ON q-DE RHAM COHOMOLOGY VIA Λ-RINGS

J.P.PRIDHAM

Abstract. We show that Aomoto’s q-deformation of de Rham cohomology arises as a natural cohomology theory for Λ-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.

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 \frac{q^{n}−1}{q−1} of the integer n. In [Sch2], 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.

We show that q-de Rham cohomology naturally arises as a functorial invariant of Λ-rings (Theorem 1.11), and that its (q − 1)-adic completion depends only on a Λ_{P} -ring structure (Theorem 2.7), for P the set of residue characteristics; a Λ_{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 complete q-de Rham cohomology depends only on a lift Ψ^{p} of absolute Frobenius locally generated by co-ordinates with Ψ^{p}(x_{i}) = x_{i}^{p}. Given such data, we construct (Proposition 2.8) a quasi-isomorphism between Hodge cohomology and q-de Rham cohomology modulo [p]_{q}, extending the local lift of the Cartier isomorphism in [Sch2, 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 3.10) that up to (q^{1/p^{∞}} − 1)-torsion, the p-adic completion of this complex depends only on the padic completion of A[ζ_{p^{∞}}] (where ζ_{n} denotes a primitive nth 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[ζ_{p^{∞}}]. As a consequence, this shows (Corollary 3.12) 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 [BMS] between q-de Rham cohomology their theory AΩ^{•}.

The cohomology theories we construct 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 [Sch2]. Some of the strongest evidence for the conjectures is provided by the lifts of the Cartier isomorphisms, which rely on a choice of Frobenius. On the other hand, the comparison theorems of [BMS] can be seen as a manifestation of q-de Rham–Witt complexes; although they do not require a lift of Frobenius, they involve all p-power roots of q.

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 Λ-ring structure, and to look at flat Λ-rings B over R[q] equipped with morphisms A → B/(q − 1) of Λ-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 §3, the key to giving a characterisation independent of lifts of Frobenius is the factorisation of the tilting equivalence for perfectoid algebra via a category of Λ_{p}-rings.

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.

Contents

Introduction 1 1. Comparisons for Λ-rings 2 1.1. The Λ-ring ℤ[q] 3 1.2. q-cohomology of Λ-rings 4 1.3. Completed q-cohomology 6 2. Comparisons for Λ_{P} -rings 8 2.1. q-cohomology for Λ_{P} -rings 8 2.2. Cartier isomorphisms in mixed characteristic 10 3. Functoriality via analogues of de Rham–Witt cohomology 11 3.1. Motivation 12 3.2. Almost isomorphisms 13 3.3. Perfectoid algebras 14 3.4. Functoriality of q-de Rham cohomology 14 References 17

1. Comparisons for Λ-rings

We will follow standard notational conventions for Λ-rings. These are commutative rings equipped with operations λ^{i} resembling alternating powers, in particular satisfying λ^{k}(a+b) = ∑_{i=0}^{k} λ^{i}(a)λ^{k−i}(b), with λ^{0}(a) = 1 and λ^{1}(a) = a. For background, see [Bor] and references therein. The Λ-rings we encounter are all torsion-free, in which case the Λ-ring structure is equivalent to giving ring endomorphisms Ψ^{n} for n ∈ ℤ_{>0} with Ψ^{mn} = Ψ^{m} ∘ Ψ^{n} and Ψ^{p}(x) ≡ x^{p} mod p for all primes p. If we write λ_{t}(f) := ∑_{i≥0} λ^{i}(f)t^{i} and Ψ_{t}(f) := ∑_{n≥1} Ψ^{n}(f)t^{n}, then the families of operations are related by the formula Ψ_{t} = −t\frac{d log λ_{−t}}{dt}.

We refer to elements x with λ^{i}(x) = 0 for all i > 1 (or equivalently Ψ^{n}(x) = x^{n} for all n) as elements of rank 1.

1.1. The Λ-ring ℤ[q].

Definition 1.1. Define ℤ[q] to be the Λ-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} ∈ ℤ[q] of the integers, with [n]_{q}! =

[n]_{q}[n − 1]_{q} … [1]_{q}, and \binom{n}{k}_{q} = \frac{[n]_{q}!}{[n−k]_{q}![k]_{q}!}.

Remark 1.2. To see the importance of regarding ℤ[q] as a Λ-ring observe that the binomial expressions

λ^{k}(n) = \binom{n}{k}, λ^{k}(−n) = (−1)^{k}\binom{n+k−1}{k}

have as q-analogues the Gaussian binomial theorems

λ^{k}([n]_{q}) = q^{k(k−1)/2}\binom{n}{k}_{q}, λ^{k}(−[n]_{q}) = (−1)^{k}\binom{n+k−1}{k}_{q},

as well as Adams operations

Ψ^{i}([n]_{q}) = [n]_{q^{i}}.

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

Lemma 1.3. For the Λ-ring structure on ℤ[x, y] with x, y of rank 1, the elements

λ^{n}(\frac{y−x}{q−1}) ∈ ℤ[q, \{(q^{n} − 1)^{−1}\}_{n≥1}, x, y]

are given by

λ^{k}(\frac{y−x}{q−1}) = \frac{(y − x)(y − qx) … (y − q^{k−1}x)}{(q − 1)^{k}[k]_{q}!},

= ∑_{j=0}^{k} \frac{q^{j(j−1)/2}(−x)^{j}y^{k−j}}{[j]_{q}![k − j]_{q}!} .

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}(\frac{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

λ^{k}(\frac{q^{n}x−x}{q−1}) = λ^{k}([n]_{q}x) = q^{k(k−1)/2}\binom{n}{k}_{q}x^{k}.

Thus λ^{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. □

Remark 1.4. Note that as q → 1, Lemma 1.3 gives (q−1)^{k}λ^{k}(\frac{y−x}{q−1}) → \frac{(x−y)^{k}}{k!} . Indeed, for any rank 1 element x in a Λ-ring we have λ_{(q−1)t}(\frac{x}{q−1}) = e_{q}(xt), the q-exponential, with multiplicativity and universality then implying that λ_{(q−1)t}(\frac{a}{q−1}) is a q-deformation of exp(at) for all a. Thus (q −1)^{k}λ^{k}(\frac{a}{q−1}) is a q-analogue of the kth divided power (a^{k}/k!). An explicit expression comes recursively from the formula

[k]_{q}(q − 1)λ^{k}(\frac{a}{q−1}) = ∑_{i>0} λ^{i}(a)λ^{k−i}(\frac{a}{q−1}).

Lemma 1.5. For elements x, y of rank 1, the Λ-subring of ℤ[q, {(q^{n} − 1)^{−1}}_{n≥1}, x, y] generated by q, x, y, \frac{y−x}{q−1} has basis λ^{k}(\frac{y−x}{q−1}) as a ℤ[q, x]-module.

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

By Lemma 1.3, we know that

λ^{i}(\frac{y−x}{q−1})λ^{j}(\frac{y−q^{i}x}{q−1}) = \binom{i+j}{i}_{q} λ^{i+j}(\frac{y−x}{q−1}).

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

λ^{i}(\frac{y−x}{q−1})λ^{j}(\frac{y−q^{i}x}{q−1}) − λ^{i}(\frac{y−x}{q−1})λ^{j}(\frac{y−x}{q−1}) ∈ M,

so the binomial expression above implies λ^{i}(\frac{y−x}{q−1})λ^{j}(\frac{y−x}{q−1}) ∈ M. □

1.2. q-cohomology of Λ-rings.

Definition 1.6. Given a Λ-ring R, say that A is a Λ-ring over R if it is a Λ-ring equipped with a morphism R → A of Λ-rings. We say that A is a flat Λ-ring over R if A is flat as a module over the commutative ring underlying R.

Definition 1.7. Given a morphism R → A of Λ-rings, we define the category Strat_{A/R}^{q} to consist of flat Λ-rings B over R[q] equipped with a compatible morphism A → B/(q − 1), such that the map A → B/(q − 1) 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, Strat_{A/R}^{q} is the Grothendieck construction of the functor

(Spec A)_{strat}^{q}: B ↦ Im (Hom_{Λ,R}(A, B) → Hom_{Λ,R}(A, B/(q − 1)))

on the category fΛ(R[q]) of flat Λ-rings over R[q].

Definition 1.8. Given a flat morphism R → A of Λ-rings, define qDR(A/R) to be the cochain complex of R[q]-modules given by taking the homotopy limit of the functor

Strat_{A/R}^{q} → Ch(R[q])
B ↦ B.

Equivalently, can we follow the approach of [Gro, Sim] towards the de Rham stack by regarding qDR(A/R) as the quasi-coherent cohomology complex of (Spec A)_{strat}^{q}. Writing 𝒪: fΛ(R[q]) → Mod_{R[q]} for the forgetful functor to the category of R[q]-modules, and [fΛ(R[q]), Set] for the category of set-valued functors on fΛ(R[q]), we have

qDR(A/R) = RHom_{[fΛ(R[q]),Set]}((Spec A)_{strat}^{q}, 𝒪),

coming from the right-derived functor of the functor Hom_{[fΛ(R[q]),Set]}((Spec A)_{strat}^{q}, −) of natural transformations with source (Spec A)_{strat}^{q}.

Definition 1.9. Given a polynomial ring R[x], recall from [Sch2] that the q-de Rham (or Aomoto–Jackson) cohomology q-Ω_{R[x]/R}^{•} is given by the complex

R[x][q] ⟶^{∇_{q}} R[x][q]dx, where ∇_{q}(f) = \frac{f(qx) − f(x)}{x(q − 1)}dx,

so ∇_{q}(x^{n}) = [n]_{q}x^{n−1}dx.

Given a polynomial ring R[x_{1}, …, x_{d}], the q-de Rham complex q-Ω_{R[x_{1},...,x_{d}]/R}^{•} is then set to be

q-Ω_{R[x_{1}]/R}^{•} ⊗_{R[q]} q-Ω_{R[x_{2}]/R}^{•} ⊗_{R[q]} … ⊗_{R[q]} q-Ω_{R[x_{d}]/R}^{•},

so takes the form

R[x_{1}, …, x_{d}][q] ⟶^{∇_{q}} Ω_{R[x_{1},...,x_{d}]/R}^{1}[q] ⟶^{∇_{q}} … ⟶^{∇_{q}} Ω_{R[x_{1},...,x_{d}]/R}^{d}[q].

Proposition 1.10. If R is a Λ-ring and x of rank 1, then qDR(R[x]/R) can be calculated by a cosimplicial module U^{•} given by setting U^{n} to be the Λ-subring

U^{n} ⊂ R[q, {(q^{m} − 1)^{−1}}_{m≥1}, x_{0}, …, x_{n}]

generated by q and the elements x_{i} and \frac{x_{i}−x_{j}}{q−1} .

Proof. For X = Spec R[x], the set-valued functor X_{strat}^{q} is not representable, but it can be resolved by the simplicial functor X̃_{strat}^{q} given by taking the Čech nerve of Hom_{Λ,R}(A, B) → Hom_{Λ,R}(A, B/(q − 1)), so

(X̃_{strat}^{q})_{n}(B) := \overbrace{Hom_{Λ,R}(A, B) ×_{Hom_{Λ,R}(A,B/(q−1))} … ×_{Hom_{Λ,R}(A,B/(q−1))} Hom_{Λ,R}(A, B)}^{n+1}

= Hom_{Λ,R}(A, \overbrace{B ×_{B/(q−1)} … ×_{B/(q−1)} B}^{n+1}).

Observe that any element of (X_{strat}^{q})_{n}(B) gives rise to a morphism f: R[q, x_{0}, …, x_{n}] → B of Λ-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 Λ-ring L over R[q, x_{0}, …, 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≥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≥1}, x_{0}, …, x_{n}]. To see that (X_{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

x_{0}^{r_{0}}λ^{r_{1}}(\frac{x_{1}−x_{0}}{q−1} ) ⋯ λ^{r_{n}}(\frac{x_{n}−x_{n−1}}{q−1} )

for U^{n} over R[q]. We therefore have qDR(R[x]/R) ≃ U^{•}. □

Theorem 1.11. If R is a Λ-ring, and the polynomial ring R[x_{1}, …, x_{d}] is given the Λring structure for which the elements x_{i} are of rank 1, then there are R[q]-linear zigzags of quasi-isomorphisms

qDR(R[x_{1}, …, x_{n}]/R) ≃ (Ω_{R[x_{1},...,x_{n}]/R}^{∗}[q], (q − 1)∇_{q})
Lη_{(q−1)}qDR(R[x_{1}, …, x_{n}]/R) ≃ q-Ω_{R[x_{1},...,x_{n}]/R}^{•},

where Lη_{(q−1)} denotes derived décalage with respect to the (q − 1)-adic filtration.

Proof. It suffices to prove the first statement, the second following immediately by décalage. Since (Spec A ⊗_{R} A′)_{strat}^{q}(B) = (Spec A)_{strat}^{q}(B) × (Spec A′)_{strat}^{q}(B), and coproduct of flat Λ-rings over R[q] is given by ⊗_{R[q]}, we have qDR((A ⊗_{R} A′)/R) ≃ qDR(A/R) ⊗_{R[q]}^{L} qDR(A′/R), so we may reduce to the case A = R[x].

Proposition 1.10 gives qDR(R[x]/R) ≃ U^{•}, and in order to compare this with q-de Rham cohomology, we now consider the cochain complexes Ω̃^{•}(U^{n}) given by

U^{n} ⟶^{(q−1)∇_{q}} ⨁_{i} U^{n}dx_{i} ⟶^{(q−1)∇_{q}} ⨁_{i<j} U^{n}dx_{i} ∧ dx_{j} ⟶^{(q−1)∇_{q}} ….

In order to see that this differential is well-defined, observe that

(q − 1)∇_{q,y}λ^{k}(\frac{y−x}{q−1}) = y^{−1}(λ^{k}(\frac{qy−x}{q−1}) − λ^{k}(\frac{y−x}{q−1}))dy
= y^{−1}(λ^{k}(y + \frac{y−x}{q−1}) − λ^{k}(\frac{y−x}{q−1}))dy
= λ^{k−1}(\frac{y−x}{q−1})dy,

and similarly

(q − 1)∇_{q,x}λ^{k}(\frac{y−x}{q−1}) = ∑_{i≥1}(−1)^{i}x^{i−1}λ^{k−i}(\frac{y−x}{q−1})dx.

The first calculation also shows that the inclusion Ω̃^{•}(U^{n−1}) ↪ Ω̃^{•}(U^{n}) is a quasiisomorphism, since

(q − 1)∇_{q,x_{n}}(f(x_{0}, …, x_{n−1})λ^{k}(\frac{x_{n}−x_{n−1}}{q−1} )) = f(x_{0}, …, x_{n−1})λ^{k−1}(\frac{x_{n}−x_{n−1}}{q−1} )dx_{n}.

By induction on n we deduce that the inclusion Ω̃^{•}(U^{0}) ↪ Ω̃^{•}(U^{n}), and hence the retraction of it given by the diagonal, is a quasi-isomorphism.

Now the complexes Ω̃^{i}(U^{•}) are all acyclic for i > 0, consisting of cosimplicial tensor products of U^{•} with cosimplicial symmetric powers of the acyclic complex given by ℤdx_{0} ⊕ ⋯ ⊕ ℤdx_{n} in level n. We therefore have quasi-isomorphisms

U^{•} ← Tot Ω̃^{•}(U^{•}) → Ω̃^{•}(U^{0})

of flat cochain complexes over R[q], so

qDR(R[x]/R) ≃ Ω̃^{•}(R[x]),

and we just observe that η_{(q−1)}Ω̃^{•}(R[x]) = (Ω_{R[x]/R}^{∗}[q], (q − 1)∇_{q}). □

Remark 1.12. Note that Theorem 1.11 implies that q-Ω_{R[x_{1},...,x_{n}]/R}^{•} naturally underlies the décalage of a cosimplicial Λ-ring over R[q]. Even the underlying cosimplicial commutative ring structure carries more information than an E_{∞}-structure when ℚ ⊈ R.

Remark 1.13. The complex (Ω_{R[x_{1},...,x_{n}]/R}^{∗}[q], (q − 1)∇_{q}) is a more fundamental object than its décalage q-Ω_{R[x_{1},...,x_{n}]/R}^{•}, since it has a vestigial memory of the Hodge filtration.

There might be a natural formulation of the theorem not involving décalage, in terms of a q-analogue of the crystalline site for a Λ-ring A over R, regarded as an R[q]-algebra via R = R[q]/(q − 1). Following Remark 1.4, this would involve extensions B → A of Λ-rings over R[q] equipped with q-analogues of divided power operations on the augmentation ideals I, looking like x ↦ (q − 1)^{k}λ^{k}(\frac{x}{q−1}).

1.3. Completed q-cohomology.

Definition 1.14. Given a morphism R → A of Λ-rings, we define the category Ŝtrat_{A/R}^{q} ⊂ Strat_{A/R}^{q} to consist of those objects which are (q − 1)-adically complete.

Definition 1.15. Given a flat morphism R → A of Λ-rings, define q̂DR(A/R) to be the cochain complex of R⟦q − 1⟧-modules given by taking the homotopy limit of the functor

Ŝtrat_{A/R}^{q} → Ch(R⟦q⟧)
B ↦ B.

The following is immediate:

Lemma 1.16. Given a flat morphism R → A of Λ-rings, the complex q̂DR(A/R) is the derived (q − 1)-adic completion of qDR(A/R).

Definition 1.17. As in [Sch2, §3], given a formally étale map □: R[x_{1}, …, x_{d}] → A, define q̂-Ω_{A/R,□}^{•} to be the complex

A⟦q − 1⟧ ⟶^{∇_{q}} Ω_{A/R}^{1}⟦q − 1⟧ ⟶^{∇_{q}} … ⟶^{∇_{q}} Ω_{A/R}^{d}⟦q − 1⟧,

where ∇_{q} is defined as follows. First note that the ring endomorphisms γ_{i} of R[x_{1}, …, x_{d}]⟦q − 1⟧ given by γ_{i}(x_{j}) = q^{δ_{ij}}x_{j} extend uniquely to endomorphisms of A⟦q − 1⟧ which are the identity modulo (q − 1), then set

∇_{q}(f) := ∑_{i} \frac{γ_{i}(f) − f}{(q − 1)x_{i}} dx_{i}.

Note that q̂-Ω_{R[x_{1},...,x_{d}]/R}^{•} is just the (q − 1)-adic completion of q-Ω_{R[x_{1},...,x_{d}]/R}^{•}.

Proposition 1.18. If R is a flat Λ-ring over ℤ and □: R[x_{1}, …, x_{d}] → A is a formally étale map of Λ-rings, the elements x_{i} having rank 1, then there are zigzags of R⟦q⟧-linear quasi-isomorphisms

q̂DR(A/R) ≃ (Ω_{A/R}^{∗}⟦q − 1⟧, (q − 1)∇_{q}),  Lη_{(q−1)}q̂DR(A/R) ≃ q̂-Ω_{A/R,□}^{•}.

The induced quasi-isomorphisms

q̂DR(A/R) ⊗_{R⟦q−1⟧}^{L} R ≃ (Ω_{A/R}^{∗}, 0), (Lη_{(q−1)}q̂DR(A/R)) ⊗_{R⟦q−1⟧}^{L} R ≃ Ω_{A/R}^{•}

are independent of the choice of framing.

Proof. This is much the same as the proof of Theorem 1.11. The complex { \atop qDR}(A/R) can be realised as a cosimplicial Λ-ring U, with U^{n} the (q − 1)-adically complete Λsubring of A^{⊗_{R}(n+1)}⟦q − 1⟧[{(q^{m} − 1)^{−1}}_{m≥1}] generated by

A^{⊗_{R}(n+1)}⟦q − 1⟧ and (q − 1)^{−1} ker(A^{⊗_{R}(n+1)} → A)⟦q − 1⟧.

Uniqueness of lifts with respect to the formally étale framing ensures that the endomorphisms γ_{i} commute with the Adams operations, so are Λ-ring endomorphisms of R. Since the formal completion of A ⊗_{R} A → A is just the Λ-ring

A⟦(x_{1} − y_{1}), (x_{2} − y_{2}), …, (x_{d} − y_{d})⟧,

the calculations of Theorem 1.11 now adapt to give quasi-isomorphisms

(Ω_{A/R}^{∗}⟦q⟧, (q − 1)∇_{q}) ← Tot Ω̃^{•}(U^{•}) → U^{•},

where Ω̃^{•}(U^{n}) is the (q − 1)-adic completion of (U^{n} ⊗_{A^{⊗(n+1)}} (Ω_{A/R}^{∗})^{⊗(n+1)}, (q − 1)∇_{q})). Reduction of this modulo (q − 1)^{2}, or of its décalage modulo (q − 1), replaces ∇_{q} with d, removing the dependence on co-ordinates. □

Remark 1.19. As in [Sch2, Definition 7.3], there is a notion of q-connections on projective A⟦q−1⟧-modules M. These give rise to modules over q̂-Ω_{A/R}^{•}, or equivalently to modules N := (M ⊗_{A⟦q−1⟧}Ω_{A/R}^{∗}⟦q−1⟧, (q−1)∇_{q}) over (Ω_{A/R}^{∗}⟦q⟧, (q−1)∇_{q}) with the décalage-type condition that N/(q − 1) = Ω_{A/R}^{∗} ⊗_{A} M_{0}, where M_{0} = M/(q − 1).

Adapting the ideas of Proposition 1.18, these will be equivalent to projective modules over X_{strat}^{q} (i.e. flat Cartesian { \atop qDR}(A/R)-modules N with N ⊗_{{ \atop qDR}(A/R)} A⟦q − 1⟧ = M)

for which the ({ \atop qDR}(A/R)/(q − 1))-module N/(q − 1) is just given by pullback of M_{0} along X_{strat}^{q}/(q − 1) → X.

Via Lemma 1.5, these data are equivalent to specifying an operator ∂^{1} : M → ˆ⨁_{k_{1},...,k_{d}}Mλ^{k_{1}}(\frac{y_{1}−x_{1}}{q−1} ) ⋯ λ^{k_{d}}(\frac{y_{d}−x_{d}}{q−1} ) satisfying a cocycle condition and congruent to the identity modulo (q − 1). Such operators then arise from q-connections (∇_{1,q}, …, ∇_{d,q}) as q-Taylor series

∂^{1}(f) := ∑_{k_{1},...,k_{d}}(q − 1)^{∑ k_{i}}(∇_{1,q}^{k_{1}} … ∇_{d,q}^{k_{d}})(f) λ^{k_{1}}(\frac{y_{1}−x_{1}}{q−1} ) ⋯ λ^{k_{d}}(\frac{y_{d}−x_{d}}{q−1} ).

2. Comparisons for Λ_{P} -rings

Since very few étale maps R[x_{1}, …, x_{d}] → A give rise to Λ-ring structures on A, Proposition 1.18 is fairly limited in its scope for applications. We now show how the construction of q̂DR and the comparison quasi-isomorphism survive when we weaken the Λ-ring structure by discarding Adams operations at invertible primes.

2.1. q-cohomology for Λ_{P} -rings. Our earlier constructions for Λ-rings all carry over to Λ_{P} -rings, as follows.

Definition 2.1. Given a set P of primes, we define a Λ_{P} -ring A to be a Λ_{ℤ,P} -ring in the sense of [Bor]. 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 Λ_{P} -ring structure on A is equivalent to giving commuting Adams operations Ψ^{p} for all p ∈ P, with Ψ^{p}(a) ≡ a^{p} mod p for all a.

Thus when P is the set of all primes, a Λ_{P} -ring is just a Λ-ring; a Λ_{∅}-ring is just a commutative ring; for a single prime p, we write Λ_{p} := Λ_{{p}}, and note that a Λ_{p}-ring is a δ-ring in the sense of [Joy].

Definition 2.2. Given a Λ_{P} -ring R, say that A is a Λ_{P} -ring over R if it is a Λ_{P} -ring equipped with a morphism R → A of Λ_{P} -rings. We say that A is a flat Λ_{P} -ring over R if A is flat as a module over the commutative ring underlying R.

Definition 2.3. Given a morphism R → A of Λ_{P} -rings, we define the category Strat_{A/R}^{q,P} to consist of flat Λ_{P} -rings B over R[q] equipped with a compatible morphism A → B/(q − 1), such that the map A → B/(q − 1) admits a lift to B. We define the category Ŝtrat_{A/R}^{q,P} ⊂ Strat_{A/R}^{q} to consist of those objects which are (q − 1)-adically complete.

More concisely, Strat_{A/R}^{q,P} (resp. Ŝtrat_{A/R}^{q,P}) is the Grothendieck construction of the functor

(Spec A)_{strat}^{q,P}: B ↦ Im (Hom_{Λ_{P} ,R}(A, B) → Hom_{Λ_{P} ,R}(A, B/(q − 1)))

of the category of flat Λ_{P} -rings (resp. (q − 1)-adically complete flat Λ_{P} -rings) over R[q].

Definition 2.4. Given a flat morphism R → A of Λ_{P} -rings, define qDR_{P} (A/R) to be the cochain complex of R[q]-modules given by taking the homotopy limit of the functor

Strat_{A/R}^{q,P} → Ch(R[q])
B ↦ B.

Define q̂DR_{P} (A/R) to be the cochain complex of R⟦q − 1⟧-modules given by the corresponding homotopy limit over Ŝtrat_{A/R}^{q,P}.

Thus when P is the set of all primes, we have qDR_{P} (A/R) = qDR(A/R). At the other extreme, for A smooth, q̂DR_{∅}(A/R) is the Rees construction of the Hodge filtration on the infinitesimal cohomology complex of A over R , with formal variable (q−1). In more detail, there is a decreasing filtration F of 𝒪_{inf} given by powers of the augmentation ideal 𝒪_{inf} → 𝒪_{Zar}, and q̂DR_{∅}(A/R) ≃ ⨁_{ν∈ℤ}(q − 1)^{−ν}RΓ(Spec A, F^{ν}𝒪_{inf})(q − 1)^{−ν}.

Lemma 2.5. For a set P of primes, the forgetful functor from Λ-rings to Λ_{P} -rings has a right adjoint W^{(∉P)}. There is a canonical ghost component morphism

W^{(∉P)}(B) → ∏_{\begin{matrix}n∈ℕ: \\ (n,p)=1 ∀p∈P \end{matrix}} B,

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 Λ-rings and Λ_{P} -rings. The ghost component morphism is given by taking the Adams operations Ψ^{n} given by the Λ-ring structure on W^{(∉P)}(B), followed by projection to B. When P contains all the residue characteristics of B, a Λ-ring structure is the same as a Λ_{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 Λ-rings, and a set P of primes, there are natural maps

qDR_{P} (A/R) → qDR(A/R),  q̂DR_{P} (A/R) → q̂DR(A/R),

and the latter map is a quasi-isomorphism when P contains all the residue characteristics of A.

Proof. We have functors

(Spec A)_{strat}^{q} ∘ W^{(∉P)} : B ↦ Im (Hom_{Λ,R}(A, W^{(∉P)}B) → Hom_{Λ,R}(A, (W^{(∉P)}B)/(q − 1)))
(Spec A)_{strat}^{q,P}: B ↦ Im (Hom_{Λ_{P} ,R}(A, B) → Hom_{Λ_{P} ,R}(A, B/(q − 1)))

on the category of flat Λ_{P} -rings over R[q]. There is an obvious map

(W^{(∉P)}B)/(q − 1) → W^{(∉P)}(B/(q − 1)),

and hence a natural transformation (Spec A)_{strat}^{q} ∘W^{(∉P)} → (Spec A)_{strat}^{q,P}, which induces the morphism qDR_{P} (A/R) → qDR(A/R) on cohomology.

When P contains all the residue characteristics of A, the map (W^{(∉P)}B)/(q − 1) → W^{(∉P)}(B/(q − 1)) is just

∏_{\begin{matrix}n∈ℕ: \\ (n,p)=1 ∀p∈P \end{matrix}} B/(q^{n} − 1) → ∏_{\begin{matrix}n∈ℕ: \\ (n,p)=1 ∀p∈P \end{matrix}} B/(q − 1),

since the morphism R[q] → W^{(∉P)}B is given by Adams operations, with Ψ^{n}(q − 1) = q^{n} − 1.

We have (q^{n} − 1) = (q − 1)[n]_{q}, and [n]_{q} is a unit in ℤ[\frac{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)_{strat}^{q} ∘ W^{(∉P)} → (Spec A)_{strat}^{q,P} is a natural isomorphism on the category of flat (q − 1)-adically complete Λ_{P} -rings over R[q], and so q̂DR_{P} (A/R) ⟶^{≃} q̂DR(A/R). □

Over ℤ[\frac{1}{P}], every Λ_{P} -ring can be canonically made into a Λ-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 Λ-ring structure will not satisfy the conditions of Proposition 1.18. We now give a more general result which does allow for meaningful comparisons.

Theorem 2.7. If R is a flat Λ_{P} -ring over ℤ and □: R[x_{1}, …, x_{d}] → A is a formally étale map of Λ_{P} -rings, the elements x_{i} having rank 1, then there is a zigzag of R⟦q⟧- linear quasi-isomorphisms

Lη_{(q−1)}q̂DR_{P} (A/R) ≃ q̂-Ω_{A/R,□}^{•}

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 Λ_{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 Λ_{P} -ring.

Similarly to Proposition 1.18, { \atop qDR}_{P} (A/R) is calculated using a cosimplicial Λ_{P} -ring given in level n by the (q − 1)-adic completion Û_{P,A}^{•} of the Λ_{P} -ring over R[q] generated by A^{⊗_{R}(n+1)}[q] and (q − 1)^{−1} ker(A^{⊗_{R}(n+1)} → A)[q]. The observation above shows that Û_{P,A}^{n} ≅ Û_{P,R[x_{1},...,x_{d}]}^{n} ˆ⊗_{R[x_{1},...,x_{d}]}A, changing base along □ applied to the first factor.

As in Proposition 2.6, Û_{P,R[x_{1},...,x_{d}]}^{•} is just the (q − 1)-adic completion of the complex U^{•} from Proposition 1.10. Further application of the key observation above then allows us to adapt the constructions of Theorem 1.11, giving the desired quasi-isomorphisms.

□

2.2. Cartier isomorphisms in mixed characteristic. The only setting in which Theorem 2.7 leads to results close to the conjectures of [Sch2] is when R = W^{(p)}(k), the p-typical Witt vectors of a field of characteristic p, and A = \underleftarrow{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 gives rise to a unique compatible lift Ψ of absolute Frobenius on A with Ψ(x_{i}) = x_{i}^{p}, so gives A the structure of a topological Λ_{p}-ring. The framing still affects the choice of Λ_{p}-ring structure, but at least such a structure is guaranteed to exist, giving rise to a complex qDR_{P} (A/R)^{∧_{p}} := \underleftarrow{lim}_{n} qDR_{p}(A/R) ⊗_{R} R_{n}

depending only on the choice of Ψ, where R_{n} = W_{n}^{(p)}(k).

Our constructions now allow us to globalise the quasi-isomorphism

(q̂-Ω_{A/R,□}^{•})^{∧_{p}}/[p]_{q} ≃ (Ω_{A/R}^{∗})^{∧_{p}}⟦q − 1⟧/[p]_{q}

of [Sch2, Proposition 3.4], where Ω_{A/R}^{∗} denotes the complex A ⟶^{0} Ω_{A/R}^{1} ⟶^{0} Ω_{A/R}^{2} ⟶^{0} . . ..

Proposition 2.8. Take a smooth formal scheme 𝔛 over R = W^{(p)}(k) equipped with a lift Ψ of Frobenius which étale locally admits co-ordinates {x_{i}}_{i} as above with Ψ(x_{i}) = x_{i}^{p}. Then there is a global quasi-isomorphism

C_{q}^{−1} : (Ω_{𝔛/R}^{∗})^{∧_{p}}⟦q − 1⟧/[p]_{q} → (Lη_{(q−1)}q̂DR_{p}(𝒪_{𝔛}/R))^{∧_{p}}/[p]_{q}

in the derived category of étale sheaves on 𝔛.

Proof. Functoriality of the construction qDR_{p} for rings with Frobenius lifts gives us a sheaf q̂DR_{p}(𝒪_{𝔛}/R)^{∧_{p}} on 𝔛. We then have maps

Ψ^{p}: qDR_{p}(𝒪_{𝔛}/R)^{∧_{p}} → qDR_{p}(𝒪_{𝔛}/R)^{∧_{p}}
qDR_{p}(𝒪_{𝔛}/R)^{∧_{p}}/(q − 1) → qDR_{p}(𝒪_{𝔛}/R)^{∧_{p}}/(q^{p} − 1),

and thus, denoting good truncation by τ,

(q − 1)^{i}Ψ^{p}: τ^{≤i}(qDR_{p}(𝒪_{𝔛}/R)^{∧_{p}}/(q − 1)) → (Lη_{(q−1)}q̂DR_{p}(𝒪_{𝔛}/R)^{∧_{p}})/[p]_{q};

the left-hand side is quasi-isomorphic to ⨁_{j≤i}(Ω_{𝒪_{𝔛}/R}^{j})^{∧_{p}}[−j] by Proposition 1.18.

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, the map Ψ^{p} necessarily corresponds via Theorem 2.7 to the chain map adx^{I} ↦ Ψ^{p}(a)x^{I(p−1)}dx^{I} on the complex (Ω_{A/R}^{∗}⟦q − 1⟧, (q − 1)∇_{q}). This gives equivalences

(q − 1)^{i}Ψ^{p} ≃ ∑_{j≤i}(q − 1)^{i−j}(C̃^{−1})^{j}

for Scholze’s locally defined lifts (C̃^{−1})^{j}: (Ω_{A/R}^{j})^{∧_{p}}[−j] → (q̂-Ω_{A/R,□}^{•})^{∧_{p}}/[p]_{q} of the Cartier quasi-isomorphism. The local calculation of [Sch2, Proposition 3.4] then ensures that C_{q}^{−1} is a quasi-isomorphism. □

3. Functoriality via analogues of de Rham–Witt cohomology

In order to obtain a cohomology theory for smooth commutative rings rather than for Λ_{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

∇_{q}(x^{m/n}) = \frac{q^{m/n} − 1}{q − 1}x^{m/n}d log x,

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}.

3.1. Motivation.

Definition 3.1. Given a Λ_{P} -ring B, define Ψ^{1/P^{∞}}B to be the smallest Λ_{P} -ring containing B on which the Adams operations are automorphisms.

In the case P = {p}, the Λ_{p}-ring Ψ^{1/p^{∞}}B is thus the colimit of the diagram

B ⟶^{Ψ^{p}} B ⟶^{Ψ^{p}} B ⟶^{Ψ^{p}} ….

The proof of Theorem 2.7 allows us to replace { \atop qDR}_{p}(A/R) with the complex (Ω_{A/R}^{∗}⟦q − 1⟧, (q − 1)∇_{q}); under this quasi-isomorphism, the Adams operations on A extend to Ω_{A/R}^{∗}⟦q − 1⟧ by setting Ψ^{n}(dx_{i}) := x_{i}^{n−1}dx_{i}. As an immediate consequence we have:

Lemma 3.2. If R is a flat Λ_{p}-ring over ℤ with Ψ^{1/p^{∞}}R = R and residue characteristic p, then Ψ^{1/p^{∞}}qDR_{p}(R[x]/R) ≃ (Ω_{R[x^{1/p^{∞}}]/R}^{∗}[q^{1/p^{∞}}], (q − 1)∇_{q}), so the décalage

Lη_{(q−1)}Ψ^{1/p^{∞}} q̂DR_{p}(R[x]/R) is quasi-isomorphic to the (q − 1)-adic completion of the complex

{a ∈ R[x^{1/p^{∞}}, q^{1/p^{∞}}] : ∇_{q}a ∈ R[x^{1/p^{∞}}, q^{1/p^{∞}}]d log x} ⟶^{∇_{q}}

{b d log x ∈ R[x^{1/p^{∞}}, q^{1/p^{∞}}]d log x : b(0, q) = 0}.

Thus in level 0 (resp. level 1), Lη_{(q−1)}Ψ^{1/p^{∞}} { \atop 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 be flat p-adically complete Λ_{p}-algebras over ℤ_{p}, with Ψ^{1/p^{∞}}R = R and, for elements x_{i} of rank 1, a map □: R[x_{1}, …, x_{d}]^{∧_{p}} → A of Λ_{p}- rings which is a flat p-adic deformation of an étale map. Then the map

(R[q^{1/p^{∞}}] ⊗_{R[q]} Lη_{(q−1)}q̂DR_{p}(A/R))^{∧_{p}} → Lη_{(q−1)}(Ψ^{1/p^{∞}} q̂DR_{p}(A/R))^{∧_{p}}

is a quasi-isomorphism.

Proof. The map Ψ^{p}: A ⊗_{R[x_{1},...,x_{d}]} R[x_{1}^{1/p}, …, x_{d}^{1/p}] → A becomes an isomorphism on p-adic completion, because □ is flat and we have an isomorphism modulo p. Thus

Ψ^{1/p^{∞}}A ≅ A[x_{1}^{1/p^{∞}}, …, x_{d}^{1/p^{∞}}]^{∧_{p}} := (A ⊗_{R[x_{1},...,x_{d}]} R[x_{1}^{1/p^{∞}}, …, x_{d}^{1/p^{∞}}])^{∧_{p}}

Combined with the calculation of Lemma 3.2, this gives us a quasi-isomorphism between (Ψ^{1/p^{∞}} { \atop qDR}_{p}(A/R))^{∧_{p}} and the (p, q − 1)-adic completion of

⨁_{α} A⟦q − 1⟧x_{1}^{α_{1}} … x_{d}^{α_{d}}dx^{I},

where α ∈ p^{−∞}ℤ^{d} such that 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

fx_{1}^{α_{1}} … x_{d}^{α_{d}}dx^{I} ↦ fx_{1}^{α_{1}} … x_{d}^{α_{d}} ∑_{i∈I} ±x_{i}[α_{i}]_{q}^{−1}dx^{(I\i)},

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

ℤ[q^{1/p^{∞}}]^{∧_{(p,q−1)}}. □

Remark 3.4. The endomorphism given on Ψ^{1/P^{∞}} q̂DR_{P} (A/R) by

a ↦ Ψ^{1/n}([n]_{q}a) = [n]_{q^{1/n}}Ψ^{1/n}a

descends to an endomorphism of H^{0}(Ψ^{1/P^{∞}} q̂DR_{P} (A/R)/(q − 1)), which we may denote by V_{n} because it mimics Verschiebung in the sense that Ψ^{n}V_{n} = n · id.

For A smooth over ℤ, we then have

H^{0}(Ψ^{1/P^{∞}} q̂DR_{P} (A/ℤ)/(q − 1))/(V_{p} : p ∈ P) ≅ A[q^{1/P^{∞}}]/([p]_{q^{1/p}} : p ∈ P)
≅ A[ζ_{P^{∞}}],

for ζ_{n} a primitive nth root of unity.

By adjunction, this gives an injective map

H^{0}(Ψ^{1/P^{∞}} q̂DR_{P} (A/ℤ)/(q − 1)) ↪ W^{(P)}A[ζ_{P^{∞}}]

of Λ_{P} -rings, which becomes an isomorphism on completing Ψ^{1/P^{∞}} q̂DR(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 [GR].

3.2. Almost isomorphisms. Combined with Lemma 3.3, Remark 3.4 allows us to regard Lη_{(q−1)}Ψ^{1/p^{∞}} q̂DR_{p}(A/ℤ_{p})^{∧_{p}} as being almost a q^{1/p^{∞}}-analogue of p-typical de Rham–Witt cohomology. (From now on, we consider only the case P = {p}.)

The ideal (q^{1/p^{∞}}−1)^{∧_{(p,q−1)}} = ker(ℤ[q^{1/p^{∞}}]^{∧_{(p,q−1)}} → ℤ_{p}) is equal to the p-adic completion of its square, since we may write it as the kernel W^{(p)}(𝔪) of W^{(p)}(𝔽_{p}[q^{1/p^{∞}}]^{∧_{(q−1)}}) → W^{(p)}(𝔽_{p}), for the idempotent maximal ideal 𝔪 = ((q − 1)^{1/p^{∞}})^{∧_{(q−1)}} in 𝔽_{p}[q^{1/p^{∞}}]^{∧_{(q−1)}}. If we set h^{1/p^{n}} to be the Teichmüller element

[q^{1/p^{n}} − 1] = lim_{r→∞}(q^{1/p^{nr}} − 1)^{p^{r}} ∈ ℤ[q^{1/p^{∞}}]^{∧_{(p,q−1)}},

then W^{(p)}(𝔪) = (h^{1/p^{∞}})^{∧_{(p,h)}}. Although W^{(p)}(𝔪)/p^{n} is not maximal in ℤ[h^{1/p^{∞}}]^{∧_{(h)}}/p^{n}, it is idempotent and flat, so gives a basic setup in the sense of [GR, 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 [GR], studying p-adically complete (ℤ[q^{1/p^{∞}}]^{∧_{(p,q−1)}})^{a}-modules (almost ℤ[q^{1/p^{∞}}]^{∧_{(p,q−1)}}-modules) given by localising at almost isomorphisms, the maps whose kernel and cokernel are W^{(p)}(𝔪)-torsion.

The obvious functor (−)^{a} from modules to almost modules has a right adjoint (−)_{∗}, given by N_{∗} := Hom_{ℤ[q^{1/p^{∞}}]^{∧_{(p,q−1)}}} (W^{(p)}(𝔪), N), the module of almost elements. 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.

Lemma 3.5. For any ℤ⟦q − 1⟧-module M, we may recover the ℤ[q^{1/p^{∞}}]^{∧_{(p,q−1)}}- module (M ⊗_{ℤ[q]} ℤ[q^{1/p^{∞}}])^{∧_{p}} as the module of almost elements of the associated almost ℤ[q^{1/p^{∞}}]^{∧_{(p,q−1)}}-module.

Proof. Since M ⊗_{ℤ[q]} ℤ[q^{1/p^{∞}}] = ⨁_{α} M ⊗ q^{α} for α ∈ p^{−∞}ℤ with 0 ≤ α < 1, calculation shows that (M ⊗_{ℤ[q]} ℤ[q^{1/p^{∞}}])^{∧_{p}} → (M ⊗_{ℤ[q]} ℤ[q^{1/p^{∞}}])_{∗}^{∧_{p}} must be an isomorphism. □

3.3. Perfectoid algebras. We now relate Scholze’s perfectoid algebras to a class of Λ_{p}-rings.

Definition 3.6. Define a perfectoid Λ_{p}-ring to be a flat p-adically complete Λ_{p}-algebra over ℤ_{p}, on which the Adams operation Ψ^{p} is an isomorphism.

For a perfectoid field K in the sense of [Sch1], there is a tilt K^{♭} (a complete perfect field of characteristic p). The subring of power-bounded elements is denoted K^{o} ⊂ K.

Lemma 3.7. Given a perfectoid field K, we have equivalences

\xymatrix{perfectoid almost K^{o}-algebras \ar[d]^{𝒜_{inf}} \\ perfectoid almost Λ_{p}-rings over 𝒜_{inf}(K^{o}) \ar[u]^{K^{oa}⊗_{(𝒜_{inf} (K^{o})^{a})} −} \ar[d]_{𝔽_{p}⊗_{ℤp}−} \\ perfectoid almost K^{♭o}-algebras \ar[u]_{W^{(p)}}}

of categories, where 𝒜_{inf}(C) := \underleftarrow{lim}_{Ψ^{p}} W^{(p)}(C).

Proof. A perfectoid Λ_{p}-ring B is a deformation of the 𝔽_{p}-algebra B/p. As in [Sch1, 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 gives the bottom pair of equivalences.

We then observe that since B := 𝒜_{inf}(C) is a perfectoid Λ_{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) ≅ \underleftarrow{lim}_{x↦x^{p}} C, from which it follows that

𝔽_{p} ⊗_{ℤ_{p}} 𝒜_{inf}(C) ≅ \underleftarrow{lim}_{Φ}(C/p) = C^{♭}

whenever C is perfectoid. Since tilting gives an equivalence of almost algebras by [Sch1, Theorem 5.2], this completes the proof. □

We will only apply Lemma 3.7 to perfectoid almost Λ_{p}-rings over ℤ[q^{1/p^{∞}}]^{∧_{(p,q−1)}}, in which case it shows that reduction modulo [p]_{q^{1/p}} (resp. p) gives an equivalence with perfectoid (ℤ[ζ_{p^{∞}}]^{∧_{p}})^{a}-algebras (resp. perfectoid (𝔽_{p}[q^{1/p^{∞}}]^{∧_{(q−1)}})^{a}-algebras),

3.4. Functoriality of q-de Rham cohomology. Since (Ψ^{1/p^{∞}} { \atop qDR}_{p}(A/ℤ_{p}))^{∧_{p}} is represented by a cosimplicial perfectoid Λ_{p} ring over ℤ[q^{1/p^{∞}}]^{∧_{(p,q−1)}} for any flat Λ_{p}-ring A over ℤ_{p}, it corresponds under Lemma 3.7 to a cosimplicial perfectoid (ℤ[ζ_{p^{∞}}]^{∧_{p}})^{a}- algebra, representing the following functor:

Lemma 3.8. For a perfectoid (ℤ[ζ_{p^{∞}}]^{∧_{p}})^{a}-algebra C, and a Λ_{p}-ring A over ℤ_{p} with X = Spec A, there is a canonical isomorphism

X_{strat}^{q}(\underleftarrow{lim}_{Ψ^{p}} W^{(p)}(C)_{∗}) ≅ Im (\underleftarrow{lim}_{Ψ^{p}} X(C_{∗}) → X(C_{∗})),

for the ring C_{∗} of almost elements.

Proof. By definition, X_{strat}^{q}(\underleftarrow{lim}_{Ψ^{p}} W^{(p)}(C)_{∗}) is the image of

Hom_{Λ_{p}}(A, \underleftarrow{lim}_{Ψ^{p}} W^{(p)}(C)_{∗}) → Hom_{Λ_{p}}(A, (\underleftarrow{lim}_{Ψ^{p}} W^{(p)}(C)_{∗})/(q − 1)).

Since right adjoints commute with limits, we may rewrite the first term as \underleftarrow{lim}_{Ψ^{p}} Hom_{Λ_{p}}(A, W^{(p)}(C_{∗})) = \underleftarrow{lim}_{Ψ^{p}} X(C_{∗}).

Setting B := \underleftarrow{lim}_{Ψ^{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 ∘ Ψ^{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

Hom_{Λ_{p}}(A, (\underleftarrow{lim}_{Ψ^{p}} W^{(p)}(C)_{∗})/(q − 1) ↪ Hom_{Λ_{p}}(A, W^{(p)}C_{∗}) = X(C_{∗}).

□

In fact, the tilting equivalence gives \underleftarrow{lim}_{Ψ^{p}} X(C_{∗}) ≅ X(C_{∗}^{♭}), so the only dependence

of ((Ψ^{1/p^{∞}} { \atop qDR}_{p}(A/ℤ_{p}))^{∧_{p}})^{a} on the Frobenius lift Ψ^{p} is in determining the image of X(C_{∗}^{♭}) → X(C_{∗}).

Although this map is not surjective, it is almost so in a precise sense, which we now use to establish independence of Ψ^{p}, showing that, up to faithfully flat descent, q̂DR_{p}(A/ℤ_{p})^{∧_{p}}/[p]_{q^{1/p}} is the best possible perfectoid approximation to A[ζ_{p^{∞}}]^{∧_{p}}.

Definition 3.9. 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

RΓ_{Pfd}(X, 𝒜) := RHom_{[Pfd((ℤ_{p}[ζ_{p^{∞}}]^{∧p})^{a}),Set]}(X, 𝒜),

where Pfd(S^{a}) denotes the category of perfectoid almost S-algebras, and [C, Set] denotes set-valued functors on C. When X is representable by a (ℤ[ζ_{p^{∞}}]^{∧_{p}})^{a}-algebra C, we simply denote this by RΓ_{Pfd}(C, 𝒜) — when C is perfectoid, this will just be 𝒜 (C).

The following gives a slight refinement of [BMS, Theorem 1.17], addressing some of the questions in [BMS, Remark 1.11]:

Theorem 3.10. If R is a p-adically complete Λ_{p}-ring over ℤ_{p}, and A a formal Rdeformation of a smooth ring over (R/p), then the complex

RΓ_{Pfd}((A[ζ_{p^{∞}}] ⊗_{R} Ψ^{1/p^{∞}}R)^{∧_{p}}, 𝒜_{inf})

of ℤ_{p}[q^{1/p^{∞}}]^{∧_{(p,q−1)}}-modules is almost quasi-isomorphic to (Ψ^{1/p^{∞}} q̂DR_{p}(A/R))^{∧_{p}} for any Λ_{p}-ring structure on A coming from a framing over R as in Theorem 2.7.

Proof. First observe that ((Ψ^{1/p^{∞}} q̂DR_{p}(A/R))^{∧_{p}})^{a} is the completion of q̂DR_{p}(A/R) with respect to the category of cosimplicial perfectoid almost Λ_{p}-rings over ℤ[q^{1/p^{∞}}]^{∧_{(p,q−1)}}. Combining the definition of { \atop qDR}_{p} with Lemma 3.7, it then follows that for X = Spec A and Y = Spec R, the complex ({ \atop qDR}_{p}(A/R)^{∧_{p}})_{∗} is given by the homotopy limit

RΓ_{Pfd}((X_{strat}^{q} ×_{Y_{strat}^{q}} Y) ∘ (𝒜_{inf})_{∗}, (𝒜_{inf})_{∗}).

Writing X^{∞}(C) := Im (\underleftarrow{lim}_{Ψ^{p}} X(C_{∗}) → X(C_{∗})), Lemma 3.8 then combines with the description above to give

({ \atop qDR}_{p}(A/R)^{∧_{p}})_{∗} ≃ RΓ_{Pfd}(X^{∞} ×_{Y^{∞}} \underleftarrow{lim}_{Ψ^{p}} Y, (𝒜_{inf})_{∗}),
≃ RΓ_{Pfd}(X^{∞} ×_{Y} \underleftarrow{lim}_{Ψ^{p}} Y, (𝒜_{inf})_{∗}).

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^{♭} = \underleftarrow{lim}_{Φ}(C/p), the functor 𝒜_{inf} satisfies descent with respect to these coverings, so the map

RΓ_{Pfd}((X^{∞} ×_{Y} \underleftarrow{lim}_{Ψ^{p}} Y)^{♯}, (𝒜_{inf})_{∗}) → RΓ_{Pfd}(X^{∞} ×_{Y} \underleftarrow{lim}_{Ψ^{p}} Y, (𝒜_{inf})_{∗})

is a quasi-isomorphism, where (−)^{♯} denotes sheafification.

In other words, the calculation of ({ \atop 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

(X^{∞})^{♯}(C) = ⋃_{C→C′} Im (X(C_{∗}) ×_{X(C′_{∗})} \underleftarrow{lim}_{Ψ^{p}} X(C′_{∗}) → X(C_{∗})),

where C → C′ runs over all covering morphisms.

Now, \underleftarrow{lim}_{Ψ^{p}} X is represented by the perfectoid algebra (Ψ^{1/p^{∞}}A)^{∧_{p}}, which is isomor-

phic to A[x_{1}^{1/p^{∞}}, …, x_{d}^{1/p^{∞}}]^{∧_{p}} as in the proof of Lemma 3.3. This allows us to appeal to André’s results [And, §2.5] as generalised in [Bha, 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 (Ψ^{1/p^{∞}}A)^{∧_{p}} → C′, so f ∈ (X^{∞})^{♯}(C). We have thus shown that (X^{∞})^{♯} = X, giving the required equivalence

((Ψ^{1/p^{∞}} q̂DR_{p}(A/R))^{∧_{p}})_{∗} ≃ RΓ_{Pfd}(X ×_{Y} \underleftarrow{lim}_{Ψ^{p}} Y, (𝒜_{inf})_{∗}).
□

Remark 3.11. Corresponding to the cohomology theory (Ψ^{1/p^{∞}} q̂DR_{p}(A/R))^{∧_{p}}, it is natural to consider q-connections on projective modules M over

η_{(q−1)}^{0}((Ψ^{1/p^{∞}} q̂DR_{p}(A/R))^{∧_{p}}) = (∑_{n}[p^{n}]_{q^{1/p^{n}}} Ψ^{1/p^{n}}A[q^{1/p^{∞}} − 1])^{∧}.

It follows from Remark 1.19 that these are equivalent to projective Ψ^{1/p^{∞}} q̂DR_{p}(A/R))^{∧_{p}}-modules N for which N/(q − 1) is the pullback of a module M_{0} over H^{0}((Ψ^{1/p^{∞}} q̂DR_{p}(A/R))^{∧_{p}}/(q − 1)).

Up to almost isomorphism, these correspond via the proof of Theorem 3.10 to pairs (M_{0}, N) with N a projective 𝒜_{inf}-module on the site of integral perfectoid algebras C over A[ζ_{p^{∞}}]^{∧_{p}}, and M_{0} a projective W^{(p)}(A[ζ_{p^{∞}}]^{∧_{p}})-module with isomorphisms

N(C) ⊗_{𝒜_{inf}(C)} W^{(p)}C ≅ M_{0} ⊗_{W^{(p)}(A[ζ_{p^{∞}}]^{∧p})} W^{(p)}C,

functorial in C.

Corollary 3.12. If R is a p-adically complete Λ_{p}-ring over ℤ_{p}, and A a formal Rdeformation of a smooth ring over (R/p), then the q-de Rham cohomology complex (q-Ω_{A/R,□}^{•}⊗_{R[q]}(Ψ^{1/p^{∞}}R)[q^{1/p^{∞}}])^{∧_{p}} is, up to quasi-isomorphism, independent of a choice of co-ordinates □. It is naturally an invariant of the commutative p-adically complete (Ψ^{1/p^{∞}}R)[ζ_{p^{∞}}]^{∧_{p}}-algebra (A[ζ_{p^{∞}}] ⊗_{R} Ψ^{1/p^{∞}}R)^{∧_{p}}.

Proof. By Theorem 3.10, we know that the complex ((Ψ^{1/p^{∞}} q̂DR_{p}(A/R))^{∧_{p}})_{∗} depends only on (A[ζ_{p^{∞}}] ⊗_{R} Ψ^{1/p^{∞}}R)^{∧_{p}}. Since

Ψ^{1/p^{∞}}qDR_{p}(A/R) = Ψ^{1/p^{∞}}qDR_{p}((A ⊗_{R} Ψ^{1/p^{∞}}R)/Ψ^{1/p^{∞}}R),

Theorem 2.7 combines with Lemmas 3.3 and 3.5 to give

(q-Ω_{A/R,□}^{•} ⊗_{R[q]} (Ψ^{1/p^{∞}}R)[q^{1/p^{∞}}])^{∧_{p}} ≃ Lη_{(q−1)}((Ψ^{1/p^{∞}} q̂DR_{p}(A/R))^{∧_{p}})_{∗},

which completes the proof. □

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