also lift uniquely. In particular, this means that the operations \(\gamma _i\) featuring in the definition of *q*-de Rham cohomology are necessarily endomorphisms of *A* as a \(\Lambda _P\)-ring.

Similarly to Theorem 1.23, \({\widehat{\mathrm {qDR}}}_P( A/R)\) is calculated using a cosimplicial \(\Lambda _P\)-ring given in level *n* by the \((q-1)\)-adic completion \({\hat{U}}_{P,A}^{\bullet }\) of the \(\Lambda _P\)-ring over *R*[*q*] generated by \(A^{\otimes _R(n+1)}[q]\) and \((q-1)^{-1}\ker ( A^{\otimes _R(n+1)}\rightarrow A)[q]\). The observation above shows that \({\hat{U}}_{P,A}^n \cong {\hat{U}}_{P,R[x_1, \ldots ,x_d]}^n\hat{\otimes }_{ R[x_1, \ldots ,x_d]}A\), changing base along applied to the first factor.

As in Proposition 2.6, \({\hat{U}}_{P,R[x_1, \ldots ,x_d]}^{\bullet }\) is just the \((q-1)\)-adic completion of the complex \(U^{\bullet }\) from Proposition 1.15. Further application of the key observation above then allows us to adapt the constructions of Theorem 1.17, giving the desired quasi-isomorphisms.\(\square \)

### 3.2 Cartier isomorphisms in mixed characteristic

In [13, Conjecture 7.1], Scholze predicted that 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 is functorial invariant of the \(\Lambda _P\)-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 \(A=\varprojlim _n A_n\) is a formal deformation of a smooth *k*-algebra \(A_0\). Then any formally étale morphism \(W^{(p)}(k)[x_1, \ldots ,x_d]\rightarrow A\) of topological rings gives rise to a unique compatible lift \(\Psi ^p\) of absolute Frobenius on *A* with \(\Psi ^p(x_i)=x_i^p\), so gives *A* the structure of a topological \(\Lambda _p\)-ring. The framing still affects the choice of \(\Lambda _p\)-ring structure, but at least such a structure is guaranteed to exist, giving rise to a complex \(\mathrm {qDR}_P(A/R)^{\wedge _p}:= \mathbf R \varprojlim _n \mathrm {qDR}_p(A/R)\otimes _R^\mathbf{L }R_n\) depending only on the choice of \(\Psi ^p\), where \(R_n=W_n^{(p)}(k)\).

Our constructions now allow us to globalise the quasi-isomorphism

of [13, Proposition 3.4], where \(\Omega ^*_{A/R}\) denotes the complex \(A \xrightarrow {0} \Omega ^1_{A/R} \xrightarrow {0} \Omega ^2_{A/R} \xrightarrow {0}\ldots \).

### Lemma 2.9

Under the quasi-isomorphism \({\widehat{\mathrm {qDR}}}_p( A/R) \simeq (\Omega ^*_{A/R}\llbracket q-1 \rrbracket , (q-1)\nabla _q)\) from Theorem 2.8, the semilinear Adams operation \(\Psi ^{p}\) on \( {\widehat{\mathrm {qDR}}}_p( A/R) \) described in Definition 1.8 corresponds to the operation on \(\Omega ^*_{A/R}\llbracket q-1 \rrbracket \) given by setting

for \(a \in A\llbracket q-1 \rrbracket \).

### Proof

Just observe that this expression defines a chain map on \((\Omega ^*_{A/R}\llbracket q-1 \rrbracket , (q-1)\nabla _q)\) (for instance \(\Psi ^p((q-1)\nabla _qx_i)= (q^p-1)\Psi ^p(dx)= (q-1)\nabla _qx_i^p\)), and that the quasi-isomorphisms in the proof of Theorem 1.23 commute with these operations. \(\square \)

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 \(\mathfrak {X}\) over \(R=W^{(p)}(k)\) equipped with a lift \(\Psi ^p\) of Frobenius which étale locally admits co-ordinates \(\{x_i\}_i\) as above with \(\Psi ^p(x_i)=x_i^p\). Then there is a global quasi-isomorphism

in the derived category of étale sheaves on \(\mathfrak {X}\).

### Proof

The unique lifting property of formally étale morphisms ensures that each affine formal scheme \(\mathfrak {U}\) étale over \(\mathfrak {X}\) has a unique lift \(\Psi ^p|_{\mathfrak {U}}\) of Frobenius compatible with the given operation \(\Psi ^p\) on \(\mathfrak {X}\). Functoriality of the construction \(\mathrm {qDR}_p\) for rings with Frobenius lifts thus gives us an étale presheaf \({\widehat{\mathrm {qDR}}}_p(\mathscr {O}_{\mathfrak {X}}/R)^{\wedge _p}\) of complexes on \(\mathfrak {X}\). As in Definition 1.8, the Adams operation \(\Psi ^p\) on \(\mathscr {O}_{\mathfrak {X}}\) then extends to \((R\llbracket q-1 \rrbracket , \Psi ^p)\)-semilinear maps

and thus, denoting good truncation by \(\tau \),

the left-hand side is quasi-isomorphic to \(\bigoplus _{j \le i} (\Omega ^j_{\mathscr {O}_{\mathfrak {X}}/R})^{\wedge _p}[-j]\) by Theorem 1.23.

Extending the construction *R*[*q*]-linearly and restricting to top summands therefore gives us the global map \(C_q^{-1}\). For a local choice of framing, Lemma 2.9 gives equivalences

for Scholze’s locally defined lifts of the Cartier quasi-isomorphism. The local calculation of [13, Proposition 3.4] then ensures that \(C_q^{-1}\) is a quasi-isomorphism. \(\square \)

## 4 Functoriality via analogues of de Rham–Witt cohomology

In order to obtain a cohomology theory for smooth commutative rings rather than for \(\Lambda _P\)-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 \([n]_{q^{1/n}}= \frac{q-1}{q^{1/n}-1}\).

### 4.1 Motivation

### Definition 3.1

Given a \(\Lambda _P\)-ring *B*, define \(\Psi ^{1/P^{\infty }}B\) to be the smallest \(\Lambda _P\)-ring which is equipped with a morphism from *B* and for which the Adams operations are automorphisms.

In the case \(P= \{p\}\), the \(\Lambda _p\)-ring \(\Psi ^{1/p^{\infty }}B\) is thus the colimit of the diagram

By Remark 2.7, \({\widehat{\mathrm {qDR}}}_p( A/R)\) naturally underlies a cosimplicial \(\Lambda _p\)-ring, so applying \(\Psi ^{1/p^{\infty }}\) levelwise gives another cosimplicial \(\Lambda _p\)-ring. For the Adams operation \(\Psi ^p\) of Definition 2.3, the underlying cochain complex is just \(\Psi ^{1/p^{\infty }}{\widehat{\mathrm {qDR}}}_p( A/R):= \varinjlim _{\Psi ^p} {\widehat{\mathrm {qDR}}}_p( A/R)\). As an immediate consequence of Lemma 2.9, we have:

### Lemma 3.2

If *R* is a flat \(\Lambda _p\)-ring over \(\mathbb {Z}_{(p)}\) with \(\Psi ^p\) an isomorphism, then \(\Psi ^{1/p^{\infty }}{\widehat{qDR}}_p( R[x]/R)\) is quasi-isomorphic to the complex

so the décalage \(\mathbf L \eta _{(q-1)}\Psi ^{1/p^{\infty }}{\widehat{\mathrm {qDR}}}_p( R[x]/R)\) and the complex

are quasi-isomorphic after \((q-1)\)-adic completion.

Thus in level 0 (resp. level 1), \(\mathbf L \eta _{(q-1)}\Psi ^{1/p^{\infty }}{\widehat{\mathrm {qDR}}}( R[x]/R)\) 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 \(q^{1/p^{\infty }}=1\) gives a complex whose *p*-adic completion is the *p*-typical de Rham–Witt complex.

### Lemma 3.3

Let *R* and *A* be flat *p*-adically complete \(\Lambda _p\)-algebras over \(\mathbb {Z}_p\), with \(\Psi ^p\) an isomorphism on *R*. For elements \(x_i\) of rank 1, take a map of \(\Lambda _p\)-rings which is a flat *p*-adic deformation of an étale map. Then the map

is a quasi-isomorphism.

### Proof

The map \(\Psi ^p : A\otimes _{R[x_1, \ldots ,x_d]}R[x_1^{1/p}, \ldots ,x_d^{1/p}] \rightarrow A\) 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 \((\Psi ^{1/p^{\infty }}{\widehat{\mathrm {qDR}}}_p(A/R))^{\wedge _p}\) and the \((p,q-1)\)-adic completion of

where *I* ranges over finite subsets of \(\{1, \ldots , d\}\) and \(\alpha \) ranges over elements of \(p^{-\infty }\mathbb {Z}^d\) with \(0 \le \alpha _i <1\) if \(i \notin I\) and \(-1< \alpha _i \le 0\) if \(i \in I\).

We then observe that the contributions to the décalage \(\eta _{(q-1)}\) from terms with \(\alpha \ne 0\) must be acyclic, via a contracting homotopy defined by the restriction to \(\eta _{(q-1)}\) of the *q*-integration map

where \([\frac{m}{p^n}]_q^{-1}= [m]_{q^{1/p^n}}^{-1}[p^n]_{q^{1/p^n}}\) for *m* coprime to *p*, noting that \([m]_{q^{1/p^n}}\) is a unit in \(\mathbb {Z}[q^{1/p^{\infty }}]^{\wedge _{(p,q-1)}}\). \(\square \)

### Remark 3.4

The endomorphism given on \(\Psi ^{1/P^{\infty }}{\widehat{\mathrm {qDR}}}_P(A/R)\) by

descends to an endomorphism of \(\mathrm {H}^0(\Psi ^{1/P^{\infty }}{\widehat{\mathrm {qDR}}}_P(A/R)/(q-1))\), which we may denote by \(V_n\) because it mimics Verschiebung in the sense that \(\Psi ^{n}V_n= n\cdot \mathrm {id}\) (since \([n]_q \equiv n \mod (q-1)\)). For *A* smooth over \(\mathbb {Z}\), we then have

for \(\zeta _n\) a primitive *n*th root of unity.

By adjunction, this gives an injective map

of \(\Lambda _P\)-rings, which becomes an isomorphism on completing \(\Psi ^{1/P^{\infty }}{\widehat{\mathrm {qDR}}}(A/\mathbb {Z})\) with respect to the system \(\{([n]_{q^{1/n}})\}_{n\in P^{\infty }}\) of ideals, where we write \(P^{\infty }\) for the set of integers whose prime factors are all in *P*. This implies that the cokernel is annihilated by all elements of \((q^{1/P^{\infty }}-1)\), so leads us to consider almost mathematics as in [7].

### 4.2 Almost isomorphisms

From now on, we consider only the case \(P=\{p\}\). Combined with Lemma 3.3, Remark 3.4 allows us to regard \(\mathbf L \eta _{(q-1)}\Psi ^{1/p^{\infty }}{\widehat{\mathrm {qDR}}}_p(A/\mathbb {Z}_p)^{\wedge _p}\) as being almost a \(q^{1/p^{\infty }}\)-analogue of *p*-typical de Rham–Witt cohomology.

The ideal \((q^{1/p^{\infty }}-1)^{\wedge _{(p,q-1)}}= \ker (\mathbb {Z}[q^{1/p^{\infty }}]^{\wedge _{(p,q-1)}} \rightarrow \mathbb {Z}_p)\) is equal to the *p*-adic completion of its square, since we may write it as the kernel \(W^{(p)}(\mathfrak {m})\) of \(W^{(p)}(\mathbb {F}_p[q^{1/p^{\infty }}]^{\wedge _{(q-1)}})\rightarrow W^{(p)}(\mathbb {F}_p)\), for the idempotent maximal ideal \(\mathfrak {m}= ( (q-1)^{1/p^{\infty }})^{\wedge _{(q-1)}}\) in \(\mathbb {F}_p[q^{1/p^{\infty }}]^{\wedge _{(q-1)}}\). If we set \(h^{1/p^n}\) to be the Teichmüller element

then \(W^{(p)}(\mathfrak {m})=(h^{1/p^{\infty }})^{\wedge _{(p,h)}}\). Although \(W^{(p)}(\mathfrak {m})/p^n\) is not maximal in \(\mathbb {Z}[h^{1/p^{\infty }}]^{\wedge _{(h)}}/p^n\), it is idempotent and flat, so gives a basic setup in the sense of [7, 2.1.1]. We thus regard the pair \((\mathbb {Z}[q^{1/p^{\infty }}]^{\wedge _{(p,q-1)}},W{(p)}(\mathfrak {m}) )\) as an inverse system of basic setups for almost ring theory.

We then follow the terminology and notation of [7], studying *p*-adically complete \((\mathbb {Z}[q^{1/p^{\infty }}]^{\wedge _{(p,q-1)}})^a\)-modules (almost \(\mathbb {Z}[q^{1/p^{\infty }}]^{\wedge _{(p,q-1)}}\)-modules) given by localising at almost isomorphisms, the maps whose kernel and cokernel are \(W^{(p)}(\mathfrak {m})\)-torsion.

### Definition 3.5

The obvious functor \((-)^a\) from modules to almost modules has a right adjoint \((-)_*\), given by \(N_*:=\mathrm {Hom}_{\mathbb {Z}[q^{1/p^{\infty }}]^{\wedge _{(p,q-1)}}}(W^{(p)}(\mathfrak {m}) ,N)\), the module of almost elements.

Since the counit \((M_*)^a\rightarrow M\) 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 \(M \rightarrow (M^a)_*\) is an isomorphism. We can define *p*-adically complete \((\mathbb {Z}[q^{1/p^{\infty }}]^{\wedge _{(p,q-1)}})^a\)-algebras similarly, forming a full subcategory of \(\mathbb {Z}[q^{1/p^{\infty }}]^{\wedge _{(p,q-1)}}\)-algebras.

### 4.3 Perfectoid algebras

We now relate Scholze’s perfectoid algebras to a class of \(\Lambda _p\)-rings, by factorising the tilting equivalence. For simplicity, we work over \(\mathbb {Z}[\zeta _{p^{\infty }}]^{\wedge _p}\), although Lemma 3.8 has natural analogues over the ring \(K^o \subset K\) of power-bounded elements of any perfectoid field *K* in the sense of [12].

### Definition 3.6

Define Fontaine’s period ring functor \(\mathscr {A}_{\inf }\) from commutative rings to \(\Lambda _p\)-rings by \(\mathscr {A}_{\inf }(C):= \varprojlim _{\Psi ^p}W^{(p)} (C)\).

### Definition 3.7

Define a perfectoid \(\Lambda _p\)-ring to be a flat *p*-adically complete \(\Lambda _p\)-algebra over \(\mathbb {Z}_p\), on which the Adams operation \(\Psi ^p\) is an isomorphism.

By analogy with [2, Notation 1.4], we say that a perfectoid \(\Lambda _p\)-ring over \(\mathbb {Z}[q^{1/p^{\infty }}]^{\wedge _{(p,q-1)}}\) is integral if the morphism \( B \rightarrow B_* \) of Definition 3.5 is an isomorphism.

### Lemma 3.8

We have equivalences of categories

### Proof

A perfectoid \(\Lambda _p\)-ring *B* is a deformation of the perfect \(\mathbb {F}_p\)-algebra *B* / *p*. As in [12, Proposition 5.13], a perfect \(\mathbb {F}_p\)-algebra *C* has a unique deformation \(W^{(p)}(C)\) over \(\mathbb {Z}_p\), to which Frobenius must lift uniquely; this shows that \(W^{(p)}\) gives an equivalence between perfect \(\mathbb {F}_p\)-algebras and perfectoid \(\Lambda _p\)-rings. To obtain the bottom equivalence of the diagram, we will show that the functor \(W^{(p)}\) commutes with the respective functors \(C \mapsto C_*\) 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 \(C_*= \bigcap _n h^{-p^{-n}}C\) in each setting. For a Teichmüller element \([c] \in W^{(p)}(C)\), the standard isomorphism \(W^{(p)}(C)\cong C^{\mathbb {N}_0}\) of sets gives an isomorphism \([c]W^{(p)}(C)\cong \prod _{m \ge 0} c^{p^m}C\). Thus the natural map \( W^{(p)}(C)_* \rightarrow W^{(p)}(C_*)\) of \(\Lambda _p\)-rings is an isomorphism, since

and taking inverse limits with respect to \(\Psi ^p\) gives \(\mathscr {A}_{\inf }(C)_*\cong \mathscr {A}_{\inf }(C_*)\) as well.

Next, we observe that since \(B:=\mathscr {A}_{\inf } (C)\) is a perfectoid \(\Lambda _p\)-ring for any flat *p*-adically complete \(\mathbb {Z}_p\)-algebra *C*, we must have \(B \cong W^{(p)}(B/p)\). Comparing rank 1 elements then gives a monoid isomorphism \((B/p)\cong \varprojlim _{x \mapsto x^p} C\), 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. \(\square \)

### 4.4 Functoriality of *q*-de Rham cohomology

Since \((\Psi ^{1/p^{\infty }}{\widehat{\mathrm {qDR}}}_p(A/\mathbb {Z}_p))^{\wedge _p}\) is represented by a cosimplicial perfectoid \(\Lambda _p\)-ring over \(\mathbb {Z}[q^{1/p^{\infty }}]^{\wedge _{(p,q-1)}}\) for any flat \(\Lambda _p\)-ring *A* over \(\mathbb {Z}_p\), it corresponds under Lemma 3.8 to a cosimplicial perfectoid \((\mathbb {Z}[\zeta _{p^{\infty }}]^{\wedge _p})^a\)-algebra, representing the following functor:

### Lemma 3.9

For a perfectoid \((\mathbb {Z}[\zeta _{p^{\infty }}]^{\wedge _p})^a\)-algebra *C*, and a \(\Lambda _p\)-ring *A* over \(\mathbb {Z}_p\) with \(X=\mathrm {Spec}\,A\), there is a canonical isomorphism

for the ring \(C_*\) of almost elements.

### Proof

By definition, \(X_{\mathrm {strat}}^{q,p}(\mathscr {A}_{\inf } (C)_*)\) is the image of

Since right adjoints commute with limits and \(\mathscr {A}_{\inf }= \varprojlim _{\Psi ^p}W^{(p)}\), we may rewrite the first term as \(\varprojlim _{\Psi ^p} \mathrm {Hom}_{\Lambda _p}(A, W^{(p)} (C_*))= \varprojlim _{\Psi ^p} X(C_*)\).

Setting \(B:=\varprojlim _{\Psi ^p}W^{(p)} (C)_*\), observe that because \([p^n]_{q^{1/p^n}}(q^{1/p^n}-1)=(q-1)\), we have \(\bigcap _n [p^n]_{q^{1/p^n}}B=(q-1)B\), 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 \(\theta : B \rightarrow C_*\) has kernel \(([p]_{q^{1/p}})\), the map \(\theta \circ \Psi ^{p^{n-1}}\) has kernel \(([p]_{q^{1/p^n}})\), and so \(B \rightarrow W^{(p)}(C)_*\) has kernel \(\bigcap _n [p^n]_{q^{1/p^n}}B\). Thus

\(\square \)

In fact, the tilting equivalence gives \(\varprojlim _{\Psi ^p} X(C_*) \cong X(C^{\flat }_*)\), so the only dependence of \(X_{\mathrm {strat}}^{q,p}(\mathscr {A}_{\inf } (C)_*)\), and hence \(((\Psi ^{1/p^{\infty }}{\widehat{\mathrm {qDR}}}_p(A/\mathbb {Z}_p))^{\wedge _p})^a\), on the Frobenius lift \(\Psi ^p\) is in determining the image of \( X(C^{\flat }_*) \rightarrow X(C_*)\) as *C* varies.

Although the map \( X(C^{\flat }_*) \rightarrow X(C_*)\) is not surjective, it is almost so in a precise sense, which we now use to establish independence of \(\Psi ^p\), showing that, up to faithfully flat descent, \({\widehat{\mathrm {qDR}}}_p(A/\mathbb {Z}_p)^{\wedge _p}/[p]_{q^{1/p}}\) is the best possible perfectoid approximation to \(A[\zeta _{p^{\infty }}]^{\wedge _p}\).

### Definition 3.10

Given a functor *X* from \((\mathbb {Z}[\zeta _{p^{\infty }}]^{\wedge _p})^a \)-algebras to sets and a functor \(\mathscr {A}\) from perfectoid \((\mathbb {Z}[\zeta _{p^{\infty }}]^{\wedge _p})^a\)-algebras to abelian groups, we write

where \(\mathrm {Pfd}(S^a) \) denotes the category of perfectoid almost *S*-algebras, and \(\mathbf R \mathrm {Hom}_{[\mathcal {C},\mathrm {Set}]}(-,-)\) is as in Definition 1.9.

When *X* is representable by a \((\mathbb {Z}[\zeta _{p^{\infty }}]^{\wedge _p})^a \)-algebra *C*, we simply denote \( \mathbf R \Gamma _{\mathrm {Pfd}}(X,\mathscr {A})\) by \( \mathbf R \Gamma _{\mathrm {Pfd}}(C,\mathscr {A})\) — when *C* is perfectoid, this will just be \(\mathscr {A}(C)\).

Thus \( \mathbf R \Gamma _{\mathrm {Pfd}}(C,\mathscr {A})\) is the homotopy limit of the functor \(\mathscr {A}\) (regarded as taking values in cochain complexes) on the category of perfectoid \((\mathbb {Z}[\zeta _{p^{\infty }}]^{\wedge _p})^a\)-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 \(\mathscr {A}=\mathscr {A}_{\inf }\) is the complex \(A\Omega \) of [4, Definition 9.1].

### Theorem 3.11

If *R* is a *p*-adically complete \(\Lambda _p\)-ring over \(\mathbb {Z}_p\), and *A* a formal *R*-deformation of a smooth ring over (*R* / *p*), then the complex

of \((\Psi ^{1/p^{\infty }}R[q])^{\wedge _{(p,q-1)}}\)-modules is almost quasi-isomorphic to \((\Psi ^{1/p^{\infty }}{\widehat{\mathrm {qDR}}}_p(A/R))^{\wedge _p} \) for any \(\Lambda _p\)-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 \(\mathrm {qDR}_p\) that the cochain complex \(((\Psi ^{1/p^{\infty }}{\widehat{\mathrm {qDR}}}_p(A/R))^{\wedge _p} )^a \) is given by \(\mathbf R \mathrm {Hom}_{[f{\hat{\Lambda }}_p(R\llbracket q-1 \rrbracket ),\mathrm {Set}]}(X_{\mathrm {strat}}^{q,p}, ((\Psi ^{1/p^{\infty }}\mathscr {O})^{\wedge _p})^a)\) in the notation of Definition 1.9, where \(f{\hat{\Lambda }}_p(R\llbracket q-1 \rrbracket ) \) denotes the category of flat \((p,q-1)\)-adically complete \(\Lambda _p\)-algebras over \(R\llbracket q-1 \rrbracket \).

Now note that \(C \mapsto ((\Psi ^{1/p^{\infty }}C)^{\wedge _p} )_*\) is left adjoint to the inclusion functor \(i : \mathrm {Pfd}\Lambda _p(R\llbracket q-1 \rrbracket ) \rightarrow f{\hat{\Lambda }}_p(R\llbracket q-1 \rrbracket )\) from the category of integral perfectoid \(\Lambda _p\)-rings over \(\Psi ^{1/p^{\infty }}(R\llbracket q-1 \rrbracket )^{\wedge _p}\). Thus \(i^* : \mathrm {Ch}([f{\hat{\Lambda }}_p(R\llbracket q-1 \rrbracket ),\mathrm {Ab}]) \rightarrow \mathrm {Ch}([\mathrm {Pfd}\Lambda _p(R\llbracket q-1 \rrbracket ),\mathrm {Ab}])\) has exact right adjoint \(\mathscr {F}\mapsto ( \mathscr {F}\circ (\Psi ^{1/p^{\infty }})^{\wedge _p})_*)\). We therefore have

It thus follows that the cochain complex \(((\Psi ^{1/p^{\infty }}{\widehat{\mathrm {qDR}}}_p(A/R))^{\wedge _p} )^a \) is the homotopy limit of the functor \((B,x,y) \mapsto B^a\) on the category of triples (*B*, *x*, *y*) for integral perfectoid \(\Lambda _p\)-rings *B* over \(\mathbb {Z}[q^{1/p^{\infty }}]^{\wedge _{(p,q-1)}}\) and

where \(X=\mathrm {Spec}\,A\) and \(Y=\mathrm {Spec}\,R\).

By Lemma 3.8, such \(\Lambda _p\)-rings *B* are uniquely of the form \(\mathscr {A}_{\inf }(C_*)\) for \(C \in \mathrm {Pfd}((\mathbb {Z}_p[\zeta _{p^{\infty }}]^{\wedge _p})^a)\), so this homotopy limit becomes

Writing \(X^{\infty }(C):= \mathrm {Im\,}( \varprojlim _{\Psi ^p} X(C_*) \rightarrow X(C_*))\), Lemma 3.9 then combines with the description above to give

We now introduce a Grothendieck topology on the category \([ \mathrm {Pfd}_{(\mathbb {Z}[\zeta _{p^{\infty }}]^{\wedge _p})^a},\mathrm {Set}]\) by taking covering morphisms to be those maps \(C \rightarrow C'\) of perfectoid algebras which are almost faithfully flat modulo *p*. Since \(C^{\flat }=\varprojlim _{\Phi } (C/p)\), the functor \(\mathscr {A}_{\inf }\) satisfies descent with respect to these coverings, so the map

is a quasi-isomorphism, where \((-)^{\sharp }\) denotes sheafification.

In other words, the calculation of \(({\widehat{\mathrm {qDR}}}_p(A/R)^{\wedge _p})^a\) is not affected if we tweak the definition of \(X^{\infty }\) by taking the image sheaf instead of the image presheaf. We then have

where \(C \rightarrow C'\) runs over all covering morphisms.

Now, \(\varprojlim _{\Psi ^p} X\) is represented by the perfectoid algebra \((\Psi ^{1/p^{\infty }}A)^{\wedge _p}\), which is isomorphic to \(A[x_1^{1/p^{\infty }}, \ldots ,x_d^{1/p^{\infty }}]^{\wedge _p}\) 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 \(f : A \rightarrow C\), there exists a covering morphism \(C \rightarrow C_i\) such that \(f(x_i)\) has arbitrary *p*-power roots in \(C_i\). Setting \(C':= C_1\otimes _C \ldots \otimes _C C_d\), this means that the composite \(A \xrightarrow {f} C \rightarrow C'\) extends to a map \((\Psi ^{1/p^{\infty }}A)^{\wedge _p} \rightarrow C'\), so \(f \in (X^{\infty })^{\sharp }(C)\). We have thus shown that \((X^{\infty })^{\sharp }=X\), giving the required equivalence

Finally, compatibility of these equivalences with the \((\Psi ^{1/p^{\infty }}R[q])^{\wedge _{(p,q-1)}}\)-module structures is given by functoriality, multiplicativity and the identification \( (\Psi ^{1/p^{\infty }}R[q])^{\wedge _{(p,q-1)}} \simeq (\Psi ^{1/p^{\infty }}{\widehat{\mathrm {qDR}}}_p(R/R))^{\wedge _p}\). \(\square \)

### Remark 3.12

Corresponding to the cohomology theory \(((\Psi ^{1/p^{\infty }}{\widehat{\mathrm {qDR}}}_p(A/R))^{\wedge _p})^a\), it is natural to consider *q*-connections on finite projective modules *M* over

It follows from the proof of Proposition 1.25 that these are equivalent, for \(X=\mathrm {Spec}\,A\), to finite projective almost \((\Psi ^{1/p^{\infty }}\mathscr {O}_{{\hat{X}}^q,\mathrm {strat}})^{\wedge _p}\)-modules \(\mathscr {N}\) for which \(\mathscr {N}/(q-1)\) is the pullback of the almost \(\mathrm {H}^0((\Psi ^{1/p^{\infty }}{\widehat{\mathrm {qDR}}}_p(A/R))^{\wedge _p}/(q-1))\)-module \(\Gamma ({\hat{X}}^q_{\mathrm {strat}}, \mathscr {N}/(q-1))=:M_0\).

Up to almost isomorphism, these correspond via the proof of Theorem 3.11 to those finite projective \(\mathscr {A}_{\inf }\)-modules *N* on the site of integral perfectoid algebras *C* over \(A[\zeta _{p^{\infty }}]^{\wedge _p}\otimes _R \Psi ^{1/p^{\infty }}R\) for which there exists a \(W^{(p)}(A[\zeta _{p^{\infty }}]^{\wedge _p})\)-module \(M_0\) with \(W^{(p)}(C)\)-linear isomorphisms

functorial in *C*.

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 \((\sum _n [p^n]_{q^{1/p^n}} \Psi ^{1/p^n}A[ q^{1/p^{\infty }}-1])^{\wedge _p})\)-modules rather than \(A\llbracket q-1 \rrbracket \)-modules.

The following gives a slight partial refinement of [4, Theorem 1.17]:

### Corollary 3.13

If *R* is a *p*-adically complete \(\Lambda _p\)-ring over \(\mathbb {Z}_p\), and *A* a formal *R*-deformation of a smooth ring over (*R* / *p*), then the *q*-de Rham cohomology complex is, up to almost quasi-isomorphism, independent of a choice of co-ordinates . As such, it is naturally an invariant of the commutative *p*-adically complete \((\Psi ^{1/p^{\infty }}R)[\zeta _{p^{\infty }}]^{\wedge _p}\)-algebra \((A[\zeta _{p^{\infty }}]\otimes _R \Psi ^{1/p^{\infty }}R)^{\wedge _p}\).

### Proof

Since

Theorem 2.8 combines with Lemma 3.3 to give

and by Theorem 3.11, we know that this depends only on \((A[\zeta _{p^{\infty }}]\otimes _R \Psi ^{1/p^{\infty }}R)^{\wedge _p}\) up to almost quasi-isomorphism. \(\square \)

### Remark 3.14

The almost quasi-isomorphism in Corollary 3.13 should be a genuine quasi-isomorphism when we impose some conditions on the base ring *R*. By [4, Lemma 8.11], it would suffice to verify that \( \mathrm {H}^*((\Psi ^{1/p^{\infty }} (\Omega ^*_{A/R}\llbracket q-1 \rrbracket , (q-1)\nabla _q))^{\wedge _p})\) and its quotient by \((q-1)\) have no \((q^{1/p^{\infty }}-1)\)-torsion, which should follow for *R* smooth by an argument similar to [4, Proposition 8.9].

### Remark 3.15

(Eliminating roots of *q*) The key feature of the comparison results in this section is that, up to faithfully flat descent, the functor \(X_{\mathrm {strat}}^{q,p}\) does not depend on Adams operations when restricted to the category of integral perfectoid \(\Lambda _p\)-rings *B* over \(\mathbb {Z}[q]\), since the proof of Theorem 3.11 gives \((X_{\mathrm {strat}}^{q,p})^{\sharp }(B) \cong X(B/[p]_{q^{1/p}})\). We can extend the latter functor to more general \(\Lambda _p\)-rings over \(\mathbb {Z}[q]\) by setting

which does not depend on any Adams operations on *X*.

When \(\mathscr {O}_X\) has a \(\Lambda _p\)-ring structure, there is then a natural map \(\alpha : X_{\mathrm {strat}}^{q,p}\rightarrow X^{q,p}\) because \(\Psi ^p((q-1)B) \subset [p]_qB\). This induces a transformation

for \(X=\mathrm {Spec}\,A\). But for integral perfectoid \(\Lambda _p\)-rings *B*, we know that \(X^{q,p}(B)= (X_{\mathrm {strat}}^{q,p})^{\sharp }(B)\), so by adjunction, as in the proof of Theorem 3.11, \(\alpha ^*\) becomes an almost quasi-isomorphism on applying a form of completed stabilisation \(\Psi ^{1/p^{\infty }}(-)^{\wedge _p}\). Thus \(\mathrm {H}^*(X^{q,p},\mathscr {O})\) might be a candidate for the co-ordinate independent *q*-de Rham cohomology theory proposed in [13]. It naturally carries an Adams operation \(\Psi ^p\), which would correspond to the operation \(\phi _p\) of [13, Conjecture 6.1].

Any \(a \in A\) defines an element of \(\mathrm {H}^0(X^{q,p},\mathscr {O}/(\Psi ^p)^{-1}([p]_q\mathscr {O}) )\) so \(\Psi ^p(a) \in \mathrm {H}^0(X^{q,p},\mathscr {O}/[p]_q)\) and applying the connecting homomorphism associated to \([p]_q : \mathscr {O}\rightarrow \mathscr {O}\) gives an element \(\beta _{[p]_q} \Psi ^p(a) \in \mathrm {H}^1(X^{q,p},\mathscr {O})\) whose image under \(\mathrm {H}^1(\alpha ^*)\) is

Moreover, to \(a \in A\) we may associate elements \(a_n \in \mathrm {H}^0(X^{q,p}, \mathscr {O}/[p^{n}]_q)\) for \(n \ge 1\), determined by the property that \(a_n \equiv \Psi ^{p^i}a^{p^{n-i}} \mod [p]_{q^{p^{i-1}}}\) for \(1 \le i \le n\), and these give rise to elements \(\beta _{[p^{n}]_q} a_n \in \mathrm {H}^1(X^{q,p},\mathscr {O})\). Explicitly, if we define operations \(\varepsilon _i\) on \(\mathscr {O}\) by \(\varepsilon _0=\mathrm {id}\) and \(\varepsilon _{i+1}(a):= (a^{p^{i+1}}- \Psi ^p(a^{p^{i}}))/p^{i+1} \), then for a local lift \({\tilde{a}} \in \mathscr {O}\) of \(a \in \mathscr {O}/(\Psi ^p)^{-1}([p]_q)\), we have

so

In particular, for \(A=R[x]\) these include all the elements \( (q-1) [m]_{q^{p^{n}}}x^{p^{n} m -1}dx \), since \(\varepsilon _i(x^m)=0\) for all \(i>0\), \(x^m\) having rank 1. This suggests that in general the image of \(\mathrm {H}^1(\alpha ^*)\) might be \((q-1)\mathrm {H}^1{\widehat{\mathrm {qDR}}}_p(A/R)^{\wedge _p}\), 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 are all natural in *A*.

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Pridham, J.P. On *q*-de Rham cohomology via \(\Lambda \)-rings.
*Math. Ann.* **375**, 425–452 (2019). https://doi.org/10.1007/s00208-019-01806-7

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DOI: https://doi.org/10.1007/s00208-019-01806-7