Nonabelian Hodge Theory in Characteristic p

Vadim Vologodsky

Contents

1 Introduction 1

2 The Cartier transform 1

3 The de Rham cohomology 6

4 Functoriality of the Cartier transform 9

5 Some examples and applications 10

1 Introduction

This talk is based on our joint work with Roman Bezrukavnikov and Arthur Ogus.

Let X be a smooth scheme over S, where S is a scheme over spec 𝔽_{p}, F_{X/S} : X → X′ = X×_{F_{S}} S is the (relative) Frobenius morphism. For any quasi-coherent O_{X′}-module E, the O_{X}-module F_{X/S}^{∗}E has a canonical integrable connection. Moreover, the theory of Cartier descent says that the above gives an equivalence between the category of quasi-coherent D_{X/S}-modules with zero p-curvature and the category of quasi-coherent O_{X′}-modules.

The purpose of this work is to extend the above correspondence to a wider class of D_{X/S}-modules. A D_{X/S}-module M is called nilpotent of level n if there exists a filtration 0 = M_{−1} ⊂ M_{0} ⊂

⋯ ⊂ M_{n} = M by submodules such that M_{i}/M_{i−1} has zero p-curvature.

The main result says that, given a lifting X → S mod p^{2} and an integer 0 ≤ n < p, we get an equivalence between the category of nilpotent level n D_{X/S}-modules and the category of O_{T_{X′/S}^{∗}}-modules on the cotangent bundle supported on the nth nilpotent neighborhood of the zero section.

2 The Cartier transform

.

1 Azumaya property of the algebra of differential operators.

The material of this subsection is borrowed from [BMR].

Let X be a smooth scheme over S, where S is a scheme over spec 𝔽_{p}. First, we observe that the ring of PD differential operators D_{X/S} has a large center Z_{X/S}. Recall that, given a vector field θ ∈ T_{X/S}, the differential operator θ^{p} acts on functions O_{X} as a derivation. We denote the corresponding vector field by θ^{[p]}. Then one can check that the operator c(θ) = θ^{p} − θ^{[p]} is in Z_{X/S}.

Denote by p the projection p : T_{X′/S}^{∗} → X′.

We define a homomorphism of sheaves of algebras

p_{∗}O_{T_{X′/S}^{∗}} = S^{·}T_{X′/S} → F_{X/S ∗}Z_{X/S}  (1)

sending f ∈ O_{X′} ⊂ p_{∗}O_{T_{X′/S}^{∗}} to F_{X/S}^{∗}f ∈ F_{X/S ∗}O_{X} ⊂ F_{X/S ∗}(D_{X/S}) and θ ∈ T_{X′/S} to c(θ).

Theorem 1 The map (1) is an isomorphism.

Via (1) we can view F_{X/S ∗}D_{X/S} as a sheaf of algebras over T_{X′/S}^{∗}.

Recall that an Azumaya algebra over a scheme Y is a coherent sheaf of algebras A which is locally for the fppf topology isomorphic to the algebra of endomorphisms of a vector bundle. It follows that the coherent sheaf A is locally free. We denote by rk A its rank. A splitting of an Azumaya algebra is a vector bundle E over Y such that A ≃ End(E). The category M(A) of quasicoherent sheaves of modules over a split Azumaya algebra is equivalent to the category M(O_{Y}) of quasi-coherent sheaves O_{Y} - modules. The functor assigning to a End(E)-module M the O_{Y} -module Hom_{End(E)}(E; M) gives the equivalence.

The following important observation is made in [BMR]

Theorem 2 F_{X/S ∗}D_{X/S} is an Azumaya algebra over T_{X′/S}^{∗} of rank p^{2 dim_{S} X}

We shall give here a sketch of the proof reffering the reader to loc. cit. for the detailes. Consider the following diagram:

T_{X′/S}^{∗} ×_{X′} X →^{δ} T_{X′/S}^{∗}
↓  ↓^{p}
X  →^{F_{X/S}} X′

We claim that the algebra F_{X/S ∗}D_{X/S} becomes split when pulled back via the map δ above. Indeed, D_{X/S} is a splitting module over δ^{∗}F_{X/S ∗}D_{X/S}.

Sending a D_{X/S}-module M to F_{X/S ∗}M we obtain an equivalence of categories between the category of quasi-coherent D_{X/S}-modules and the category of quasi-coherent modules over the Azumaya algebra F_{X/S ∗}D_{X/S}.

Notice that the Azumaya algebra F_{X/S ∗}D_{X/S} is never split (unless dim_{S}X = 0). In effect, D_{X/S} has no zero divisors, whereas the matrix algebra End(E) does have.

Let i : X′ ↪ T_{X′/S}^{∗} be the zero section.

Lemma 3 The Azumaya algebra i^{∗}F_{X/S ∗}D_{X/S} splits canonically:

i^{∗}F_{X/S ∗}D_{X/S} ≃ End_{O_{X′}} (F_{X/S ∗}O_{X})

Indeed, the action of D_{X/S} on O_{X} gives the homomorphism i^{∗}F_{X/S ∗}D_{X/S} → End_{O_{X′}} (F_{X/S ∗}O_{X}). Since rk(i^{∗}F_{X/S ∗}D_{X/S}) = rk(End_{O_{X′}} (F_{X/S ∗}O_{X})), it is an isomorphism.

As a corollary we get an important result of Cartier.

Corollary 4 (”Frobenius descent”) The functor sending a D_{X/S}-module M to the O_{X′}-module F_{X/S ∗}Hom_{D_{X/S}}(O_{X}; M) defines an equivalence of categories between the category of quasi-coherent D_{X/S}-modules with zero p-curvature and the category of quasi-coherent O_{X′}-modules.

In effect, the category of D_{X/S}-modules with zero p-curvature is equivalent to the category of i^{∗}F_{X/S ∗}D_{X/S}-modules.

2. Let S̃ be a flat scheme over spec ℤ_{p}/p^{2}, S = S̃ × spec𝔽_{p}, F̃_{S} : S̃ → S̃ a lifting of the Frobenius F_{S} : S → S (i.e. a morphism whose restriction to S is F_{S}).

A lifting of X → S is a smooth scheme X̃ → S̃ over S̃ such that X = X̃ ×S̃ S.

We are going show that given a lifting X̃ → S̃, the splitting of F_{X/S ∗}D_{X/S} over the zero section i : X′ ↪ T_{X′/S}^{∗} extends to a splitting over a larger subscheme.

First, assume that, in addition to the lifting X̃ → S̃, we are given a lifting of the Frobenius: F̃_{X̃/S̃} : X̃ → X̃′.

The above lifting gives a map:

CF̃_{X̃/S̃} : T_{X/S} → F_{X/S}^{∗}T_{X′/S}  (2)

Given a vector field θ ∈ Γ(T_{U/S}) on a open subset U ∈ X we can locally choose a lifting θ̃ of

θ on Ũ and put CF̃_{X̃/S̃}(θ) = \frac{dF̃_{X̃/S̃}(θ̃)}{p}|_{U′}. It is easy to check that the right hand side of the above

formula does not depend on the choice of lifting θ̃, hence it gives a well defined map. Next, we construct a functor:

ΦF̃_{X̃/S̃} : M(O_{T_{X′/S}^{∗}}) → M(D_{X/S})  (3)

from the category of quasi-coherent O_{T_{X′/S}^{∗}}-modules to the category of quasi-coherent D_{X/S}-modules.

Given a O_{T_{X′/S}^{∗}}-module E we define the O_{X}-module ΦF̃_{X̃/S̃}(E) to be F_{X/S}^{∗}(p_{∗}E). The sheaf ΦF̃_{X̃/S̃}(E) is a module over the commutative sheaf of algebras F_{X/S}^{∗}(S^{·}T_{X′/S}). We want to endow F_{X/S}^{∗}(p_{∗}E) with an action of the Lie algebra of vector fields. Let θ ∈ Γ(T_{U/S}) and h = ∑_{i} f_{i} ⊗ s_{i} be a section of F_{X/S}^{∗}(p_{∗}E) over U, f_{i} ∈ O(U), s_{i} ∈ p_{∗}E(U′). Put

θ(h) = ∑_{i}(f_{i}(CF̃_{X̃/S̃}(θ)(s_{i})) + θ(f_{i}) ⊗ s_{i})  (4)

It is easy to see that the formula (4) gives F_{X/S}^{∗}(p_{∗}E) a D_{X/S}-module structure.

We shall compute the action of the center Z_{X/S} ⊂ D_{X/S} on ΦF̃_{X̃/S̃}(E) more explicitly. Notice that O_{X′}-module

F_{X/S ∗}ΦF̃_{X̃/S̃}(E) = F_{X/S ∗}O_{X} ⊗_{O_{X′}} p_{∗}E

has a natural action of S^{·}T_{X′/S}: the one which comes from the S^{·}T_{X′/S}-module structure on p_{∗}E. However, in general, it does not coincide with the action of F_{X/S ∗}Z_{X/S} identified with S^{·}T_{X′/S} via (1).

To describe the picture we shall introduce a (non-linear) X′- endomorphism of the cotangent space: hF̃_{X̃/S̃} : T_{X′/S}^{∗} → T_{X′/S}^{∗}. or, equivalently, an endomorphism of the sheaf of algebras:

h∗ ; F̃_{X̃/S̃} : S^{·}T_{X′/S} → S^{·}T_{X′/S}  (5)

Given a vector field v ∈ T_{X′/S} we define

h∗ ; F̃_{X̃/S̃}(v) = −v + (PF̃_{X̃/S̃}(v))^{p} ∈ T_{X′/S} ⊕ S^{p}T_{X′/S}

where PF̃_{X̃/S̃} : T_{X′/S} → (F_{X/S}π)^{∗}T_{X′/S} is the pullback of CF̃_{X̃/S̃} with respect to π : X′ → X.

Notice that hF̃_{X̃/S̃} is identical on the zero section X′ ↪ T_{X′/S}^{∗} and induces an automorphism of X′_{n} ↪ T_{X′/S}^{∗}.

Lemma 5 Let E be a quasi-coherent O_{T_{X′/S}^{∗}}-module. A central element θ ∈ F_{X/S ∗}Z_{X/S} acts on F_{X/S ∗}ΦF̃_{X̃/S̃}(E) = F_{X/S ∗}O_{X} ⊗_{O_{X′}} p_{∗}E as 1 ⊗ h∗ ; F̃_{X̃/S̃}(θ). In the other words we have a canonical

isomorphism of O_{T_{X′/S}^{∗}}-modules

F_{X/S ∗}ΦF̃_{X̃/S̃}(E) ≃ h_{F̃_{X̃/S̃} ∗}(p^{∗}F_{X/S ∗}O_{X} ⊗_{O_{T{∗ \atop X′/S}}} E)

We omit the proof.

Let î: T̂_{X′/S}^{∗} ↪ T_{X′/S}^{∗} be the formal completion of T_{X′/S}^{∗} along the zero section.

Theorem 6 Let X̃ → S̃ be a lifting of X and F̃_{X̃/S̃} : X̃ → X̃′ a lifting of the Frobenius F_{X/S}. Then the Azumaya algebra î^{∗}F_{X/S ∗}D_{X/S} splits canonically.

Proof:

Let i_{n} : X′_{n} ↪ T_{X′/S}^{∗} be the n-th infinitesimal neighborhood of X′ in T_{X′/S}^{∗} (i.e. O_{X′_{n}} = i^{·}O_{T_{X′/S}^{∗}}/i^{·}J^{n+1}, where J is the sheaf of ideals of X′ ↪ T_{X′/S}^{∗} ).

We shall see that the i_{n}^{∗}F_{X/S ∗}D_{X/S}-module i_{n}^{∗}F_{X/S ∗}ΦF̃_{X̃/S̃}(i_{n ∗}O_{X′_{n}}) is a splitting of the Azumaya algebra i_{n}^{∗}F_{X/S ∗}D_{X/S}.

Indeed, according to the above lemma i_{n}^{∗}F_{X/S ∗}ΦF̃_{X̃/S̃}(i_{n ∗}O_{X′_{n}}) is a vector bundle over X′_{n} of

rank p^{dim_{S} X}.

Therefore i_{n}^{∗}F_{X/S ∗}D_{X/S}-module structure on i_{n}^{∗}F_{X/S ∗}ΦF̃_{X̃/S̃}(i_{n ∗}O_{X′_{n}}) gives an isomorphism

i_{n}^{∗}F_{X/S ∗}D_{X/S} → End_{O_{X′_{n}}} (i_{n}^{∗}F_{X/S ∗}ΦF̃_{X̃/S̃}(O_{X′_{n}}))

It follows that

lim_{←} F_{X/S ∗}ΦF̃_{X̃/S̃}(i_{n ∗}O_{X′_{n}})

is a splitting of î^{∗}F_{X/S ∗}D_{X/S}.

Remarks:1) In fact, the Azumaya algebra F_{X/S ∗}D_{X/S} becomes split when pulled back via the map hF̃_{X̃/S̃} : T_{X′/S}^{∗} → T_{X′/S}^{∗}. Indeed, thanks to Lemma 5 F_{X/S ∗}ΦF̃_{X̃/S̃}(O_{T_{X′/S}^{∗}}) is a splitting of

h∗ ; F̃_{X̃/S̃}F_{X/S ∗}D_{X/S}.

2) The map hF̃_{X̃/S̃} is etale and surjective. Indeed, hF̃_{X̃/S̃} is a group endomorphism, where T_{X′/S}^{∗} is viewed as a group scheme over X′. Also, the differential dhF̃_{X̃/S̃} at any point at the zero section X′ ⊂ T_{X′/S}^{∗} is equal to −Id. Hence, the image of hF̃_{X̃/S̃} is an open group subscheme and therefore it coincides with T_{X′/S}^{∗}.

Next, we shall study the dependence of the functor ΦF̃_{X/S} of the choice of lifting: F̃_{X/S} : X̃ → X̃′. Notice that the set of such liftings is a torser over Γ(F_{X/S}^{∗}T_{X′/S}). Given a section w ∈ Γ(F_{X/S}^{∗}S^{·}T_{X′/S}) we define an automorphism of the algebra of differential

operators:

e^{w} : D_{X/S} → D_{X/S}  (6)

If w is written on an open subscheme U ⊂ X in the form w = ∑_{i} f_{i} ⊗ v_{i}, where f_{i} ∈ Γ(O_{U}), v_{i} ∈ Γ(S^{·}T_{U′/S}) we put:

e^{w}(f) = f, for f ∈ Γ(O_{U}), e^{w}(u) = u + ∑_{i} u(f_{i}) v_{i}, for u ∈ Γ(T_{U′/S})

Here u(f_{i}) denotes the Lie derivative and v_{i} are viewed as elements of the center of D_{U/S} via (1). The other ingredient we will need is the endomorphism (5). Recall, that h∗ ; F̃_{X̃/S̃} induces an

automorphism of the completed symmetric algebra. We consider its pullback on X:

h∗ ; F̃_{X̃/S̃} : F_{X/S}^{∗}Ŝ^{·}T_{X′/S} → F_{X/S}^{∗}Ŝ^{·}T_{X′/S}

Let D̂_{X/S} denote the completed algebra of differential operators:

D̂_{X/S} = D_{X/S} ⊗_{Z_{X/S}} F_{X/S}^{·}Ŝ^{·}T_{X′/S}

If M is a D̂_{X/S}-module and σ : D_{X/S} → D_{X/S} is an endomorphism, we shall denote by M^{σ} the twisted D̂_{X/S}-module.

Proposition 7 Let E be a nilpotent quasi-coherent OT̂_{X′/S}^{∗}-module ( i.e. E gets killed by some

power of the ideal of the zero section) and F̃_{X̃/S̃}, \overline{F}_{X̃/S̃} : X̃ → X̃′ two liftings of the Frobenius, \overline{F}_{X̃/S̃} − F̃_{X̃/S̃} = w ∈ ΓF_{X/S}^{∗}T_{X′/S}. Then

Φ_{\overline{F}_{X̃/S̃}}(E) = (ΦF̃_{X̃/S̃}(E))^{σ}

where σ = e^{h∗−1 ; F̃_{X̃/S̃} (w)}.

We shall not prove this assertion: it is an easy straightforward computation.

3. Splitting of the Azumaya algebra over the universal PD-envelope of the zero section.

Denote by î_{pd} : T̂_{X′/S,pd}^{∗} → T_{X′/S}^{∗} the universal PD-envelope of the zero section and by D̂_{X/S,pd} the sheaf of algebras D_{X/S} ⊗_{Z_{X/S}} F_{X/S}^{·}p_{∗}OT̂_{X′/S,pd}^{∗}. A sheaf of OT̂_{X′/S,pd}^{∗}-modules is called nilpotent if it gets killed by some divided power of the ideal of the zero section. Similarly, we define the notion of nilpotent D̂_{X/S,pd}-modules

First, we shall see that the functor (3) extends to a functor:

ΦF̃_{X̃/S̃} : M(OT̂_{X′/S,pd}^{∗}) → M(D̂_{X/S,pd})

from the category of nilpotent quasi-coherent OT̂_{X′/S,pd}^{∗}-modules to the category of nilpotent ^{1}

quasi-coherent D̂_{X/S,pd}-modules.

Indeed, given a nilpotent OT̂_{X′/S,pd}^{∗}-module E, we already have a D_{X/S}-module ΦF̃_{X̃/S̃}(E) and it

remains to show that the action of the center Z_{X/S} on ΦF̃_{X̃/S̃}(E) naturally extends to an action of the divided power algebra F_{X/S}^{·}p_{∗}OT̂_{X′/S,pd}^{∗} which commutes with D_{X/S}-action.

Notice that we have an F_{X/S}^{·}p_{∗}OT̂_{X′/S,pd}^{∗}- action on ΦF̃_{X̃/S̃}(E) = F_{X/S}^{∗}p_{∗}E since p_{∗}E is naturally a p_{∗}OT̂_{X′/S,pd}^{∗}-module. Let us twist this action by the endomorphism

h∗ ; F̃_{X̃/S̃} : p_{∗}OT̂_{X′/S,pd}^{∗} → p_{∗}OT̂_{X′/S,pd}^{∗}^{2}

Lemma (5) implies that the new action has all the required properties.

In fact, the map hF̃_{X̃/S̃} : T̂_{X′/S,pd}^{∗} → T̂_{X′/S,pd}^{∗} is just multiplication by −1. It immediately follows from the definition.

Moreover, the above functors, for any two liftings of the Frobenius are canonically isomorphic. Indeed, for any w ∈ Γ(F_{X/S}^{∗}T_{X′/S}) = O_{X} ⊗_{F_{X/S}^{·}O_{X′}} Z_{X/S} ⊂ D̂_{X/S,pd} the automorphism

e^{w} : D̂_{X/S,pd} → D̂_{X/S,pd}

is inner: it is the conjugation by exp(w) = ∑_{i} \frac{w^{i}}{i!} ∈ Γ(D̂_{X/S,pd}). Therefore, given any two liftings, F̃_{X̃/S̃} and \overline{F}_{X̃/S̃}, with \overline{F}_{X̃/S̃} − F̃_{X̃/S̃} = w, and a quasi-coherent nilpotent OT̂_{X′/S,pd}^{∗}-module E, we have the isomorphism:

Φ_{\overline{F}_{X̃/S̃}}(E) ⟶^{exp(−w)} (ΦF̃_{X̃/S̃}(E))

Since exp(w + w′) = exp(w)exp(w′), the above isomorphisms satisfy the transitivity condition, for any three lifting.

It has a very important corollary: given just a lifting X̃ → S̃ of X → S, we get a functor :

Φ_{X̃/S̃} : M(OT̂_{X′/S,pd}^{∗}) → M(D̂_{X/S,pd})  (7)

In effect, we can choose an open covering X̃ = ∪_{i}Ũ_{i} together with local liftings F̃_{Ũ_{i}/S̃} and then glue Φ_{X/S} out from ΦF̃_{Ũ_{i}/S̃}.

Theorem 8 i) A lifting X̃ → S̃ gives rise to a splitting of the Azumaya algebra î_{pd}^{∗}F_{X/S ∗}D_{X/S} = F_{X/S ∗}D̂_{X/S,pd}

ii) The functor Φ_{X̃/S̃} is the equivalence of categories.

1The nilpotency condition is technical and can be suppressed. But then one has use a different definition of quasicoherent D̂_{X/S,pd}-modules: the problem is that D̂_{X/S,pd} is not a quasi-coherent O_{X}-module. For example, we can define the category of quasi-coherent D̂_{X/S,pd}-modules simply to be the category modules over the Azumaya algebra F_{X/S ∗}D̂_{X/S,pd}, which are quasi-coherent as T̂_{X′/S,pd}^{∗}-modules.

2Recall that the endomorphsm hF̃_{X̃/S̃} : T_{X′/S}^{∗} → T_{X′/S}^{∗} is identical on the zero section. Hence, it extends canonically to the universal PD-envelope

Proof: i) In effect,

M_{pd} = lim_{←} F_{X/S ∗}Φ_{X̃/S̃}(OT̂_{X′/S,pd}^{∗}/J^{[n]})

where J^{[n]} is the divided power of the ideal of the zero section, is a splitting.

ii) It is easy to see that for any quasi-coherent nilpotent OT̂_{X′/S,pd}^{∗}-module E the F_{X/S ∗}D̂_{X/S,pd}-

module F_{X/S ∗}Φ_{X̃/S̃}(E) is equal to E ⊗_{OT̂{∗ \atop X′/S,pd}} M_{pd}. Hence, by i) and the general property of

Azumaya algebras F_{X/S ∗}Φ_{X̃/S̃} gives an equivalence of categories between the category of nilpotent OT̂_{X′/S,pd}^{∗}-modules and the one of nilpotent F_{X/S ∗}D̂_{X/S,pd}-modules.

We shall call the inverse functor:

C_{X̃/S̃} = Φ_{X̃/S̃}^{−1} : M(D̂_{X/S,pd}) → M(OT̂_{X′/S,pd}^{∗})

the Cartier transform.

4. Geometric construction of the splitting. The above we constructed a splitting F_{X/S ∗}M_{pd} of the Azumaya algebra F_{X/S ∗}D̂_{X/S,pd}, first, locally, choosing a lifting of the Frobenius, and then glued the global splitting from those defined on small open subsets. Here we give an alternative geometric construction of M_{pd} ” in one step”.

We start with a smooth morphism X → S and a lifting S̃. We shall show that a lifting X̃′ gives rise to a splitting of F_{X/S ∗}D̂_{X/S,pd}.

Given a PD-thickining (i : U ↪ T) ∈ Cris(X/S) there is a canonical morphism f_{T/S} : T → X′ such that f_{T/S}i = F_{U/S}: the Frobenius F_{T/S} factors uniquely through U′ ↪ T′ and f_{T/S} is the composition of the map to U′ and the embedding i′ : U′ ↪ X′.

Theorem 9 A lifting X̃′ gives rise to a crystal L_{X̃′/S̃} of F_{X/S}^{∗}T_{X′/S}-torsers on Cris(X/S) uniquely characterized by the following property:

for any object (U ↪ T̃) of Cris(X/S̃) with T̃ flat over S̃ and T = T̃ ×S̃ S the F_{X/S}^{∗}T_{X′/S}(T) -set L_{X̃′/S̃}(T) is identified with the set of liftings f̃_{T/S} : T̃ → X̃′ of f_{T/S}.

The sheaf L_{X̃′/S̃} is represented by an affine bundle over X equipped with an integrable connection; the associated vector bundle is F_{X/S}^{∗}T_{X′/S} together with the p-flat connection on it. We shall use the same name q : L_{X̃′/S̃} → X for the affine bundle.

Let M_{pd} = Hom_{O_{X}} (q_{∗}O_{L_{X̃′/S̃}}; O_{X}). By construction, M_{pd} has a D_{X/S}-module structure. On the other hand the F_{X/S}^{∗}T_{X′/S}-action on L_{X̃′/S̃} gives the embedding

F_{X/S}^{∗}p_{∗}OT̂_{X′/S,pd}^{∗} = Ŝ_{pd}^{·}(F_{X/S}^{∗}T_{X′/S}) ↪ D̂_{L_{X′/S}/X}^{∞}

whose image consists of all F_{X/S}^{∗}T_{X′/S}-invariant hyper-differential operators. Hence, the algebra the algebra F_{X/S}^{∗}p_{∗}OT̂_{X′/S,pd}^{∗} acts naturally on M_{pd}. In particular, we have an action of F_{X/S}^{∗}p_{∗}O_{T_{X′/S}^{∗}}

Proposition 10 The above action, twisted by the automorphism −id : T_{X′/S}^{∗} → T_{X′/S}^{∗}, coincides with the one that comes from the D_{X/S}-module structure and the canonical map F_{X/S}^{∗}p_{∗}O_{T_{X′/S}^{∗}} ↪ D_{X/S}.

Corollary 11 The sheaf M_{pd} has a canonical D̂_{X/S,pd}-module structure. Moreover, F_{X/S ∗}M_{pd} is a splitting of F_{X/S ∗}D̂_{X/S,pd}.

3 The de Rham cohomology

1. Let X̃ → S̃ be, as in the previous section.

Given an integer n ≥ 0 we denote by HIG^{n}(X′/S) the category of quasi-coherent O_{X′_{n}}-modules and by MIC^{n}(X/S) the category of quasi-coherent modules over D_{X/S,n} = D_{X/S}⊗_{Z_{X/S}} F_{X/S}^{·}p_{∗}O_{X′_{n}}.

Theorem (8) has the following important corollary Corollary 12 For any p > n ≥ 0, we have a canonical ( depending on the lifting) equivalence of categories:

C_{X̃/S̃} : MIC^{n}(X/S) ≃ HIG^{n}(X′/S)

Indeed, for any such n, there is a natural morphism X′_{n} → T̂_{X′/S,pd}^{∗} → T_{X′/S}^{∗}, thus from Theorem (8) we get a splitting of the Azumaya algebra on X′_{n}. It defines the equivalence.

The purpose of this section is to prove the following result:

Theorem 13 Let M be an object of MIC^{n}(X/S), for p > n ≥ 0. Put m = p − n − 1. There is a canonical isomorphism in the derived category of quasi-coherent O_{X′}- modules:

γ : τ_{≤m}R^{·}Hom_{O_{T{∗ \atop X′/S}}} (O_{X′}; C_{X̃/S̃}(M)) ≃ τ_{≤m}F_{X/S ∗}R^{·}Hom_{D_{X/S}}(O_{X}; M)

In particular, if n + dim_{S}X < p we have an isomorphism

R^{·}Hom_{O_{T{∗ \atop X′/S}}} (O_{X′}; C_{X̃/S̃}(M)) ≃ F_{X/S ∗}R^{·}Hom_{D_{X/S}}(O_{X}; M)

A proof is given in Subsection 4. In the next two subsections we establish some auxiliary results.

2. Let Y be a scheme and p : L → Y a vector bundle on Y. We denote by i_{n} : Y_{n} ↪ L the n-th infinitesimal neighborhood of of the zero section i_{0} : Y ↪ L (i.e. O_{Y_{n}} = i_{0}^{·}(O_{L}/J^{n+1})). For n ≤ m, we have the natural immersion i_{n,m} : Y_{n} ↪ Y_{m}.

There is a canonical morphism:

α_{n,m} : R^{·}i_{n}^{∗}(i_{0 ∗}O_{Y}) → R^{·}i_{n,m}^{∗}(i_{0 ∗}O_{Y})

The purpose of this section is to show that, for m ≥ n + rk L, there exists a canonical morphism β_{n,m} in the opposite direction such that β_{n,m}α_{n,m} = Id. In the other words, R^{·}i_{n}^{∗}(i_{0 ∗}O_{Y}) is a direct summand of R^{·}i_{n,m}^{∗}(i_{0 ∗}O_{Y}). More completely, for any n ≤ m, we construct the morphism

β_{n,m} : τ_{≥n−m}R^{·}i_{n,m}^{∗}(i_{0 ∗}O_{Y}) → τ_{≥n−m}R^{·}i_{n}^{∗}(i_{0 ∗}O_{Y})

such that β_{n,m}(τ_{≥n−m}α_{n,m}) = Id

The construction makes use of the 𝔾_{m}-action on L and the fact that both the objects R^{·}i_{n}^{∗}(i_{0 ∗}O_{Y}), R^{·}i_{n,m}^{∗}(i_{0 ∗}O_{Y}) and the morphism α_{n,m} live in the derived category of 𝔾_{m}-equivariant coherent sheaves on Y_{n}.

Notice the pushforward of a complex of 𝔾_{m}-equivariant sheaves E to Y is endowed with a grading:

R^{·}p_{∗}E = ⨁_{i∈ℤ}(R^{·}p_{∗}E)_{i}

For example, (p_{∗}O_{Y_{n}})_{i} = S^{−i}L^{∗}, for −n ≤ i ≤ 0, and 0 otherwise. E is said to have weights greater (resp. less ) than k if (R^{·}p_{∗}E)_{i} = 0, for all i ≤ k (resp. i ≥ k). We have the natural truncation functor w_{>k} (resp. w_{<k} ) from the category of derived category of 𝔾_{m}-equivariant sheaves to the subcategory of complexes whose weight are greater (resp. less) than k. For any object E,there is a natural morphism E → w_{>k}E (resp. w_{<k}E → E).

Lemma 14 i) The sheaf R^{j}i_{n}^{∗}(i_{0 ∗}O_{Y}) is pure of weight j − n, for any j < 0, and it has weight 0 if j = 0.

ii) The morphism α_{n,m} induces an isomorphism w_{>−m−1}R^{·}i_{n}^{∗}(i_{0 ∗}O_{Y}) ≃ w_{>−m−1}R^{·}i_{n,m}^{∗}(i_{0 ∗}O_{Y}).

Proof: i)Since the statement is local on Y, we may assume L to be the trivial vector bundle. For n = 0, it can be checked directly with help of the Koszul resolution. In general, we may proceed by induction on n using the exact sequence

S^{n}L^{∗} ⊗ R^{j}i_{0}^{∗}(i_{0 ∗}O_{Y}) → p_{∗}R^{j}i_{n}^{∗}(i_{0 ∗}O_{Y}) → p_{∗}R^{j}i_{n−1}^{∗}(i_{0 ∗}O_{Y})

ii)We have to show that cone(α_{n,m}) has weights less than −m. By i) the cone C of the canonical morphism R^{·}i_{m}^{∗}(i_{0 ∗}O_{Y}) → i_{0 ∗}O_{Y} has weights less than −m. On the other hand, we have cone(α_{n,m}) = R^{·}i_{n,m}^{∗}C. Thus the statement follows from the fact that R^{·}i_{n,m}^{∗} does not increase weights.

As a corollary of i) we obtain that the complex τ_{≥k}R^{·}i_{n}^{∗}(i_{0 ∗}O_{Y}) has weights greater than k−n−1. Using the lemma we can define β_{n,m} to be the composition of the following arrows

τ_{≥n−m}R^{·}i_{n,m}^{∗}(i_{0 ∗}O_{Y}) → w_{>−m−1}τ_{≥n−m}R^{·}i_{n,m}^{∗}(i_{0 ∗}O_{Y}) ≃ w_{>−m−1}τ_{≥n−m}R^{·}i_{n}^{∗}(i_{0 ∗}O_{Y})

≃ τ_{≥n−m}R^{·}i_{n}^{∗}(i_{0 ∗}O_{Y})

3. Given a D_{X/S}-module M (resp. O_{T_{X′/S}^{∗}}-module E ), we put M̂ = D̂_{X/S} ⊗_{D_{X/S}} M (resp.

Ê = OT̂_{X′/S}^{∗} ⊗_{O_{T{∗ \atop X′/S}}} E).

Lemma 15 i) For any D_{X/S}-module M, there are canonical isomorphisms

R^{·}Hom_{D_{X/S}}(O_{X}; M) = R^{·}HomD̂_{X/S}(O_{X}; M̂)

ii) For any O_{T_{X′/S}^{∗}}-module E, we have

R^{·}Hom_{O_{T{∗ \atop X′/S}}} (O_{X′}; M) = R^{·}HomÔ_{T{∗ \atop X′/S}} (O_{X′}; Ê)

Proof:i) Assigning to a D_{X/S}-module M the D̂_{X/S}-module M̂ we get an exact functor from the category of D_{X/S}-modules to the category of D̂_{X/S}-modules. In turn, it gives a morphism: R^{·}Hom_{D_{X/S}}(O_{X}; M) → R^{·}HomD̂_{X/S}(O_{X}; M̂). To complete the proof it suffices to check that, for

any M, Hom_{D_{X/S}}(O_{X}; M) = HomD̂_{X/S}(O_{X}; M̂) which is fairly obvious. A similar argument gives ii).

4. Proof of the Theorem. We define the morphism γ to be the composition of the following morphisms:

τ_{≤m}R^{·}Hom_{O_{T{∗ \atop X′/S}}} (O_{X′}; C_{X̃/S̃}(M)) ≃ τ_{≤m}R^{·}Hom_{O_{X′_{n}}} (R^{·}i_{n}^{∗}O_{X′}; C_{X̃/S̃}(M)) ≃

τ_{≤m}R^{·}Hom_{O_{X′_{n}}} (τ_{≥−m}R^{·}i_{n}^{∗}O_{X′}; C_{X̃/S̃}(M)) →^{β_{n,p−1}} τ_{≤m}R^{·}Hom_{O_{X′_{n}}} (τ_{≥−m}R^{·}i_{n,p−1}^{∗}O_{X′}; C_{X̃/S̃}(M)) ≃

τ_{≤m}R^{·}Hom_{O_{X{′ \atop p−1}}} (O_{X′}; C_{X̃/S̃}(M)) ≃ τ_{≤m}F_{X/S ∗}R^{·}Hom_{D_{X/S,p−1}}(O_{X}; M) →
τ_{≤m}F_{X/S ∗}R^{·}Hom_{D_{X/S}}(O_{X}; M)

We shall prove that γ is a quasi-isomorphism. Since the statement is local we may assume that there exists a Frobenius lifting: F̃_{X/S} : X̃ → X̃′. According to Theorem (6) the lifting F̃_{X/S} gives a splitting of the Azumaya algebra F_{X/S ∗}D̂_{X/S} over the formal completion T̂_{X′/S}^{∗}. The above combined with the lemma from the previous subsection implies that for any nilpotent D_{X/S}-module there is an isomorphism

R^{·}Hom_{O_{T{∗ \atop X′/S}}} (O_{X′}; Φ−1 ; F̃_{X̃/S̃}(M)) ≃ F_{X/S ∗}R^{·}Hom_{D_{X/S}}(O_{X}; M)

Consider the following commutative diagram

\begin{matrix}τ_{≤m}R^{·}Hom_{O_{T{∗ \atop X′/S}}} (O_{X′}; C_{X̃/S̃}(M))→ & τ_{≤m}F_{X/S ∗}R^{·}Hom_{D_{X/S,p−1}}(O_{X}; M)→ \\ ↓ & ↓ \\ τ_{≤m}R^{·}Hom_{O_{T{∗ \atop X′/S}}} (O_{X′}; C_{X̃/S̃}(M))→ & τ_{≤m}R^{·}Hom_{O_{X{′ \atop p−1}}} (O_{X′}; C_{X̃/S̃}(M))→ \end{matrix}

→τ_{≤m}F_{X/S ∗}R^{·}Hom_{D_{X/S}}(O_{X}; M)
↓
→τ_{≤m}R^{·}Hom_{O_{T{∗ \atop X′/S}}} (O_{X′}; C_{X̃/S̃}(M))

All the vertical arrows in the diagram are isomorphisms while the composition of the morphisms in the second row is the identity morphism. It completes the proof.

4 Functoriality of the Cartier transform

1. Definition of the functors R^{·}f_{DR∗},R^{·}f_{Hig∗}, f^{∗} Let f : X → Y be a morphism of smooth schemes over S

df : T_{X/S} → f^{∗}T_{Y/S}

be the differential of f. To fix notation we recall the definition of the direct and inverse of a D-module.

First, let M be a left D_{Y/S}-module. Then there exists a unique D_{X/S}- module structure on f^{∗}M = f^{·}M ⊗_{f^{·}O_{Y}} O_{X} with the property that:

v(f^{∗}(m)) = ∑_{i} f^{∗}(w_{i}(m)) ⊗ h_{i}

where m ∈ M(U), U ⊂ Y, v ∈ T_{f^{−1}(U)/S} and w_{i} ∈ T_{U/S}, h_{i} ∈ O_{f^{−1}(U)} such that df(v) = ∑_{i} f^{∗}(w_{i}) ⊗ h_{i}.

The definition of the pushforward is more involved. First, recall that, for a left D_{Y/S}-module M, M ⊗_{O_{Y}} ω_{Y/S} has a structure of a right D_{Y/S}-module. Here ω_{Y/S} is the canonical sheaf on Y. The action is given by the formula

w(m ⊗ u) =: −m ⊗ Lie_{w}(u) − w(m) ⊗ u

where w ∈ T_{U/S}, m ∈ M(U), u ∈ ω_{U/S}. The inverse functor from the category of right D-modules to the one of left D-modules sends M to M ⊗_{O_{Y}} ω_{Y/S}^{−1}.

In particular, we have the right D_{X/S}-module structure on f^{∗}(D_{Y/S} ⊗ω_{Y/S}^{−1})⊗ω_{X/S}. In addition, the latter is endowed with a left action of f^{·}(D_{Y/S}. Now, given an object N ∈ D^{b}(D_{X/S} − modules) of the bounded derived category of D_{X/S}-modules, we define it pushforward to be

R^{·}f_{DR∗}N := R^{·}f_{·}(f^{∗}(D_{Y/S} ⊗ ω_{Y/S}^{−1}) ⊗ ω_{X/S} ⊗_{D_{X/S}} N)[dim Y − dim X]

In the case, when f is smooth, R^{·}f_{DR∗}N can be computed by the relative De Rham complex:

R^{·}f_{DR∗}N = R^{·}f_{·}(N ⊗_{O_{X}} Ω_{X/Y}^{·})

The D_{Y/S}-module structure on the left hand side corresponds to the ”Gauss-Manin connection” on the right one.

Lemma 16 Assume that f is smooth. Then the functor R^{·}f^{∗} is left adjoint to R^{·}f_{DR∗}

Next, we define the functors R^{·}f^{∗}, R^{·}f_{Hig∗} for the category of Higgs modules i.e. O-modules on the cotangent bundle.

Given a morphism f : X → Y we have the following diagram

f^{∗}T_{Y/S}^{∗} →^{i} T_{X/S}^{∗}
↓^{p}
T_{Y/S}^{∗}

Denote by D^{b}(O_{T_{X/S}^{∗}} − modules) the bounded derived category of Higgs modules. Given complexes E in D^{b}(O_{T_{Y /S}^{∗}} − modules) and G in D^{b}(O_{T_{X/S}^{∗}} − modules) we define:

R^{·}f^{∗}E = R^{·}i_{∗}R^{·}p^{∗}E
R^{·}f_{Hig∗}G = p_{Y}^{∗} ω_{Y/S}^{−1} ⊗_{O_{T{∗ \atop Y /S}}} R^{·}p_{∗}R^{·}i^{∗}(G ⊗_{O_{T{∗ \atop X/S}}} p_{X}^{∗}ω_{Y/S})[dim Y − dim X]

Here p_{X} (resp. p_{Y}) denotes the projection T_{X/S}^{∗} → X (resp. T_{Y/S}^{∗} → Y). The properties of R^{·}f^{∗}, R^{·}f_{Hig∗} for Higgs modules are completely parallel to those for D-modules:

Lemma 17 Assume that f is smooth. Then i) R^{·}f_{Hig∗}G = R^{·}f_{·}(N ⊗_{O_{X}} Ω_{X/Y}^{·}) ii)The functor R^{·}f^{∗} is left adjoint to R^{·}f_{Hig∗}

2. Compatibility with the Cartier transform. Let f̃: X̃ → Ỹ be a morphism of smooth schemes over S̃. Denote by f̃′ : X̃′ → Ỹ′ the corresponding morphism of the Frobenius twists.

Theorem 18 For any Higgs module E in HIG^{p−1}(Y′/S), R^{i}f^{′∗}E are in HIG^{p−1}(X′/S) and there is a natural isomorphism C^{−1}(R^{i}f^{′∗}E) ≃ R^{i}f^{∗}C^{−1}(E)

To formulate the second main result, we assume, in addition, that a lifting F̃_{Ỹ /S̃} : Ỹ → Ỹ′ is given.

Theorem 19 Let M be an object of MIC^{n}(X/S), for p > n ≥ 0. Put m = p − n − 1. There is a canonical isomorphism in the derived category of O_{T_{Y ′/S}^{∗}}-modules:

τ_{≤m}Φ−1 ; F̃_{Ỹ /S̃}R^{·}f_{DR ∗}M ≃ τ_{≤m}R^{·}f_{Hig ∗}C_{X̃/S̃}M

5 Some examples and applications

1. Motivic D-modules.

In [F] Faltings introduced certain category MF_{0;k}^{∇}(X), an analog of the category of variations of Hodge structure in characteristic p. Using the notion of Cartier transform his definition can be reformulated as follows.

Definition 20 An object of the category MF_{0;k}^{∇}(X), (0 ≤ k < p) consists of the following data: 1) A coherent sheaf M on X with a integrable connection ∇ : M → M ⊗ Ω_{X/S}^{1}

2) A filtration 0 = F^{k+1} ⊂ F^{1} ⊂ ⋯ ⊂ F^{0} = M by O_{X}-submodules satisfying Griffithstransversality: ∇(F^{i}) ⊂ F^{i−1} ⊗ Ω_{X/S}^{1}

The above gives the O_{X}-linear maps F^{i}/F^{i+1} → F^{i−1}/F^{i} ⊗ Ω_{X/S}^{1}. That makes ⨁_{i} F^{i}/F^{i+1} a p_{∗}O_{T_{X/S}^{∗}}-module.

3)An isomorphism of p_{∗}O_{T_{X′/S}^{∗}}-modules:

ϕ : p_{∗}C_{X̃/S̃}(M) ≃ π_{∗}(⨁_{i} F^{i}/F^{i+1})

It is proven in [F] that the coherent sheaf M which admits the above structure is a vector bundle and that the category MF_{0;k}^{∇}(X) is abelian.

Let X̃ →^{f̃} Ỹ be smooth proper morphism of smooth schemes over S̃ and M an object of MF_{0;k}^{∇}(X). Recall that the D_{Y/S}-module R^{i}f_{DR∗}M is equipped with the filtration 0 = G^{k+1+i} ⊂ G^{1} ⊂ ⋯ ⊂ G^{0} = R^{i}f_{DR∗}M satisfying Griffiths-transversality.

Theorem 21 Assume that k + i < p. Then there is a canonical isomorphism

ϕ : p_{∗}C_{Ỹ /S̃}(R^{i}f_{DR∗}M) ≃ π_{∗}(⨁_{j} G^{j}/G^{j+1})

i.e. (R^{i}f_{DR∗}M; G^{·}) is an object of MF_{0;k+i}^{∇}(Y)

A proof can be found in [F]. It also follows from Theorem (19) combined with the following lemma.

Lemma 22 Let f : X → Y be smooth proper morphism, (M; F^{·}) a coherent sheaf together with a integrable connection and a filtration satisfying Griffiths-transversality. Assume that the Hodge spectral sequence associated to (M; F^{·}) degenerates. Then there is a canonical isomorphism of p_{∗}O_{T_{Y /S}^{∗}}- modules:

⨁_{j} G^{j}/G^{j+1} ≃ R^{i}f_{Hig ∗}(⨁_{j} F^{j}/F^{j+1})

2. The theorem of Barannikov and Kontsevich

Recall the following striking result of Barannikov and Kontsevich.

Theorem 23 Let X_{ℂ} be a quasi-projective smooth scheme over ℂ. Suppose f ∈ O_{X_{ℂ}}(X_{ℂ}) defines a proper morphism to 𝔸_{ℂ}^{1}. Then the hypercohomologies of the complexes

O_{X_{ℂ}} ⟶^{d+df} Ω_{X_{ℂ}}^{1} ⟶^{d+df} ⋯,

O_{X_{ℂ}} ⟶^{df} Ω_{X_{ℂ}}^{1} ⟶^{df} ⋯

have the same finite dimension in every degree.

We make use the theory of Cartier transform to prove the following more precise result.

Let K be a finite extension of ℚ, O_{K} the ring of integers in K, and S ⊂ spec O_{K} an open subscheme.

Theorem 24 Let X be a smooth scheme over S and f ∈ O_{X} (X) a function. For a point s ∈ S, denote by X_{s} the fiber X_{s} = X ×_{S} spec k(s). Then, for almost all closed points s ∈ S, the complexes

F_{∗}(O_{X_{s}} ⟶^{d+df} Ω_{X_{s}}^{1} ⟶^{d+df} ⋯),

O_{X′_{s}} ⟶^{df} Ω_{X′_{s}}^{1} ⟶^{df} ⋯

are quasi-isomorphic.

In particular, it gives an alternative proof of the Barannikov-Kontsevich theorem.

References

[BMR] Roman Bezrukavnikov, Ivan Mirkovic, and Dmitry Rumynin, Localization of modules for a semisimple Lie algebra in prime characteristic. arXiv:math RT/0205144 v.5

[F] Gerd Faltings, Crystalline cohomology and p-adic Galois representation. In Jun-Ichi Igusa, editor, Algebraic Analysis, Geometry, and Number Theory, pages 25-80, 1989