CRYSTALLINE COHOMOLOGY AND DE RHAM COHOMOLOGY

BHARGAV BHATT AND AISE JOHAN DE JONG

Abstract. The goal of this short paper is to give a slightly different perspective on the comparison between crystalline cohomology and de Rham cohomology. Most notably, we reprove Berthelot’s comparison result without using pd-stratifications, linearisations, and pd-differential operators.

Crystalline cohomology is a p-adic cohomology theory for varieties in characteristic p created by Berthelot [Ber74]. It was designed to fill the gap at p left by the discovery [SGA73] of ℓ-adic cohomology for ℓ ≠ p. The construction of crystalline cohomology relies on the crystalline site, which is a better behaved positive characteristic analogue of Grothendieck’s infinitesimal site [Gro68]. The motivation comes from Grothendieck’s theorem [Gro66] identifying infinitesimal cohomology of a complex algebraic variety with its singular cohomology (with C-coefficients); in particular, infinitesimal cohomology gives a purely algebraic definition of the “true” cohomology groups for complex algebraic varieties. In fact, the fundamental structural result of Berthelot [Ber74, Theorem V.2.3.2] is a direct p-adic analogue of this reconstruction result: the crystalline cohomology of a smooth variety X over Z/p is canonically identified with the de Rham cohomology of a lift of X to Z_{p}, provided one exists. In particular, crystalline cohomology produces the “correct” Betti numbers, at least for liftable smooth projective varieties (and, in fact, even without liftability by [KM74]). We defer to [Ill94] for a detailed introduction, and connections with p-adic Hodge theory.

Our goal in this note is to give a different perspective on the relationship between de Rham and crystalline cohomology. In particular, we give a short proof of the aforementioned comparison result [Ber74, Theorem V.2.3.2]; see Theorem 3.6. Our approach replaces Berthelot’s differential methods (involving stratifications and linearisations) with a resolutely Čech-theoretic approach. It seems that Theorem 3.2 is new, although it may have been known to experts in the field. This theorem also appears in forthcoming work by Beilinson [Bei].

Conventions. Throughout this note, p is a fixed prime number. Our base scheme will be typically be Σ = Spec(Z_{p}), though occasionally we discuss the theory over Σ_{e} = Spec(Z_{p}/p^{e}) as well (for some e ≥ 1). All divided powers will be compatible with the divided powers on pZ_{p}. Modules of differentials on divided power algebras are compatible with the divided power structure. A general reference for divided powers and the crystalline site is [Ber74].

1. Review of modules on the crystalline site

Let S be a Σ-scheme such that p is locally nilpotent on S. The (small) crystalline site of S is denoted (S/Σ)_{cris}. Its objects are triples (U, T, δ) where U ⊂ S is an open subset, U ⊂ T is a nilpotent thickening of Σ-schemes, and δ is a divided power structure on the ideal of U in T; the morphisms are the obvious ones, while coverings of (U, T, δ) are induced by Zariski covers of T. The structure sheaf O_{S/Σ} of (S/Σ)_{cris} is defined by O_{S/Σ}((U, T, δ)) = Γ(T, O_{T}).

Given a Z_{p}/p^{e}-algebra B and an ideal J ⊂ B endowed with divided powers δ, the module of differentials compatible with divided powers is the quotient of the module of Σ-linear differentials by the relations dδ_{n}(x) = δ_{n−1}(x)d(x), for x ∈ J and n ≥ 1. We simply write Ω_{B}^{1} for this module as confusion is unlikely. The formation of Ω_{B}^{1} commutes with localisation on B, so the formula Ω_{S/Σ}^{1}((U, T, δ)) = Γ(T, Ω_{T}^{1} ) defines a sheaf Ω_{S/Σ}^{1} on (S/Σ)_{cris}. Like its classical analogue, the sheaf Ω_{S/Σ}^{1} can also be described via the diagonal as follows. Given an object (U, T, δ) of (S/Σ)_{cris}, let (U, T (1), δ(1)) be the product of (U, T, δ) with itself in (S/Σ)_{cris}: the scheme T (1) is simply the divided power envelope of U ⊂ T ×_{Σ} T, with δ(1) being the induced divided power structure. The diagonal map Δ : T → T (1) is a closed immersion corresponding to an ideal sheaf I with divided powers, and we have

Ω_{S/Σ}^{1}((U, T, δ)) = Γ(T, I/I^{[2]}),

where I^{[2]} denotes the second divided power of I. For i ≥ 0 we define Ω_{S/Σ}^{i} as the i-th exterior power of Ω_{S/Σ}^{1}.

An O_{S/Σ}-module F on (S/Σ)_{cris} is called quasi-coherent if for every object (U, T, δ), the restriction F_{T} of F to the Zariski site of T is a quasi-coherent O_{T} -module. Examples include O_{S/Σ} and Ω_{S/Σ}^{i} for all i > 0.

An O_{S/Σ}-module F on (S/Σ)_{cris} is called a crystal in quasi-coherent modules if it is quasi-coherent and for every morphism f : (U, T, δ) → (U′, T′, δ′) the comparison map

c_{f} : f^{∗}F_{T′} → F_{T}

is an isomorphism. For example, the sheaf O_{S/Σ} is a crystal (by fiat), but the sheaves Ω_{S/Σ}^{i}, i > 0 are not crystals.

Given a crystal F in quasi-coherent modules and an object (U, T, δ), the projections define canonical isomorphisms

pr_{1}^{∗}F_{T} ⟶^{c_{1}} F_{T (1)} ⟵^{c_{2}} pr_{2}^{∗}F_{T} .

These comparison maps are functorial in the objects of the crystalline site. Hence we obtain a canonical map

(1.0.1)  ∇ : F ⟶ F ⊗_{O_{S/Σ}} Ω_{S/Σ}^{1}

such that for any object (U, T, δ) and any section s ∈ Γ(T, F_{T}) we have

c_{1}(s ⊗ 1) − c_{2}(1 ⊗ s) = ∇(s) ∈ I/I^{[2]} ⊗_{O_{S/Σ}} F_{T (1)}.

Transitivity of the comparison maps implies this connection is integrable, hence defines a de Rham complex

F → F ⊗_{O_{S/Σ}} Ω_{S/Σ}^{1} → F ⊗_{O_{S/Σ}} Ω_{S/Σ}^{2} → · · · .

We remark that this complex does not terminate in general.

2. The de Rham-crystalline comparison for affines

In this section, we discuss the relationship between de Rham and crystalline cohomology (with coefficients) when S is affine. First, we establish some notation that will be used throughout this section.

Notation 2.1. Assume S = Spec(A) for a Z/p^{N}-algebra A (and some N > 0). Choose a polynomial algebra P over Z_{p} and a surjection P → A with kernel J. Let D = D_{J}(P)^{∧} be the p-adically completed divided power envelope of P → A. We set Ω_{D}^{1} := Ω_{P}^{1} ⊗_{P}^{∧} D, so Ω_{D}^{1}/p^{e} ≃ Ω_{D/p^{e}D}^{1}. We also set D(0) = D and let

D(n) = D_{J(n)}(P ⊗_{Z_{p}} ⋯ ⊗_{Z_{p}} P)^{∧}

where J(n) = Ker(P ⊗ ⋯ ⊗ P → A) and where the tensor product has (n + 1)-factors. For each e ≥ N and any n ≥ 0, we have a natural object (S, Spec(D(n)/p^{e}(n)), δ(n)) of (S/Σ)_{cris}. Using this, for an abelian sheaf F on (S/Σ)_{cris}, we define

F(n) := lim_{e≥N} F((S, Spec(D(n)/p^{e}(n)), δ(n))).

Each (S, Spec(D(n)/p^{e}(n)), δ(n)) is simply the (n+1)-fold self-product of (S, Spec(D/p^{e}), δ) in (S/Σ)_{cris}. Letting n vary, we obtain a natural cosimplicial abelian group (or a cochain complex)

F(•) := (F(0) → F(1) → F(2) ⋯),

that we call the Čech-Alexander complex of F associated to D.

2.2. Some generalities on crystalline cohomology. This subsection collects certain basic tools necessary for working with crystalline cohomomology; these will be used consistently in the sequel. We begin with a brief review of the construction of homotopy-limits in the only context where they appear in this paper.

Construction 2.3. Let C be a topos. Fix a sequence T_{1} ⊂ T_{2} ⊂ ⋯ T_{n} ⊂ ⋯ of monomorphisms in C. We will construct the functor R lim_{i} RΓ(T_{i}, −); here we follow the convention that G(U) = Γ(U, G) = Hom_{C}(U, G) for any pair of objects U, G ∈ C. Let Ab^{N} denote the category of projective systems of abelian groups indexed by the natural numbers. The functor F ↦ lim_{i} F(T_{i}) can be viewed as the composite

Ab(C) →^{{Γ(T_{i},−)}_{i}} Ab^{N} →^{lim_{i}} Ab.

Each of these functors is a left exact functor between abelian categories with enough injectives, so we obtain a composite of (triangulated) derived functors

D^{+}(Ab(C)) →^{{RΓ(T_{i},−)}_{i}} D^{+}(Ab^{N}) →^{R lim_{i}} D^{+}(Ab).

We use R lim_{i} RΓ(T_{i}, −) to denote the composite functor. To identify this functor, observe that if we set T = colim_{i} T_{i}, then F(T) = lim_{i} F(T_{i}) by adjunction. Moreover, for any injective object I of Ab(C), the projective system i ↦ I(T_{i}) has surjective transition maps I(T_{i+1}) → I(T_{i}): the maps T_{i} → T_{i+1} are injective, and I is an injective object. Since projective systems in Ab^{N} with surjective transition maps are acyclic for the functor lim_{i} (by the Mittag-Leffler condition), there is an identification of triangulated functors

RΓ(T, −) ≃ R lim_{i} RΓ(T_{i}, −).

Thus, the value R lim_{i} RΓ(T_{i}, F) is computed by I^{•}(T) = lim_{i} I^{•}(T_{i}), where F → I^{•} is an injective resolution. An observation that will be useful in the sequel is the following: if each RΓ(T_{i}, F) is concentrated in degree 0, then R lim_{i} RΓ(T_{i}, F) coincides with R lim_{i} F(T_{i}), and thus has only two non-zero cohomology groups (as R^{j} lim_{i} A_{i} = 0 for j > 1 and any N-indexed projective system {A_{i}}_{i} of abelian groups).

We use Construction 2.3 to show that the Čech-Alexander complex often computes crystalline cohomology (compare with [Ber74, Theorem V.1.2.5]).

Lemma 2.4. Let F be a quasi-coherent O_{S/Σ}-module. Assume that for each n > 0, the group R^{1} lim_{e≥N} vanishes for the projective system e ↦ F((S, Spec(D(n)/p^{e}D(n)), δ(n))). Then the complex F(•) computes RΓ(S/Σ, F).

Proof. As representable functors are sheaves on (S/Σ)_{cris} (by Zariski descent), we freely identify objects of (S/Σ)_{cris} with the corresponding sheaf on (S/Σ)_{cris}. One can easily check that the map colim_{e≥N} ((S, Spec(D/p^{e}), δ)) → ∗ is an effective epimorphism is the topos of sheaves on (S/Σ)_{cris}. Since filtered colimits and the Yoneda embedding both commute with finite products, the (n + 1)-fold self-product of colim_{e≥N} ((S, Spec(D/p^{e}), δ)) is simply

colim_{e≥N} ((S, Spec(D(n)/p^{e}(n)), δ(n))). General topos theory (see Remark 2.5) shows that RΓ(S/Σ, F) is computed by

RΓ(colim_{e≥N} ((S, Spec(D(0)/p^{e}(0)), δ(0))), F) → RΓ(colim_{e≥N} ((S, Spec(D(1)/p^{e}(1)), δ(1))), F) → · · · .

The discussion in Construction 2.3 and the vanishing of quasi-coherent sheaf cohomology on affine schemes then identify the above bicomplex with the bicomplex

R lim_{e≥N} (F((S, Spec(D(0)/p^{e}D(0)), δ(0)))) → R lim_{e≥N} (F((S, Spec(D(1)/p^{e}D(1)), δ(1)))) → · · · .

The R^{1} lim_{e} vanishing hypothesis ensures that the bicomplex above collapses to F(•) proving the claim. □

Remark 2.5. The following fact was used in the proof of Lemma 2.4: if C is a topos, and X → ∗ is an effective epimorphism in C, then for any abelian sheaf F in C, the object RΓ(∗, F) is computed by a bicomplex

RΓ(X, F) → RΓ(X × X, F) → RΓ(X × X × X, F) → ⋯,

i.e., the choice of an injective resolution F → I^{•} defines a bicomplex I^{•}(X^{•}) whose totalisation computes RΓ(∗, F); this follows from cohomological descent since the augmented simplicial object X^{•} → ∗ is a hypercover. In particular, there is a spectral sequence with E_{1}-term given by H^{q}(X^{p}, F) that converges to H^{p+q}(∗, F).

Remark 2.6. The R^{1} lim_{e≥N} vanishing assumption of Lemma 2.4 will hold for all sheaves appearing in this paper. For quasi-coherent crystals F, this assumption clearly holds as

F((S, Spec(D(n)/p^{e}(n)), δ(n))) → F((S, Spec(D(n)/p^{e−1}(n)), δ(n)))

is surjective for all e > N and all n ≥ 0. By direct computation, the same is also true for the sheaves Ω_{S/Σ}^{i}.

Next, we formulate and prove a purely algebraic lemma comparing p-adically complete Z_{p}-modules with compatible systems of Z/p^{e}-modules; the result is elementary and well-known, but recorded here for convenience. We remind the reader that a Z_{p}-module M is said to be p-adically complete if the natural map M → lim_{e} M/p^{e} is an isomorphism.

Lemma 2.7. The functor M ↦ (M/p^{e}, can_{e} : (M/p^{e+1})/p^{e} ≃ M/p^{e}) defines an equivalence between the category of p-adically complete Z_{p}-modules M and the category of projective systems (M_{e}, ϕ_{e}) (indexed by e ∈ N) with M_{e} a Z/p^{e}-module, and ϕ_{e} : M_{e+1}/p^{e} ≃ M_{e} an isomorphism.

Proof. A left-inverse functor is given by (M_{e}, ϕ_{e}) ↦ M := lim_{e} M_{e}, the limit being taken along the maps ϕ_{e}. To check that this is also a right inverse, it suffices to show that (lim_{e} M_{e})/p^{n} ≃ M_{n} for any system (M_{e}, ϕ_{e}) as in the lemma. Projection defines a natural map (lim_{e} M_{e})/p^{n} → M_{n} which is surjective as the ϕ_{e}’s are all surjective. For injectivity, it suffices to show that any element m = (m_{e}) ∈ lim_{e} M_{e} with m_{n} = 0 is divisible by p^{n} in lim_{e} M_{e}. The hypothesis implies that there exists an m′ ∈ lim_{e} M_{e} such that m − p^{n}m′ maps to 0 in M_{n+1}: we can simply take m′ to be an arbitrary lift of an element in M_{n+1} which gives m_{n+1} on multiplication by p^{n} (which exists since ϕ_{e+1} maps M_{n+1}/p^{n} isomorphically onto M_{n}). Continuing this process, for each i > 0, we can find an element m_{i} ∈ lim_{e} M_{e} such that m − p^{n}m_{i} maps to 0 in M_{n+i}. Taking the limit i → ∞ proves the desired claim. □

The next lemma is a standard result in crystalline cohomology (see [Ber74, Chapter IV] and [BO78, Theorem 6.6]). We sketch the proof to convince the reader that this result is elementary.

Lemma 2.8. The category of crystals in quasi-coherent O_{S/Σ}-modules is equivalent to the category of pairs (M, ∇) where M is a p-adically complete D-module and ∇ : M → M ⊗_{D}^{∧} Ω_{D}^{1} is a topologically quasi-nilpotent integrable connection.

Proof. Given a crystal in quasi-coherent modules F we set

M = F(0) := lim_{e≥N} F((S, Spec(D/p^{e}D), δ))

and ∇ is as in (1.0.1). Conversely, suppose that (M, ∇) is a module with connection as in the statement of the lemma. Then, given an affine object (S ↪ T, δ) of the crystalline site corresponding to the divided power thickening (B → A, δ), we set

F((S, T, δ)) = M ⊗_{D} B

where D → B is any divided power map lifting id_{A} : A → A. Note that p^{m}B = 0 for some m ≥ 0 by the definition of the crystalline site, so completion isn’t needed in the formula. To see that this is well defined suppose that φ_{1}, φ_{2} : D → B are two maps lifting id_{A}. Then we have an isomorphism

M_{ϕ_{1},ϕ_{2}} : M ⊗_{D,φ_{1}} B ⟶ M ⊗_{D,φ_{2}} B

which is B-linear and characterized by the (Taylor) formula

m ⊗ 1 ⟼ ∑_{E=(e_{i})} (∏(∇_{ϑ_{i}})^{e_{i}}) (m) ⊗ ∏ δ_{e_{i}}(h_{i})

where the sum is over all multi-indices E with finite support. The notation here is: P = Z_{p}[{x_{i}}_{i∈I}], ϑ_{i} = ∂/∂x_{i} and h_{i} = φ_{2}(x_{i}) − φ_{1}(x_{i}). Since h_{i} ∈ Ker(B → A) it makes sense to apply the divided powers δ_{e} to h_{i}. The sum converges precisely because the connection is topologically quasi-nilpotent (this can be taken as the definition). For three maps φ_{1}, φ_{2}, φ_{3} : D → B lifting id_{A}, the resulting isomorphisms satisfy the cocycle condition

M_{ϕ_{2},ϕ_{3}} ∘ M_{ϕ_{1},ϕ_{2}} = M_{ϕ_{1},ϕ_{3}}

by the flatness of ∇. Hence, the above recipe defines a sheaf on (S/Σ)_{cris}. □

Remark 2.9. Lemma 2.8 remains valid if we replace the polynomial algebra P appearing in Notation 2.1 with any smooth Z_{p}-algebra P equipped with a surjection to A (and D with the corresponding p-adically completed divided power envelope). The only non-obvious point is to find a replacement for the Taylor series appearing in the formula for M_{ϕ_{1},ϕ_{2}} in the proof of Lemma 2.8. However, note first that the Taylor series makes sense as soon as there is a polynomial algebra F and an étale map F → P. Moreover, a “change of variables” computation shows that the resulting map is independent of choice of étale chart F → P. The general case then follows by Zariski glueing.

We use Lemma 2.8 to show that crystals on (S/Σ)_{cris} are determined by their restriction to the special fibre.

Corollary 2.10. Reduction modulo p gives an equivalence between categories of crystals of quasi-coherent modules over the structure sheaves of (S/Σ)_{cris} and (S ⊗_{Σ} Spec(F_{p})/Σ)_{cris}

Proof. This follows from Lemma 2.8 by identifying, for any e ≥ N, the divided power envelopes of P/p^{e} → A and P/p^{e} → A → A/p (by [Ber74, Proposition I.2.8.2]). □

2.11. The main theorem in the affine case. The goal of this section is to prove the following theorem.

Theorem 2.12. Suppose that F corresponds to a pair (M, ∇) as in Lemma 2.8. Then there is a natural quasiisomorphism

RΓ(S/Σ, F) ≃ (M → M ⊗_{D}^{∧} Ω_{D}^{1} → M ⊗_{D}^{∧} Ω_{D}^{2} → ⋯).

For F and M as in Theorem 2.12, we use M(n) and M(•) instead of F(n) and F(•) from Notation 2.1. Each M(n) is a D(n)-module with integrable connection as in (1.0.1), so it defines a de Rham complex

M(n) → M(n) ⊗_{D(n)}^{∧} Ω_{D(n)}^{1} → M(n) ⊗_{D(n)}^{∧} Ω_{D(n)}^{2} → · · · .

As n varies, these complexes fit together to define a bicomplex, which we call the de Rham complex of M(•). Our proof of Theorem 2.12 hinges on the observation that both sides of the quasi-isomorphism occurring in the statement of Theorem 2.12 appear as the 0-th rows and columns of the de Rham complex of M(•). Thus, the proof of Theorem 2.12 is reduced to certain acyclicity results for the de Rham complex of M(•), which we show next. The following lemma shows that the “columns” of this bicomplex are all quasi-isomorphic.

Lemma 2.13. The map of complexes

M ⊗_{D}^{∧} Ω_{D}^{∗} → M(n) ⊗_{D(n)}^{∧} Ω_{D(n)}^{∗}

induced by any of the maps D → D(n) is a quasi-isomorphism.

Proof. This is the “naive” poincare lemma. More precisely, each natural map D → D(n) defines an isomorphism of D(n)-modules M(n) ≃ M ⊗_{D}^{∧} D(n) compatible with ∇ by the crystalline nature of F. Thus, there is a filtration of M(n) ⊗_{D(n)}^{∧} Ω_{D(n)}^{∗} whose graded pieces are M ⊗_{D}^{∧} Ω_{D}^{i} ⊗_{D}^{∧} Ω_{D(n)/D}^{∗}. Thus it suffices to show the natural map

D → (D(n) → Ω_{D(n)/D}^{1} → Ω_{D(n)/D}^{2} → ⋯)

is a quasi-isomorphism. This can be checked explicitly as D(n) is a divided power polynomial algebra over D (see [Ber74, Lemma V.2.1.2]). □

Next, we identify the “rows” of the de Rham complex of M(•).

Lemma 2.14. The complex

M ⊗_{D}^{∧} Ω_{D}^{i} → M(1) ⊗_{D(1)}^{∧} Ω_{D(1)}^{i} → M(2) ⊗_{D(2)}^{∧} Ω_{D(2)}^{i} ⋯

computes RΓ(S/Σ, F ⊗_{O_{S/Σ}} Ω_{S/Σ}^{i}).

Proof. By Lemma 2.7, we have

M ⊗_{D}^{∧} Ω_{D}^{i} ≃ lim_{e≥N} (M/p^{e} ⊗_{D/p^{e}D} Ω_{D/p^{e}D}^{i}) ≃ lim_{e≥N} ((F ⊗_{O_{S/Σ}} Ω_{S/Σ}^{i})((S, Spec(D/p^{e}), δ))),

and similarly for the terms over D(n). The claim now follows from Lemma 2.4, the fact that Ω_{S/Σ}^{i} is quasi-coherent, and the fact that the transition maps M/p^{e+1}M ⊗_{D/p^{e+1}D} Ω_{D/p^{e+1}D}^{i} → M/p^{e}M ⊗_{D/p^{e}D} Ω_{D/p^{e}D}^{i} are surjective. □

To finish the proof of Theorem 2.12, we need an acyclicity result about the “rows” of the de Rham complex of M(•). First, we handle the case M = D, i.e., when F = O_{S/Σ}.

Lemma 2.15. The complex

Ω_{D}^{1} → Ω_{D(1)}^{1} → Ω_{D(2)}^{1} → ⋯

is homotopic to zero as a D(•)-cosimplicial module.

Proof. This complex is equal to the base change of the cosimplicial module

M_{∗} = (Ω_{P}^{1} → Ω_{P ⊗P}^{1} → Ω_{P ⊗P ⊗P}^{1} → ⋯)

via the cosimplicial ring map P^{⊗n+1} → D(n). Hence it suffices to show that the cosimplicial module M_{∗} is homotopic to zero. Let P = Z_{p}[{x_{i}}_{i∈I}]. Then P^{⊗n+1} is the polynomial algebra on the elements

x_{i}(e) = 1 ⊗ ⋯ ⊗ x_{i} ⊗ ⋯ ⊗ 1

with x_{i} in the eth slot. The modules of the complex are free on the generators dx_{i}(e). Note that if f : [n] → [m] is a map then we see that

M_{∗}(f)(dx_{i}(e)) = dx_{i}(f(e))

Hence we see that M_{∗} is a direct sum of copies of Example 2.16 indexed by I, and we win. □

Example 2.16. Suppose that A_{∗} is any cosimplicial ring. Consider the cosimplicial module M_{∗} defined by the rule

M_{n} = ⨁_{i=0,...,n} A_{n}e_{i}

For a map f : [n] → [m] define M_{∗}(f) : M_{n} → M_{m} to be the unique A_{∗}(f)-linear map which maps e_{i} to e_{f(i)}. We claim the identity on M_{∗} is homotopic to 0. Namely, a homotopy is given by a map of cosimplicial modules

h : M_{∗} ⟶ Hom(Δ[1], M_{∗})

where Δ[1] denote the simplicial set whose set of n-simplices is Mor([n], [1]), see [Mey90]. Let α_{j}^{n} : [n] → [1] be defined by α_{j}^{n}(i) = 0 ⇔ i < j. Then we define h in degree n by the rule

h_{n}(e_{i})(α_{j}^{n}) = \begin{cases}e_{i} & if & i < j \\ 0 & else \end{cases}

We first check h is a morphism of cosimplicial modules. Namely, for f : [n] → [m] we will show that

(2.16.1)  h_{m} ∘ M_{∗}(f) = Hom(Δ[1], M_{∗})(f) ∘ h_{n}

This is equivalent to saying that the left hand side of (2.16.1) evaluted at e_{i} is given by

h(e_{f(i)})(α_{j}^{m}) = \begin{cases}e_{f(i)} & if & f(i) < j \\ 0 & else \end{cases}

Note that α_{j}^{n} ∘ f = α_{j′}^{m} where 0 ≤ j′ ≤ n is such that f(a) < j if and only if a < j′. Thus the right hand side of (2.16.1) evaluted at e_{i} is given by

M_{∗}(f)(h(e_{i})(α_{j}^{m} ∘ f) = M_{∗}(f)(h(e_{i})(α_{j′}^{n})) = \begin{cases}e_{f(i)} & if & i < j′ \\ 0 & else \end{cases}

It follows from our description of j′ that the two answers are equal. Hence h is a map of cosimplicial modules. Let 0 : Δ[0] → Δ[1] and 1 : Δ[0] → Δ[1] be the obvious maps, and denote ev_{0}, ev_{1} : Hom(Δ[1], M_{∗}) → M_{∗} the corresponding evaluation maps. The reader verifies readily that the the compositions

ev_{0} ∘ h, ev_{1} ∘ h : M_{∗} ⟶ M_{∗}

are 0 and 1 respectively, whence h is the desired homotopy between 0 and 1.

We now extend Lemma 2.15 to allow non-trivial coefficients.

Lemma 2.17. For all i > 0 cosimplicial module

M ⊗_{D}^{∧} Ω_{D}^{i} → M(1) ⊗_{D(1)}^{∧} Ω_{D(1)}^{i} → M(2) ⊗_{D(2)}^{∧} Ω_{D(2)}^{i} ⋯

is homotopy equivalent to zero.

Proof. The cosimplicial D(•)-module above is a (termwise) completed tensor product of the cosimplicial D(•)- modules M(•) and Ω_{D(•)}^{i}. Lemma 2.15 shows that Ω_{D(•)}^{1} is homotopy equivalent to zero as a cosimplicial D(•)- module. The claim now follows as the following three operations preserve the property of being homotopy equivalent to 0 for cosimplicial D(•)-modules: termwise application of ∧^{i}, tensoring with another cosimplicial D(•)-module, and termwise p-adic completion. □

The material above gives a rather pleasing proof that crystalline cohomology is computed by the de Rham complex:

Proof of Theorem 2.12. We look at the first quadrant double complex M^{•,•} with terms

M^{n,m} = M(n) ⊗_{D(n)}^{∧} Ω_{D(n)}^{m}.

The horizontal differentials are given determined by the Čech-Alexander complex, while the vertical ones are given by the de Rham complex. By Lemma 2.13, each column complex M^{n,•} is quasi-isomorphic to de Rham complex M ⊗_{D}^{∧} Ω_{D}^{•}. Hence H^{m}(M^{n,•}) is independent of n and the differentials are

H^{m}(M^{0,•}) ⟶^{0} H^{m}(M^{1,•}) ⟶^{1} H^{m}(M^{2,•}) ⟶^{0} H^{m}(M^{3,•}) ⟶^{1} ⋯

We conclude that Tot(M^{•,•}) computes the cohomology of the de Rham complex M ⊗_{D}^{∧} Ω_{D}^{∗} by the first spectral sequence associated to the double complex. On the other hand, Lemma 2.4 shows that the “row” complex M^{•,0} computes the cohomology of F. Hence if we can show that each M^{•,m} for m > 0 is acyclic, then we’re done by the second spectral sequence. The desired vanishing now follows from Lemma 2.17. □

Remark 2.18. Lemma 2.8and Theorem 2.12remain valid when D is taken to be the p-adic completion of the divided power envelope of a surjection P → A with P any smooth Z_{p}-algebra. For Lemma 2.8, this was discussed in Remark 2.9. Thus, the only non-obvious point now is whether an analogue of Lemma 2.15 is valid. However, at least Zariski locally on Spec(P), there is an étale map F → P with F a polynomial Z_{p}-algebra. Thus, the cotangent bundle of P (and hence that of D) is obtained by base change from that of F, so the required claim follows from the proof of Lemma 2.15. This shows that the assertion of Theorem 2.12 is true Zariski locally on S, and hence globally by the Čech spectral sequence for a suitable affine cover.

Remark 2.19. Let Σ_{e} = Spec(Z/p^{e}), and let S be an affine Σ_{e}-scheme. One can define the crystalline site (S/Σ_{e})_{cris} and crystals in the obvious way. The arguments given in this section work mutatis mutandis to show that the cohomology RΓ(S/Σ_{e}, F) of a crystal F of quasi-coherent O_{S/Σ_{e}}-modules is computed by the de Rham complex

M_{e} → M_{e} ⊗_{D/p^{e}} Ω_{D/p^{e}}^{1} → M_{e} ⊗_{D/p^{e}} Ω_{D/p^{e}}^{2} ⋯,

where D/p^{e} is as in Notation 2.1, and M_{e} = F((S, Spec(D/p^{e}), δ)) is the D/p^{e}-module that is the value of the crystal F on Spec(D/p^{e}), equipped with the integrable connection as in (1.0.1).

3. Global analogues

Our goal in this section is to prove a global analogue (Theorem 3.6) of the results of §2, and deduce some geometric consequences (Corollaries 3.8 and 3.10). In order to do so, we first conceptualize the work done in §2 as a vanishing result on arbitrary schemes in Theorem 3.2; this formulation gives us direct access to certain globally defined maps, which are then used to effortlessly reduce global statements to local ones.

3.1. A vanishing statement. Our vanishing result is formulated terms of the “change of topology” map relating the crystalline site to the Zariski site, whose construction we recall first. Let f : S → Σ be a map with p locally nilpotent on S. There is a morphism of ringed topoi

u_{S/Σ} : (Shv((S/Σ)_{cris}), O_{S/Σ}) → (Shv(S_{zar}), f^{−1}O_{Σ})

characterised by the formula

u_{S/Σ}^{∗}(F)((U, T, δ)) = F (U)

for any sheaf F ∈ Shv(S_{zar}) and object (U, T, δ) ∈ (S/Σ)_{cris} (see [Ber74, §III.3.2]). The associated pushforward Ru_{S/Σ} : D(O_{S/Σ}) → D(f^{−1}O_{Σ}) is a localised version of crystalline cohomology, i.e., for any U ∈ S_{zar}, we have

RΓ(U, Ru_{S/Σ}(F)) ≃ RΓ(U/Σ, F);

see [Ber74, Corollary III.3.2.4] for the corresponding statement at the level of cohomology groups. With this language, our main result is the following somewhat surprising theorem.

Theorem 3.2. Let S be a scheme over Σ such that p is locally nilpotent on S. Let F be a crystal in quasi-coherent O_{S/Σ}-modules. The truncation map of complexes

(F → F ⊗_{O_{S/Σ}} Ω_{S/Σ}^{1} → F ⊗_{O_{S/Σ}} Ω_{S/Σ}^{2} → ⋯) ⟶ F[0],

while not a quasi-isomorphism, becomes a quasi-isomorphism after applying Ru_{S/Σ}. In fact, for any i > 0, we have

Ru_{S/Σ}(F ⊗_{O_{S/Σ}} Ω_{S/Σ}^{i}) = 0.

Proof. This follows from the vanishing of the cohomology of the sheaves F ⊗_{O_{S/Σ}} Ω_{S/Σ}^{i} over affines for i > 0, see Lemmas 2.14 and 2.17. □

Remark 3.3. The proof of Theorem 3.2 shows that the conclusion Ru_{S/Σ}(F ⊗_{O_{S}} Ω_{S/Σ}^{i}) = 0 for i > 0 is true for any quasi-coherent O_{S/Σ}-module which, locally on S, satisfies the R^{1} lim_{e≥N} vanishing condition of Lemma 2.4. This applies to the following non-crystals: Ω_{S/Σ}^{i} for all i, and any O_{S/Σ}-module of the form u_{S/Σ}^{∗}F, where F is an O_{S}-module on S_{zar}. In particular, it applies to the sheaf O_{S} on (S/Σ)_{cris} defined by O_{S}((U, T, δ)) = Γ(U, O_{U}).

3.4. Global results. We now explain how to deduce global consequences from Theorem 3.2, such as the identification of crystalline cohomology with de Rham cohomology. First, we establish notation used in this section.

Notation 3.5. Let S be a Σ-scheme such that p^{N} = 0 on S. Assume there is a closed immersion i : S → X of Σschemes with X finitely presented and smooth over Σ. For each e ≥ N, set D_{e} to be the divided power hull of the map O_{X}/p^{e} → O_{S}. Each D_{e} is supported on S, and letting e vary defines a p-adic formal scheme T with underlying space S and structure sheaf lim_{e≥N} D_{e}; there is an equivalence between the category of quasi-coherent O_{T} -modules and the category of compatible systems of D_{e}-modules on S (with compatibilities as in Lemma 2.7), and we only use T as a tool for talking about such compatible systems. The sheaves D_{e} define (honest) subschemes T_{e} = Spec(D_{e}) ⊂ T containing S. The quasi-compactness of S then gives objects (S, T_{e}, δ) of (S/Σ)_{cris}. Following our conventions, let Ω_{T_{e}}^{i} be the pullback of the corresponding sheaf on X, and set Ω_{T}^{i} to be the result of glueing the Ω_{T_{e}}^{i}.

Let S, T_{e}, and T be as in Notation 3.5, and let F be a crystal in quasi-coherent O_{S/Σ}-modules. Restricting F to (S, T_{e}, δ) defines quasi-coherent O_{T_{e}}-modules F_{T_{e}} for e ≥ N, and hence a quasi-coherent O_{T} -module M by glueing. The integrable connections F_{T_{e}} → F_{T_{e}} ⊗_{O_{Te}} Ω_{T_{e}}^{1} coming from (1.0.1) glue to give an integrable connection

∇ : M ⟶ M ⊗_{O_{T}} Ω_{T}^{1} ,

which then defines a de Rham complex on T.

Theorem 3.6. Let S → X, T, F, and M be as in Notation 3.5 and the following discussion. The hypercohomology on T of the complex

M → M ⊗_{O_{T}} Ω_{T}^{1} → M ⊗_{O_{T}} Ω_{T}^{2} → ⋯

computes RΓ(S/Σ, F).

Proof. First, we construct the map. By basic formal scheme theory, we have a formula

R lim_{e≥N} RΓ((S, T_{e}, δ), F ⊗_{O_{S/Σ}} Ω_{S/Σ}^{i}) = R lim_{e≥N} RΓ(T_{e}, F_{T_{e}} ⊗_{O_{Te}} Ω_{T_{e}}^{i})
= RΓ(T, R lim_{e≥N} (F_{T_{e}} ⊗_{O_{Te}} Ω_{T_{e}}^{i}))
= RΓ(T, M ⊗_{O_{T}} Ω_{T}^{i})

for each i ≥ 0. Here the first equality follows from the definition of cohomology in the crystalline site; the second equality follows from the identification of the (derived functors of the) composite functors

Ab(C)^{N} →^{Γ(T,−)} Ab^{N} →^{lim} Ab and Ab(C)^{N} →^{lim} Ab(C) →^{Γ(T,−)} Ab;

the last equality follows from the vanishing of

R^{j} lim_{e} (F_{e} ⊗_{O_{Te}} Ω_{T_{e}}^{i})

for all j > 0, which follows from the vanishing of higher quasi-coherent sheaf cohomology for affines. Using this formula, and applying

RΓ(∗, −) ⟶ R lim_{e≥N} RΓ((S, T_{e}, δ), −)

to the morphism in Theorem 3.2, gives the desired map

RΓ(S/Σ, F) → RΓ(T, M → M ⊗_{O_{T}} Ω_{T}^{1} → M ⊗_{O_{T}} Ω_{T}^{2} → ⋯).

Moreover, this map is an isomorphism for affine S by Theorem 2.12 (see Remark 2.18) and functorial in S. The Čech spectral sequence for an affine open cover then immediately implies the claim for X is quasi-compact and separated (as the E_{2} terms involve cohomology of affines). For an X only assumed to be quasi-compact compact and quasiseparated, another application of the spectral sequence for an affine open cover finishes the proof (as the E_{2} terms involve cohomology on quasi-compact and separated schemes). □

Remark 3.7. The arguments that go into proving Theorem 3.6 also apply mutatis mutandis to reprove Grothendieck’s theorem from [Gro68]: if S is a variety over C, and S ⊂ X is a closed immersion into a smooth variety, and T denotes the formal completion of X along S, then the cohomology of the structure sheaf on the infinitesimal site (S/Spec(C))_{inf} is computed by the hypercohomology on S of the de Rham complex of T (defined suitably). The only essential change is that the proof of Lemma 2.13, which relies on the vanishing of the higher de Rham cohomology of a divided power polynomial algebra, must be replaced by its formal analogue, i.e., the vanishing of the higher de Rham cohomology of a formal power series ring in characteristic 0.

In certain situations, Theorem 3.6 can be algebraised to get a statement about classical schemes. For example:

Corollary 3.8. Let f : X → Σ be a proper smooth morphism, and set S = X ×_{Σ} Spec(F_{p}). Then the hypercohomology of the de Rham complex

O_{X} → Ω_{X/Σ}^{1} → Ω_{X/Σ}^{2} → ⋯,

computes RΓ(S/Σ, O_{S/Σ}). In particular, the de Rham cohomology of X → Σ is determined functorially by the fibre S ↪ X of f, and thus admits a Frobenius action.

Proof. This follows from Theorem 3.6 and the formal functions theorem as T is just the p-adic completion of X: the ideal ker(O_{X} → O_{S}) = (p) ⊂ O_{X} already has specified divided powers, so O_{X}/p^{e} = D_{e} for any e. □

Remark 3.9. One can upgrade Corollary 3.8to a statement that incorporates coefficients as follows. There is an equivalence of categories between crystals in quasi-coherent sheaves on (S/Σ) and quasi-coherent sheaves on X equipped with a flat connection relative to Σ (see Lemma 2.8 and Corollary 2.10). This equivalence respects cohomology, i.e., the crystalline cohomology of a crystal in quasi-coherent sheaves on S/Σ is computed by the de Rham cohomology of the corresponding quasi-coherent sheaf with flat connection on X.

We conclude with a brief discussion of the base change behaviour. For an Z/p^{N}-scheme S as above, a natural question is whether crystalline cohomology relative to Σ_{e} = Spec(Z/p^{e}) (for e ≥ N) can be recovered from crystalline cohomology relative to Σ via (derived) base change along Z_{p} → Z/p^{e}. In general, the answer is “no,” see Example 3.11. However, under suitable flatness conditions, the answer is “yes”:

Corollary 3.10. Let S → X, T and F be as in Notation 3.5 and the following discussion. Assume that for each e ≥ N, the O_{T_{e}}-module F_{T_{e}} is flat over Z/p^{e}. Then for each e ≥ N, we have a base change isomorphism

Z/p^{e} ⊗_{Z}^{L} RΓ(S/Σ, F) ≃ RΓ(S/Σ_{e}, F|_{S/Σ_{e}}),

where F|_{S/Σ_{e}} denotes the restriction of F along (S/Σ_{e})_{cris} ⊂ (S/Σ)_{cris}.

Proof. By Theorem 3.6, we have

RΓ(S/Σ, F) ≃ RΓ(T, M → M ⊗_{O_{T}} Ω_{T}^{1} → M ⊗_{O_{T}} Ω_{T}^{2} → ⋯).

Note that since each Ω_{T}^{i} is flat over O_{T}, the tensor products M ⊗_{O_{T}} Ω_{T}^{i} appearing in the de Rham complex above are automatically derived tensor products. Applying Z/p^{e} ⊗_{Z}^{L} − (and observing that this operation commutes with applying RΓ(T, −)) then shows that Z/p^{e} ⊗_{Z}^{L} RΓ(S/Σ, F) is computed as the hypercohomology on T of the complex

K := (Z/p^{e} ⊗_{Z}^{L} M → Z/p^{e} ⊗_{Z}^{L} M ⊗_{O_{T}} Ω_{T}^{1} → Z/p^{e} ⊗_{Z}^{L} M ⊗_{O_{T}} Ω_{T}^{2} → ⋯).

The flatness assumption on F implies that M is flat over Z_{p}. Lemma 2.7 then shows that

Z/p^{e} ⊗_{Z}^{L} M ≃ Z/p^{e} ⊗_{Z} M ≃ F_{T_{e}},

and so

Z/p^{e} ⊗_{Z}^{L} M ⊗_{O_{T}} Ω_{T}^{i} ≃ F_{T_{e}} ⊗_{O_{Te}} Ω_{T_{e}}^{i}.

In other words, the complex K of sheaves appearing above is identified with the de Rham complex of the O_{T_{e}}-module F_{T_{e}}. The claim now follows from the modulo p^{e} version of Theorem 3.6 (see also Remark 2.19). □

The hypotheses of Corollary 3.10 are satisfied, for example, when S is a flat local complete intersection over Z/p^{N}, and F is a crystal in locally free quasi-coherent O_{S/Σ}-modules; the smooth case is discussed in [Ber74, §V.3.5]. Moreover, there are extremely simple examples illustrating the sharpness of the lci assumption:

Example 3.11. Let S = F_{p}[x, y]/(x^{2}, xy, y^{2}). In [BO83, Appendix (A.2)], Berthelot-Ogus exhibit a non-zero ptorsion class τ ∈ H^{0}(S/Σ, O_{S/Σ}) by constructing a non-zero p-torsion ∇-horizontal element of the p-adically completed divided power envelope of the natural surjection Z_{p}[x, y] → F_{p}[x, y]/(x^{2}, xy, y^{2}). Via the exact triangle

RΓ(S/Σ, O_{S/Σ}) →^{p} RΓ(S/Σ, O_{S/Σ}) → RΓ(S/Σ, O_{S/Σ}) ⊗_{Z}^{L} Z/p,

τ defines a non-zero class in H^{−1}(RΓ(S/Σ, O_{S/Σ})⊗_{Z}^{L}Z/p). In particular, RΓ(S/Σ, O_{S/Σ})⊗_{Z}^{L}Z/p has cohomology in negative degrees, so it cannot be equivalent to RΓ(S/Σ_{1}, O_{S/Σ_{1}}) (or the cohomology of any sheaf on any site).

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