CRYSTALS AND DE RHAM-WITT CONNECTIONS

SPENCER BLOCH

Abstract. For a smooth variety X over a perfect field k in characteristic p, an equivalence of category is established between the category of crystals on X/W (k) and the category of p-adically nilpotent, integrable de Rham-Witt connections.

“Let me make one thing crystal clear...” R. Nixon.

1. Introduction

Let X be a smooth variety over a perfect field k of characteristic p > 0. Let W (k) denote the ring of p-Witt vectors over k. It is a complete discrete valuation ring with W (k)/pW (k) = k. Recall ([I1], [I2]) one has defined a pro-sheaf of W (k)-algebras

(1.1)  W_{X} = {W_{n,X}}_{n=1,2,...}

where W_{n,X} is just the sheafified functor of p-Witt vectors of length n, applied to O_{X}. One has a surjection of algebras W_{n} ↠ W_{1}, (r_{1}, r_{2}, …) ↦ r_{1}, and the formula

(0, r_{2}, …) · (0, s_{1}, …) = p(0, r_{1}s_{1}, …)

gives a canonical divided power structure on the kernel. It follows that for U = Spec (R) ⊂ X an affine open, the embedding

U ↪ Spec (Γ(U, W_{n,X}))

with the above divided powers represents an object in the crystalline site of X/W. By definition, a crystal 𝔼 on X/W is a sheaf of O_{X,crys}- modules on the crystalline site, where O_{X,crys}(U ↪ V) := Γ(V, O_{V}). The rule which associates to 𝔼 the Zariski sheaf

(1.2)  E := {U ↦ 𝔼(U ↪ Spec (Γ(U, W_{n,X})))_{n=1,2,...}}

defines a functor from crystals to W_{X}-modules.

Date: October 8, 2001.

1991 Mathematics Subject Classification. Primary 14C40 19E20 14C99.

In fact, E has more structure. Quite generally, if U ↪ V is an object in the crystalline site and if V^{(2)} ⊂ V ×_{Spec W (k)} V is the first infinitesimal neighborhood of the diagonal, we have a diagram

\begin{matrix}U & ⟶ & V^{(2)} \\ ∥ & ^{pr_{1}}↓ & ↓^{pr_{2}} \\ U & ⟶ & V \end{matrix}

From the crystalline property we deduce an isomorphism

(1.3)  pr_{1}^{∗}𝔼(U ↪ V) ≅ pr_{2}^{∗}𝔼(U ↪ V),

i.e. a connection on the O_{V} -module. This connection is integrable ([Be]).

Recall the de Rham-Witt complex is a complex of pro-sheaves for the Zariski topology on X

(1.4)  {W_{n,X} →^{d} W_{n}Ω_{X}^{1} → ⋯ → W_{n}Ω_{X}^{m}}_{n=1,2,...},

where m = dim X. At level n, the complex is a quotient of the de Rham complex of the sheaf of rings W_{n,X}. In particular, the integrable connection on the W -module E deduced from (1.3) permits one to couple E to W Ω^{•}:

(1.5)  E ⟶^{∇} E ⊗_{W} W Ω_{X}^{1} ⟶^{∇} E ⊗_{W} W Ω_{X}^{2} ⟶^{∇} …

We refer to ∇ as a de Rham-Witt connection on E. More generally, for a W -module E, a de Rham-Witt connection on E will be a map ∇ : E → E ⊗_{W} W Ω_{X}^{1} satisfying the Leibniz rule ∇(we) = w∇(e) + e ⊗ dw. The connection is integrable if ∇^{2} = 0 as in (1.5). Connections arising from crystals are integrable.

The functor 𝔼 ↦ (E, ∇) was studied in [E]. Let u_{n} : (X/W_{n})_{crys} → X_{zar} be the natural map. Assume the crystal 𝔼 is locally free as an O_{X,crys}-module. (This is the only case to be considered in the present note.) The main result in (loc. cit., thm. 2.1) is an isomorphism in the derived category of Zariski sheaves on X:

(1.6)  ℝu_{n,∗}(𝔼) ≅ {E ⊗ W_{n} → E ⊗ W_{n}Ω_{X}^{1} → …}.

Taking ℍ^{0}, it follows easily that the functor 𝔼 ↦ (E, ∇) is fully faithful. Indeed, for locally free crystals 𝔼_{i}, i = 1, 2

ℝu_{n,∗}\underline{Hom}(𝔼_{1}, 𝔼_{2}) ≅ \{E_{1}^{∨} ⊗ E_{2} ⊗ W_{n} → E_{1}^{∨} ⊗ E_{2} ⊗ W_{n}Ω^{1} → …\} so

Hom_{crys}(𝔼_{1}, 𝔼_{2}) ≅ \underleftarrow{lim}_{n} ℍ^{0}ℝu_{n,∗}(𝔼_{1}^{∨} ⊗ 𝔼_{2}) ≅

\underleftarrow{lim}_{n} ker (H^{0}(X, E_{1}^{∨} ⊗ E_{2} ⊗ W_{n}) → H^{0}(X, E_{1}^{∨} ⊗ E_{2} ⊗ W_{n}Ω^{1})) ≅

\underleftarrow{lim}_{n} Hom_{∇}(E_{1} ⊗ W_{n}, E_{2} ⊗ W_{n}) ≅ Hom_{∇}(E_{1}, E_{2}).

The main result in the following is that this functor is an equivalence of category. More precisely, (E, ∇) is said to be quasi-nilpotent if the connection (note W_{1}Ω_{X}^{•} = Ω_{X}^{•} is the de Rham complex over X) E ⊗ O_{X} → E ⊗ Ω_{X}^{1} is quasi-nilpotent in the sense of ([Be], Def. 4.10 and 4.14).

Theorem 1.1. The functor 𝔼 ↦ (E, ∇) defines an equivalence of category between the category of locally-free crystals on X and the category of locally-free W_{X}-modules E with a quasi-nilpotent, integrable connection ∇.

The Witt vector sheaf W_{X} has a canonical Frobenius endomorphism σ given by raising coordinates to the p-th power. The differential dσ : W Ω_{X}^{1} → W Ω_{X}^{1} is defined. In fact dσ = pF where F : W Ω_{X}^{1} → W Ω_{X}^{1} is the Frobenius endomorphism [I2]. Let (E, ∇) be as above. A Frobenius structure on (E, ∇) is by definition a map Φ : σ^{∗}E → E which is horizontal for the connection. When 𝔼 is an F -crystal [K], then (E, ∇) has a Frobenius structure. As a consequence of theorem 1.1 one gets

Corollary 1.2. The functor 𝔼 ↦ (E, ∇) defines an equivalence of category between the category of locally-free F -crystals on X and the category of locally-free W_{X}-modules E with a quasi-nilpotent, integrable connection ∇ and a Frobenius structure Φ : σ^{∗}E → E.

The de Rham-Witt complex is globally defined and functorial in X. Heretofore, much of the work on crystals has involved choosing local liftings of X and Frobenius, and then studying connections on the local lifting. My hope is that the possibility of doing this canonically will simplify and clarify the picture.

It seems natural to ask whether one has some sort of D-module interpretation of de Rham-Witt connections. the naive idea of defining derivations to be Hom_{W} (W Ω^{1}, W) does not seem to work, but perhaps there is something more clever.

The author acknowledges considerable help from conversations and correspondence with Pierre Berthelot, Luc Illusie, Kirti Joshi, Minhyong Kim, and Arthur Ogus.

2. The Structure of W Ω^{•} over a Polynomial Ring

Much of the material in this section is taken from [I2]. Let A = k[T_{1}, …, T_{n}] be a polynomial ring where k is a perfect field as above. Let K be the quotient field of W (k), and let C := ⋃_{r≥0} K[T_{1}^{p^{−r}}, …, T_{n}^{p^{−r}}]. An element x ∈ Ω_{C/K}^{m} can be written uniquely in the form

(2.1)  x = ∑_{1≤i_{1}<...<i_{m}≤n} a_{i_{1},...,i_{m}}(T)d log(T_{i_{1}}) ∧ … ∧ d log(T_{i_{m}})

where a_{I} ∈ C is divisible by (T_{i_{1}} ⋯ T_{i_{m}})^{p^{−r}} for some r ≥ 0. We say that such an x is integral if the a_{I}(T) have coefficients in W (k) ⊂ K. Define a subcomplex E^{•} ⊂ Ω_{C}^{•} by

(2.2)  E^{i} = {x ∈ Ω_{C}^{i} | x, dx integral}.

For example, since dT^{p^{−r}} = p^{−r}T^{p^{−r}}d log(T), it follows that

(2.3)  E^{0} = ⋃_{r≥0} p^{r}W [T_{1}^{p^{−r}}, …, T_{n}^{p^{−r}}].

The subcomplex E^{•} is compatible with the natural grading on Ω_{C}^{•} where the I = (i_{1}, …, i_{n})-graded piece is spanned as a K-vector space by T_{1}^{i_{1}} · T_{1}^{i_{2}} ⋯ T_{n}^{i_{n}}d log(T_{j_{1}}) ∧ ⋯ ∧ d log(T_{j_{r}}) for varying {j_{1}, …, j_{r}}. We write E^{•}(I) for the corresponding grading.

The main point ([I1], §2, p. 550) is that W Ω_{A}^{•} is a suitable completion of E^{•}. Indeed, define an operator F on Ω_{C}^{•} from the ring automorphism F^{∗} : C ≅ C, F^{∗}(T) = T^{p}, F^{∗}|_{W (k)} = frobenius via the formula

F ( ∑ a_{I}(T)d log(T_{j_{1}}) ∧ ⋯ ∧ d log(T_{j_{r}})) =

∑ F^{∗}(a_{I})d log(T_{j_{1}}) ∧ ⋯ ∧ d log(T_{j_{r}}).

Then F stabilizes E^{•}, as does V := pF^{−1}. Define fil^{r}E^{i} := V^{r}E^{i} + d(V^{r}E^{i−1}). One shows W_{r}Ω_{A}^{i} = E^{i}/fil^{r}E^{i}.

It will be convenient to write

(2.4)  E^{i} := E^{i}/pE^{i},

so for example E^{0} is an augmented A-algebra

(2.5)  E^{0} = A ⊕ J .

Here J ⊂ E^{0} is generated over k by the classes p^{s}T^{I} where I = (i_{1}, …, i_{n}) with i_{j} ∈ ℤ[1/p], i_{j} ≥ 0, s := max_{j}{−ord_{p}(i_{j})} > 0.

Lemma 2.1. There is a split-exact sequence

0 → J →^{d} E^{1}/J E^{1} → Ω_{A}^{1} → 0

Proof. Here d is induced from the differential E^{0} → E^{1}. The definition of the maps and the splitting are clear. It remains to show exactness.

One has a grading E^{•} = ⊕E^{•}(I) as above.

We claim

(2.6)  E^{1} = Ω_{A}^{1} + J E^{1} + dJ .

To see this, let T^{I} ∑_{j} b_{j}d log(T_{j}) ∈ E^{1}(I) represent an element in E^{1}(I). In other words, writing I = (i_{1}, …, i_{n}), we have b_{j}, i_{r}b_{j} − i_{j}b_{r} ∈ W (k). Let s = s(I) := max(−ord_{p}(i_{j})) > 0 as above, and write ι = p^{s}(i_{1}, …, i_{n}). Let f := min(ord_{p}(b_{j})), and write β = p^{−f}(b_{1}, …, b_{n}). Write ῑ and β̄ for the images of these vectors mod p in k^{n}. By construction, these vectors are non-zero. We have

(2.7)  β̄ ∧ ῑ = p^{s−f}(…, b_{j}i_{r} − b_{r}i_{j}, …)

If f < s, we conclude β ≡ cι mod p for some c ∈ W (k). This implies

(2.8)  T^{I} ∑_{j} b_{j}d log(T_{j}) ≡ p^{f}c · d(p^{s}T^{I}) mod p^{f+1}

(Note the RHS lies in dJ.) Continuing in this way, we may assume p^{s}|b_{j} for all j, which implies that

(2.9) T^{I} ∑_{j} b_{j}d log(T_{j}) = ∑_{j} p^{s}T^{I−{i_{j}}}(p^{−s}b_{j}T_{j}^{i_{j}}d log(T_{j})) ∈ J E^{1}.

The claim (2.6) follows. In particular, we see that Ω_{E^{0}}^{1} ↠ E^{1}, from which Ω_{A}^{1} ⊕ J ↠ E^{1}/J E^{1}.

It remains to show the map d in the exact sequence is injective. This can be done one graded piece at a time, so it suffices to show d(p^{s}T^{I}) ≢ 0 mod J E^{1}. By the assumption on s, d(p^{s}T^{I}) = T^{I} ∑_{j} p^{s}i_{j}d log(T_{j}) and p^{s}i_{j} is a unit for some j. On the other hand, elements in the I-graded piece of J E^{1} are of the form T^{I} ∑ c_{j}d log(T_{j}) where all the c_{j} ≡ 0 mod p (cf. (2.9)). It follows that d : J ↪ E^{1}/J E^{1}, proving the lemma. □

In what follows we shall need detailed information about the de Rham-Witt complex. For a careful exposition of the de Rham-Witt complex, see [I1], §1, 2.

Corollary 2.2. Let W(A) = W (A)/pW (A) and let

I_{A} := ker(W(A) → A); WΩ_{A}^{1} = W Ω_{A}^{1}/pW Ω_{A}^{1}

Then there is a split-exact sequence

(2.10)  0 → I_{A} →^{d} WΩ_{A}^{1}/I_{A}WΩ_{A}^{1} → Ω_{A}^{1} → 0.

Proof. Using the lemma and the identity

W_{r}Ω_{A}^{i} = E^{i}/(V^{n}E^{i} + d(V^{n}E^{i−1})),

one reduces to showing

(V^{n}E^{1} + J E^{1}) ∩ dJ = (0).

A non-zero element in dJ of graded degree I is represented by an element T^{I} ∑_{j} a_{j}d log(T_{j}) ∈ E^{1} where one at least of the a_{j} ∈ W (k) is a unit. On the other hand, elements in V^{n}E^{1} + J E^{1} are represented by elements where all the a_{j} are divisible by p. (For example, V (T^{I} ∑_{j} b_{j}d log(T_{j})) = pT^{I/p} ∑ b_{j}d log(T_{j})).) The corollary follows from this. □

Lemma 2.3. The composition

J E^{1} →^{d} E^{2} → E^{2}/(Ω_{A}^{2} + J E^{2} + V E^{2})

is injective.

Proof. We can work with I-graded pieces for I non-integral and ignore Ω_{A}^{2}. The map sends

(2.11) ∑ τ_{μ}w^{μ} ↦ ∑ dτ_{μ} ∧ w^{μ} mod J E^{2}; τ_{μ} ∈ J , w^{μ} ∈ E^{1}.

If we lift elements of J E^{2} + V E^{2} to E^{2} and expand as

T^{I} ∑_{q,r} c_{q,r}d log(T_{q}) ∧ d log(T_{r})

we find all coefficients c_{q,r} ≡ 0 mod p. (Indeed, J is generated by elements p^{s}T^{I} for s = s(I) > 0, and V = pF^{∗−1}.) On the other hand, if we lift ∑ τ_{μ}w^{μ} to an element ω ∈ E^{1} and we expand dω = T^{I} ∑_{q,r} C_{q,r}d log(T_{q}) ∧ d log(T_{r}), then this class is trivial in E^{2} if and only if the C_{q,r} are divisible by p. Indeed, the condition for a form to lie in E^{•} is that both it and its differential have integral coefficients. Since dω is closed and since the E^{•} have no p-torsion, the assertion follows. But notice the expansion of ω ∈ E^{1} also has all coefficients divisible by p because the τ_{μ} do, so if the coefficients of dω are divisible by p then ω/p is integral so ω ≡ 0 mod p. □

3. The Structure of W Ω_{R}^{•} for general R

Let R be a commutative ring in which every rational prime ℓ ≠ p is invertible. We write W (R) for the ring of p-Witt vectors with coefficients in R. Recall [I1] an element a ∈ W (R) has Witt coordinates a = (a_{0}, a_{1}, …). The ghost coordinates w_{n}(a) := ∑_{j=0}^{n} p^{j}a_{j}^{p^{n−j}}, n = 0, 1, … define ring homomorphisms W (R) → R. The Frobenius F : W (R) → W (R) (resp. F_{n} : W_{n}(R) → W_{n−1}(R)) is a ring homomorphism satisfying w_{n} ∘ F = w_{n+1}. When pR = 0, F (a_{0}, a_{1}, …) = (a_{0}^{p}, a_{1}^{p}, …).

Lemma 3.1. Let S be a W (k)-algebra and assume S has no p-torsion. Let f : S → S be a ring homomorphism lifting the p-th power frobenius map on S/pS. Write W (S) = lim_{⟵} W_{n}(S) for the ring of p-Witt vectors on S. Then there exists a unique ring homomorphism ρ = ρ_{f} : S → W (S) such that ρ ∘ f = F ∘ ρ. One has w_{n} ∘ ρ = f^{n}, n = 0, 1, …

Proof. The referee suggests this lemma is due to Cartier. For a proof, see [L], VII, §4. □

With notation as above, let R = S/pS. We will be more interested in the composed map, which we also denote by ρ = ρ_{f}

(3.1)  ρ = ρ_{f} : S → W (S) → W (R)

(This composed map still depends on the choice of f lifting frobenius.) Write

(3.2) W(R) = lim_{⟵} W_{n}(R) = W (R)/pW (R); WΩ_{R}^{r} = W Ω_{R}^{r}/pW Ω_{R}^{r}.

Let I ⊂ W be the kernel of the projection W → R. Let

(3.3)  σ := ρ mod p : R → W

We have a split-exact sequence

(3.4)  0 → I → W →^{←^{σ}} R → 0

Note I^{2} = (0) because I = V W and V (x)V (y) = pV (xy). In particular, I has a natural R-module structure which is independent of the choice of σ. (Recall σ depends on the choice of f.) Let

(3.5)  d : I → WΩ_{R}^{1}

be induced by the de Rham-Witt differential.

Lemma 3.2. With notation as above, assume R is smooth over k. There is a split-exact sequence of R-modules

0 → I ⟶^{d} WΩ_{R}^{1}/IWΩ_{R}^{1} →^{←^{σ}} Ω_{R}^{1} → 0.

Proof. the maps are all defined. To check exactness, it suffices to work locally on Spec (R), so we may assume R is an étale algebra over a polynomial ring A := k[T_{1}, …, T_{n}]. One knows ([I1], prop. 1.14, p.

549) that in this case W_{m}Ω_{R}^{i} ≅ W_{m}Ω_{A}^{i} ⊗_{W_{m}(A)} W_{m}(R), which implies

(3.6) W_{m}Ω_{R}^{i}/I_{R}W_{m}Ω_{R}^{i} ≅ W_{m}Ω_{A}^{i} ⊗_{W_{m}(A)} R ≅ W_{m}Ω_{A}^{i}/I_{A}W_{m}Ω_{A}^{i} ⊗ R

Let I_{m,R} = ker(W_{m}(R) → R). Since R = R^{p}A and V (r^{p}a) = rV (a) we see that

(3.7)  I_{m,R} = I_{m,A} ⊗_{A} R.

Also, of course, Ω_{R}^{1} = Ω_{A}^{1}⊗_{A}R. combining these identities, exactness in lemma 3.2 reduces to the case R = A = k[T_{1}, …, T_{n}]. This is corollary 2.2. □

4. Descent modulo p

Let R be a smooth k-algebra as above, and let (M, ∇) be an integrable de Rham-Witt connection. We assume M ≅ W (R)^{r} as a W (R)-module. (For the complicated calculations which follow, it is convenient to work with W (R)-modules rather than pro-objects of W_{n}(R)-modules. Because our W -modules are assumed to be locally free in theorem 1.1, this is legitimate. I do not know what is true more generally.)

Write M = M/pM and let ∇ = ∇_{M} denote the corresponding connection on M as well. Define

(4.1)  (N_{0}, Ξ_{0}) := (M, ∇) ⊗_{W (R)} R = (M, ∇) ⊗_{W} R

where W = W (R)/pW (R).

Assume we are given a flat, p-adically complete W(k)-algebra S, an isomorphism of k-algebras R ≅ S/pS, and a lifting f : S → S of frobenius. Let σ : R → W be the corresponding map (3.3). Corresponding to σ, we have the direct sum decomposition from lemma 3.2

(4.2)  WΩ_{R}^{1}/IWΩ_{R}^{1} ≅ Ω_{R}^{1} ⊕ I.

Define a map ∇_{I} : M → N_{0} ⊗_{R} I to be the composition

(4.3) M →^{∇} M ⊗_{W} WΩ_{R}^{1} → M ⊗_{W} (WΩ_{R}^{1}/IWΩ_{R}^{1})

≅ M ⊗_{W} R ⊗_{R} (Ω_{R}^{1} ⊕ I) →^{proj} N_{0} ⊗_{R} I.

Let π : M → N_{0} = M/IM be the reduction map, and let

(4.4)  θ = (π, ∇_{I}) : M → N_{0} ⊕ (N_{0} ⊗_{R} I) ≅ N_{0} ⊗_{R,σ} W.

Proposition 4.1. The map θ is a horizontal isomorphism of connections

(M, ∇_{M}) ≅ (N_{0}, Ξ_{0}) ⊗_{R,σ} W.

Proof. Note first that θ is a homomorphism of W-modules. In other words, given x ∈ R, m ∈ M, and ι ∈ I, we have θ(σ(x)m) = xθ(m) and θ(ιm) = π(m) ⊗ ι. In particular, θ is the identity mod I, so it is an isomorphism.

We must show θ is compatible with connections, i.e. the diagram

(4.5)  \begin{matrix}M & ⟶^{θ} & N_{0} ⊗_{(R,σ)} W \\ ↓^{∇} & & ↓^{Ξ_{0}⊗1+1⊗d} \\ M ⊗_{W} WΩ_{R}^{1} & ⟶^{θ⊗1} & N_{0} ⊗_{(R,σ)} WΩ_{R}^{1} \end{matrix}

commutes.

Identify N_{0} = ker(∇_{I}) ⊂ M. Let {m_{i}} be an R-basis for the free module N_{0}. By assumption,

(4.6)  ∇(m_{ℓ}) = ∑_{i} m_{i} ⊗ η_{ℓ}^{i} + ∑_{i,μ} m_{i} ⊗ τ_{μ}w_{ℓ}^{i,μ}

where η_{ℓ}^{i} ∈ Ω_{R}^{1} ↪^{σ} WΩ_{R}^{1}, τ_{μ} runs through an R-basis of I, and w_{ℓ}^{i,μ} ∈ WΩ_{R}^{1}. It follows from integrability that

(4.7) 0 = ∇^{2}(m_{ℓ}) ≡ ∑_{i,μ} m_{i} ⊗ dτ_{μ} ∧ w_{ℓ}^{i,μ} mod N_{0} ⊗ (Ω_{R}^{2} + IWΩ_{R}^{2}).

I claim the composition map

(4.8)  IWΩ_{R}^{1} →^{d} WΩ_{R}^{2} → WΩ_{R}^{2}/(Ω_{R}^{2} + IWΩ_{R}^{2})

is injective. Granting this, it follows from (4.6) that

(4.9)  ∇(m_{ℓ}) = ∑_{i} m_{i} ⊗ η_{ℓ}^{i} ∈ N_{0} ⊗_{R} Ω_{R}^{1},

which shows that the diagram (4.5) commutes on the m_{ℓ}. Since these form a W-basis for M, the diagram commutes, proving the proposition.

To see that (4.8) is injective, one can localize R and assume R is étale over a polynomial ring A. The assertion then follows by tensoring with R from the corresponding assertion for A. This is lemma 2.3. □

5. Descent modulo p^{N}; proof of theorem 1.1

We keep the same notations as above. In particular, we view Ω_{R}^{•} ⊂ WΩ_{R}^{•} via the map σ : R ↪ W. Define C^{•} = WΩ_{R}^{•}/Ω_{R}^{•}. We define a decreasing filtrations P^{∗}WΩ_{R}^{•} and P^{j}C^{•} by

(5.1) P^{j}WΩ_{R}^{i} = Image(WΩ_{R}^{i−j} ∧Ω_{R}^{j}) ⊂ WΩ_{R}^{i}; P^{j}C^{i} = P^{j}WΩ_{R}^{i}/Ω_{R}^{i}

Lemma 5.1. We have H^{i}(gr_{P}^{j} C^{•}) = (0) for i = 0, 1 and all j.

Proof. Again, as above we reduce to the case R = A = k[T_{1}, …, T_{n}] is a polynomial ring and we work with I-graded pieces. For I integral, we are in Ω_{A}^{•} so C^{•}(I) = (0). Assume s = s(I) = max{−ord_{p}(i_{j})} > 0. Then

(5.2)  H^{0}(gr_{P}^{0} C^{•}(I)) = ker(d : C^{0}(I) → C^{1}(I)/P^{1}C^{1}(I)).

Suppose −ord_{p}(i_{j}) = s(I). Then d(p^{s}T^{I}) contains the term T^{I}d log(T_{j}) which does not lie in P^{1}C^{1}(I), from which it follows that this kernel is zero.

For H^{1} there are two complexes to consider

C^{0}(I) → C^{1}(I)/P^{1}C^{1}(I) → C^{2}(I)/P^{1}C^{2}(I); (5.3)

0 → P^{1}C^{1}(I) → P^{1}C^{2}(I)/P^{2}C^{2}(I). (5.4)

We write s = s(I) > 0. Note P^{m}C^{m}(I) consists of elements of the form T^{I} ∑_{|J|=m} a_{J}d log(T_{J}) with ord_{p}(a_{J}) ≥ s > 0. (Here, of course, d log(T_{j}) := d log(T_{j_{1}}) ∧ d log(T_{j_{2}}) ∧ …) In particular, if w ∈ P^{1}C^{1}(I) and dw ∈ P^{2}C^{2}(I), then w/p is integral so w ≡ 0 mod p. It follows that the complex (5.4) has H^{1} = (0).

Finally, we consider the complex (5.3). Start with w ∈ C^{1}(I) represented by T^{I} ∑ v_{j}d log(T_{j}) ∈ E^{1}. Integrality means

(5.5)  v_{j} ∈ W (k), ∀j; x_{mℓ} := v_{ℓ}i_{m} − v_{m}i_{ℓ} ∈ W (k), ∀m, ℓ.

The assumption dw ∈ P^{1}C^{2} means

(5.6)  ∑ x_{mℓ}d log(T_{mℓ}) = ∑_{ν} ( ∑_{μ} y_{μν}d log(T_{μ})) ∧ d log(T_{ν})

where

(5.7)  y_{μν} ∈ W (k); y_{μν}i_{τ} − y_{τν}i_{μ} ∈ W (k), ∀μ, ν, τ.

Further, if y_{μν} ≠ 0, then because of the denominators in the d log terms, we must have i_{μ}i_{ν} ≠ 0. Thus we can define θ_{ab} := y_{ab}i_{a}^{−1} with the understanding that θ_{ab} = 0 if y_{ab} = 0. Note, by (5.7)

(5.8)  θ_{ab} − θ_{cb} ∈ i_{a}^{−1}i_{b}^{−b}W (k).

Fix ϵ so ord_{p}(i_{ϵ}) = −s = −s(I) < 0. Define w′ := T^{I} ∑_{ℓ} θ_{ϵℓ}d log(T_{ℓ}). Because p^{s}|θ_{ϵℓ} we see w′ ∈ P^{1}C^{1}. We have (defining z_{ℓm})

(5.9) d(w − w′) = T^{I} ∑_{m,ℓ} ((v_{ℓ} − θ_{ϵℓ})i_{m} − (v_{m} − θ_{ϵm})i_{ℓ})d log(T_{ℓm}) =

T^{I} ∑_{m,ℓ} ((v_{ℓ}−θ_{mℓ}+(θ_{mℓ}−θ_{ϵℓ}))i_{m}−(v_{m}−θ_{ℓm}+(θ_{ℓm}−θ_{ϵm}))i_{ℓ})d log(T_{ℓm}) =

T^{I} ∑_{m,ℓ}(x_{mℓ} −(y_{mℓ} −y_{ℓm})+p^{s}z_{mℓ})d log(T_{mℓ}) = p^{s}T^{I} ∑_{m,ℓ} z_{mℓ}d log(T_{mℓ}).

Here the z_{mℓ} ∈ W (k).

Replacing w by w − w′, it follows from the above calculation that we may suppose

(5.10)  v_{ℓ}i_{m} − v_{m}i_{ℓ} ∈ p^{s}W (k).

Recall we had chosed ϵ so that −ord_{p}(i_{ϵ}) = s = s(I). Since the v_{ℓ} ∈ W (k), we can write

(5.11)  v_{ϵ} = cp^{s}i_{ϵ}, c ∈ W (k).

Then, for any m,

(5.12)  v_{m} = cp^{s}i_{m} − i_{ϵ}^{−1}p^{s}x, x ∈ W (k).

In particular, cp^{s}T^{I} ∈ C^{0}(I), and

(5.13)  d(cp^{s}T^{I}) = w + p^{2s}T^{I} ∑_{j} b_{j}d log(T_{j}); b_{j} ∈ W (k).

It follows that w = d(cp^{s}T^{I}) ∈ C^{1}(I). □

Our final task will be to “lift” (so to speak) our mod p descent to a descent mod p^{N}. We keep the notation from section 4. Recall we have chosen a lifting S of R and a frobenius f : S → S which induces ρ : S ↪ W (R). We view S as a subring of W (R) via ρ. We fix a trivialization M = W (R)^{⊕r}, and we write G for the connection matrix. G is an r × r matrix with entries in W Ω_{R}^{1}. We assume inductively that the entries of G lie in Ω_{S}^{1} + p^{s}W Ω_{R}^{1} for some s ≥ 1,

(5.14)  G = β + p^{s}γ

Lemma 5.2. There exists a matrix U_{s} ∈ Mat_{r}(W (R)) such that the coordinate change I + p^{s}U_{s} leads to a connection matrix with entries in Ω_{S}^{1} + p^{s+1}W Ω_{R}^{1}. The matrix U_{s} is unique modulo p.

Proof. The new connection matrix is

(5.15) (I + p^{s}U_{s})G(I + p^{s}U_{s})^{−1} + p^{s}dU_{s}(I + p^{s}U_{s})^{−1} ≡

β + p^{s}(γ + [U_{s}, β] + dU_{s}) mod p^{s+1}.

the curvature equation dG − G^{2} = 0 yields

(5.16)  dβ − β^{2} + p^{s}(dγ − βγ − γβ) ≡ 0 mod p^{2s}.

Recall we have defined (C^{•}, d) = (WΩ_{R}^{•}/Ω_{R}^{•}, d). One has an action of Ω_{R}^{•} on C^{•} in an obvious sense. Let b := β mod p ∈ Mat_{r}(Ω_{R}^{1}). Note db − b^{2} = 0. Define a complex (Mat_{r}(C^{•}), δ_{b}) by

(5.17)  δ_{b}(e) = de + (−1)^{deg e}[e, b]

where the bracket is the graded bracked [e, b] = eb − (−1)^{deg e}be. Note

(5.18) δ_{b}^{2}(e) = (−1)^{deg e}(d[e, b] − [de + (−1)^{deg e}[e, b], b]) =

(−1)^{deg e}(de)b + e(db) − (db)e + b(de) − (−1)^{deg e}(de)b − b(de) − eb^{2}

+ (−1)^{deg e}beb + (−1)^{deg e+1}beb + b^{2}e = 0.

By (5.16), γ represents an element in H^{1}(Mat_{r}(C^{•}), δ_{b}). By (5.15), the desired matrix U_{s} exists if and only if the cohomology class of γ is trivial. If H^{0} = (0) then U_{s} is unique mod p.

The filtration P^{∗}Mat_{r}(C^{•}) := Mat_{r}(P^{∗}C^{•}) is stabilized by δ_{b}, and the differential on gr_{P} Mat_{r}(C^{•}) is just given by d. Using the exact sequences associated to the filtration, we see for any i that

H^{i}(gr_{P}^{∗} Mat_{r}(C^{•}), d) = (0) ⇒ H^{i}(Mat_{r}(C^{•}), δ_{b}) = (0).

But the complex gr_{P}^{∗} Mat_{r}(C^{•}), d) is just a direct sum of copies of (gr_{P}^{∗} C^{•}, d), so the desired vanishing follows from lemma 5.1. □

We turn now to the proof of theorem 1.1. As explained in the introduction (see equation (1.2)) one has a functor 𝔼 ↦ (E, ∇) associating a de Rham-Witt connection to a crystal on X. By [E], II, théorème 2.1, there is a canonical isomorphism between the cohomologies of 𝔼 and (E, ∇). In particular, looking at H^{0}, we see that the above functor is fully faithful. This reduces the problem of essential surjectivity to a local problem (gluing data lifts canonically).

We may assume X = Spec (R) and we have S, f lifting R and the frobenius as above, so S ↪ W (R). Given an integrable de Rham-Witt connection (M, ∇), we have from proposition 4.1 and lemma 5.2 that there exists a canonical descent (N, Ξ) to an S-module with an integrable connection such that (M, ∇) = (N, Ξ) ⊗_{S} W (R). Recall in section 1 we defined (M, ∇) to be quasi-nilpotent if the connection M/V W ·M on Spec (R) was quasi-nilpotent. This connection coincides with the connection on N/pN, so it follows that the functor

(5.19)  N ↦ N ⊗_{S} W (R),

from quasi-nilpotent connections on S to quasi-nilpotent de Rham-Witt connections, is essentially surjective. One knows (theorem of Berthelot. For a precise statement, see [K], section (2.4), p. 145) that the category of crystals on Spec (R) is equivalent to the category of quasi-nilpotent connections on S. The functor associating a de Rham-Witt connection to a crystal is easily identified with (5.19) (use the crystal property). It follows that the functor from crystals to quasi-nilpotent de Rham-Witt connections is essentially surjective, and hence, by the result of Etesse cited above, is an equivalence of category.

References

[Be] Berthelot, P., Ogus, A., Notes on Crystalline Cohomology, Princeton Univ. Press (1978).

[B] Bloch, S., Algebraic K-theory and Crystalline Cohomology, Pub. math. de l’IHES 47, 187-268 (1978).

[E] Etesse, J-Y., Complexe de De Rham-Witt à coefficients dans un cristal, Comp. Math. 66, 57-120 (1988).

[I1] Illusie, L., Complexe de De Rham-Witt et cohomologie cristalline, Ann. Scient. Ec. Norm. Sup. 4e série, t. 12, 501-661 (1979).

[I2] Illusie, L., Complexe de De Rham-Witt, in Journées de Géometrie Algébrique de Rennes, Astérisque 63, Société Math. de France (1979).

[K] Katz, N., Slope Filtration of F -Crystals, in Journées de Géometrie Algébrique de Rennes, Astérisque 63, Société Math. de France (1979).

[L] Lazard, M., Commutative Formal Groups, LNM 443, Springer Verlag (1975).

Dept. of Mathematics, University of Chicago, Chicago, IL 60637, USA

  E-mail address: bloch@math.uchicago.edu