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An introduction to the Riemann-Hilbert Correspondence for Unit F -Crystals.

Matthew Emerton and Mark Kisin

In memory of Bernie Dwork

Introduction

Let X be a smooth scheme over ℂ, and (M, ∇) an integrable differential equation on X. That is, M is a locally free O_{X}-module of finite rank, equipped with an integrable connection ∇ : M → M ⊗_{O_{X}} Ω_{X}^{1}. The sheaf of local solutions of this differential equation forms a local system L of ℂ-vector spaces on X. One can recover (M, ∇) as L ⊗_{ℂ} O_{X}.

The Riemann-Hilbert correspondence extends the correspondence (M, ∇) ↔ L to the situation where the differential equation (M, ∇) may have singular points. The desirability of such an extension may be seen in geometry. If f : Y ⟶ X is a smooth proper map of smooth ℂ-schemes, then for any non-negative integer i, the relative De Rham cohomology R^{i}f_{∗}Ω_{Y/X}^{•} carries a connection called the Gauss-Manin connection, with sheaf of solutions naturally isomorphic to the higher direct image sheaf R^{i}f_{∗}ℂ. If f is not smooth, or not proper, however, then the sheaves R^{i}f_{∗}ℂ are not local systems in general; they are merely constructible sheaves. One would like to construct a “singular differential equation” on X, of which R^{i}f_{∗}ℂ is the sheaf of solutions.

The notion of singular differential equations is introduced in a precise way through the category of regular holonomic D-modules. The solutions of a regular holonomic D-module are indeed a constructible sheaf. To obtain an equivalence of categories, however, one must pass to derived categories. Making this transition, one finds that the derived solutions functor induces an equivalence of triangulated categories between the derived category of regular holonomic D-modules and the derived category of constructible sheaves. Furthermore, this equivalence respects

Grothendieck’s six operations f^{!}, f_{!}, f^{∗}, f_{∗}, \underline{Rhom}, ⊗^{𝕃} . (In particular, if f : Y → X is a map of smooth ℂ-schemes, not necessarily smooth or projective, then the complex Rf_{∗}ℂ is naturally isomorphic to the derived solutions of a complex of regular holonomic D-modules, usually denoted f_{+}O_{Y} .) This result, known as the “Riemann-Hilbert correspondence”, was first proved by Kashiwara and by Mebkhout (See [Bo] for an algebraic proof.)

The purpose of this paper is to describe some results whose proof is the subject of the papers [EK1], [EK2], and which provide a p-adic analogue of the Riemann-Hilbert correspondence. Our starting point is a theorem of Nick Katz [Ka, Prop. 4.1.1], which provides the analogue of the correspondence between differential equations and local systems.

Theorem 0.1. (Katz) Let k be a perfect field of characteristic p. If X is a smooth scheme over W_{n}(k), the ring of Witt vectors of length n, and if F_{X} is a lift to X of the Frobenius on W_{n}(k), then there is an equivalence of categories between the category of étale sheaves of locally free ℤ/p^{n} modules E of finite rank, and the category of locally free O_{X} modules E of finite rank, equipped with an O_{X} linear isomorphism F_{X}^{∗}E ˜ ⟶E. The equivalence is realized by associating E = E ⊗_{W_{n}(k)} O_{X} to E.

The sheaves E appearing in Katz’s theorem are the analogues of differential equations. The analogue of 𝒟-modules turn out to be sheaves over certain rings O_{F,X} (when n = 1) and 𝒟_{F,X} (in general). The first three sections of this paper outline the Riemann-Hilbert correspondence for unit O_{F,X}-modules proved in [EK 1]. We assume that n = 1, so that X is a smooth k-scheme. As in the case of differential equations, we obtain not just an (anti-)equivalence of triangulated categories, but also its compatibility with certain cohomological functors.

A consequence of the Riemann-Hilbert correspondence over ℂ is that one obtains a distinguished class of complexes of constructible ℂ-vector spaces, which correspond to a single regular holonomic 𝒟-module. These are called “perverse sheaves.” They form an abelian category, and have a description in term of the so called “middle extension”. In §4, we complete the results of [EK, §11] regarding the structure of “perverse sheaves” in our context, and show that they have a similar description in terms of middle extension.

In §5 we sketch the applications to L-functions that are the subject of [EK, §12]. Finally in §6 we explain the results of [EK 2] on the Riemann-Hilbert correspondence for a smooth W_{n}(k)-scheme. This relies heavily on the theory of arithmetic 𝒟-modules developed by Berthelot [Be].

To keep the paper a reasonable length, we have had to omit many details, and refinements. However we have tried to give indications of proofs where possible, and have included some typical calculations.

The study of unit root F -crystals and in particular their degeneration and Lfunctions was pioneered by Bernard Dwork. We dedicate this paper to his memory.

1. O_{F,X}-modules

An introduction to the Riemann Hilbert correspondence for unit F -crystals 3

1.1. Let k be a perfect field of characteristic p. For any k-algebra R, we denote by R[F] the non-commutative polynomial ring in a formal variable F, which satisfies the relation F a = a^{p}F for a ∈ R. For any k-scheme X, we denote by O_{F,X} the quasi-coherent sheaf of non-commutative rings which over on an affine open Spec R takes the value R[F]. As an (O_{X}, O_{X})-bimodule O_{F,X} is isomorphic to ⊕_{i=0}^{∞}F^{i∗}O_{X}, where F denotes the absolute Frobenius on X. Using this description, one sees that to give an O_{F,X}-module M is the same as giving the underlying O_{X}- module M together with the map

ϕ_{M} : F^{∗}M = F^{∗}O_{X} ⊗_{O_{X}} M → M  (1.1.1)

1.2. An O_{F,X}-module M is called a unit O_{F,X}-module if M is quasi-coherent as an O_{X}-module, and if the map ϕ_{M} is an isomorphism. There is a general technique for constructing unit O_{F,X}-modules, due to Lyubeznik [Lyu, 1.9]. Consider a quasi-coherent O_{X}-module M, and a map β : M → F^{∗}M. Set M = lim_{→}F^{i∗}M

where the transition maps are given by F^{i∗}β. Then M has a natural structure of a unit O_{F,X}-module, obtained by taking ϕ_{M} to be the limit of the maps

F^{∗}(F^{i∗}M) ˜ ⟶F^{i+1∗}M

Conversely, any unit O_{F,X}-module can be obtained in this way, since we may take M = M and β = ϕ_{M}^{−1}. We say that the morphism β generates M.

For our purposes the category of unit O_{F,X} which are, locally on X, finitely generated over O_{F,X} will be very important. They are the analogue, in our present situation, of holonomic 𝒟-modules. Over smooth schemes, one has the following characterisation of such modules [EK 1, 6.1.3]

Proposition 1.2.1. Let X be a smooth k-scheme, and M an O_{F,X}-module. Then M is a locally finitely generated unit O_{F,X}-module if and only if there exists an injective generator β : M → F^{∗}M for M, with M a coherent O_{X}-module.

In the situation of (1.2.1) an injective generator β : M → F^{∗}M with M coherent is called a “root” of M. From now on we will assume that X is a smooth k-scheme. We abbreviate the phrase “locally finitely generated unit” to “lfgu”. As an application of (1.2.1), we have the following structural result, which often enables one to make excision arguments.

Corollary 1.2.2. Let M be an lfgu O_{F,X}-module. There exists a dense open U ⊂ X, such that M|_{U} is a finite, locally free O_{X}-module.

Proof. Let β : M → F^{∗}M be a root of M. Then we may choose U so that M is locally free over U and β is an isomorphism. ⊓ ⊔

The following result shows that the subset U ⊂ X can be taken to be the biggest open subset over which M is coherent.

Proposition 1.2.3. Let M be a unit O_{F,X}-module, which is coherent as an O_{X}- module. Then M is locally free over O_{X}.

Proof. One proof may be found in [EK 1, 6.9.3]. Another, slicker, proof is as follows: We may work locally, and assume that X = Spec R for some integral domain R. The isomorphism ϕ_{M} implies that the Fitting ideals of M and F^{∗}M are equal. But if I^{k} is the k-th Fitting ideal of M then the k-th Fitting ideal of F^{∗}M is F (I^{k})R. This implies that I^{k} is either 0 or R. If k_{0} is the smallest integer such that I^{k_{0}} = R, then M is locally free of rank k_{0}. ⊓ ⊔

A coherent unit O_{F,X}-module is called an F -crystal.

1.3. Suppose that M is an O_{F,X} module. Then we have the O_{X}-linear map

−ϕ_{M} ⊕ id : F^{∗}M → M ⊕ F^{∗}M ⊂ ⊕_{i=0}^{∞}F^{∗i}M = O_{F,X} ⊗_{O_{X}} M

Extending this map O_{F,X}-linearly one gets a map O_{F,X} ⊗_{O_{X}} F^{∗}M → O_{F,X} ⊗_{O_{X}} M and it is not hard to check that this map sits in an exact sequence

0 → O_{F,X} ⊗_{O_{X}} F^{∗}M → O_{F,X} ⊗_{O_{X}} M → M → 0  (1.3.1)

Thus (1.3.1) provides a canonical resolution of any O_{F,X}-module, by O_{F,X} modules of the form O_{F,X} ⊗_{O_{X}} N. This resolution is an important tool in cohomological calculations. As a first simple application we have (cf. [EK 1, 1.8.4])

Lemma 1.3.2. Let M be an O_{F,X}-module. Then M has Tor-dimension at most dim X + 1.

Proof. The assertion is that for any right O_{F,X}-module N , Tor_{O_{F,X}}^{i} (M, N) vanishes for i > d + 1. These Tor groups may be computed by taking a resolution of M by locally free O_{F,X} modules [EK, 1.6.2]. To see that they vanish, it suffices, using the resolution (1.3.1), to check that they vanish for i > d when we replace M by O_{F,X} ⊗_{O_{X}} M. However, this follows because M has Tor-dimension at most d as an O_{X}-module (Serre’s theorem), together with the fact that O_{F,X} is flat as a right O_{X}-module. ⊓ ⊔

1.3.3. When M is a unit module, we may identify M with F^{∗}M via the isomorphism ϕ_{M}, and replace F^{∗}M by M in (1.3.1). Then the map −ϕ_{M} ⊕ id becomes −id ⊕ β : M → M ⊕ F^{∗}M where β = ϕ_{M}^{−1}. There is a useful generalisation of this construction: Let β : M → F^{∗}M be any generator of M. Then we have a map −id ⊕ β : M → M ⊕ F^{∗}M, which induces a map O_{F,X} ⊗_{O_{X}} M → O_{F,X} ⊗_{O_{X}} M as above, sitting in a short exact sequence

0 → O_{F,X} ⊗_{O_{X}} M → O_{F,X} ⊗_{O_{X}} M → M → 0  (1.3.4)

In particular, if M is an lfgu O_{F,X}-module, then M can be taken to be coherent.

2. Cohomological Operations

2.1. Let • be one of +, −, b or ∅. Denote by D^{•}(O_{F,X}) the derived category of complexes of O_{F,X} modules with the indicated boundedness condition. We denote by D_{qc}^{•}(X, O_{F,X}) (resp. D_{u}^{•}(O_{F,X}), resp. D_{lfgu}^{•}(O_{F,X})) the full subcategory of D^{•}(O_{F,X}) consisting of complexes with quasi-coherent (resp. unit, resp. lfgu) cohomology sheaves. If is easy to see that D_{qc}^{•}(O_{F,X}) is a triangulated subcategory of D^{•}(O_{F,X}). For D_{u}^{•}(O_{F,X}) and D_{lfgu}^{•}(O_{F,X}) this follows from the following result [EK 1, 5.2, 6.2.3]

Lemma 2.1.1. Let

M_{1} → M_{2} → M_{3} → M_{4} → M_{5}

be an exact sequence of O_{F,X}-modules. If each of M_{1}, M_{2}, M_{4}, and M_{5} is a unit (resp. lfgu) O_{F,X}-module, then M_{3} is a unit (resp. lfgu) O_{F,X}-module.

2.2. Let M and N be O_{F,X}-modules. Then we have a natural map

F^{∗}(M ⊗_{O_{X}} N) ˜ ⟶F^{∗}M ⊗_{O_{X}} F^{∗}N ⟶^{ϕ_{M}⊗ϕ_{N}} M ⊗ N

Thus M ⊗_{O_{X}} N has a natural structure of O_{F,X}-module.

We already remarked in the proof of (1.3.2) above, that any O_{F,X} module has a resolution by locally free O_{F,X}-modules, and that the terms of such a resolution are flat over O_{X}. Thus we can define the derived functor

⊗^{𝕃}_{O_{X}}: D^{−}(O_{F,X}) × D^{−}(O_{F,X}) → D^{−}(O_{F,X})

and (1.3.2) implies (in particular) that this restricts to

⊗^{𝕃}_{O_{X}}: D^{b}(O_{F,X}) × D^{b}(O_{F,X}) → D^{b}(O_{F,X})

This bifunctor respects the subcategories D_{qc}, D_{u} and D_{lfgu} [EK 1, 1.9.4, 5.5.2, 6.4.1]

Proposition 2.2.1. Let ∗ be one of qc, u or lfgu. Then ⊗^{𝕃}_{O_{X}} restricts to

⊗^{𝕃}_{O_{X}}: D_{∗}^{−}(O_{F,X}) × D_{∗}^{−}(O_{F,X}) → D_{∗}^{−}(O_{F,X})

2.3. Let f : Y → X be a morphism of smooth k-schemes. There is a map of rings f^{−1}(O_{F,X}) → O_{F,Y} induced by the natural map f^{−1}O_{X} → O_{Y} . Using this map we regard O_{F,Y} as a right f^{−1}O_{F,X}-module, and it has a natural structure as a left O_{F,Y} -module We denote by O_{F,Y →X} the ring O_{F,Y} considered as a (O_{F,Y} , f^{−1}O_{F,X})-bimodule.

We denote by d_{X} the function which to a point x ∈ X assigns the dimension of the connected component containing x. We set d_{Y/X} = d_{Y} − d_{X} ∘ f

For M in D^{−}(O_{F,X}), we set f^{!}M^{•} = O_{F,Y →X} ⊗^{𝕃}_{f^{−1}O_{F,X}} f^{−1}M^{•}[d_{Y/X}]. In

fact (1.3.2) can be used to show that f^{!} is even well defined for M^{•} in D(O_{F,X}). It also implies that f^{!} restricts to a functor

f^{!} : D^{b}(O_{F,X}) → D^{b}(O_{F,Y})

We have [EK 1, 2.3.2, 5.8, 6.7]

Proposition 2.3.1. Let ∗ be one of qc, u or lfgu. Then f^{!} induces a functor

f^{!} : D_{∗}^{−}(O_{F,X}) → D_{∗}^{−}(O_{F,Y})

Sketch of proof. The case ∗ = qc follows from the fact that on underlying com-

plexes of O_{X}-modules, f^{!} is given by O_{Y} ⊗^{𝕃}_{O_{X}} – followed by a shift. For the case ∗ = u, or lfgu one reduces to the case of a single module M, and X affine. If β : M → F^{∗}M is an injective generator for M, we may take a resolution P^{•} of M by finite free O_{X}-modules, and lift β to a map β′ : P^{•} → F^{∗}P^{•}. If P^{•} denotes the complex of unit O_{F,X} modules generated by β′, then one easily checks that P^{•} is a resolution of M and that, up to a shift, f^{!}P^{•} is the complex generated by f^{∗}β. In particular, f^{!}M has unit cohomology sheaves. If in addition, M is locally finitely generated, then (1.2.1) implies that we may take M, and hence the terms of P^{•}, to be coherent. In this case f^{!}P^{•} is a complex of lfgu O_{F,X}-modules, and in particular has cohomology sheaves with are lfgu O_{F,Y} -modules ⊓ ⊔

We record two more properties of the functor f^{!} [EK 1, 2.4, 2.5]

Proposition 2.3.2. 1. If g : Z → Y and f : Y → X are maps of smooth k-schemes, then there exists a natural isomorphism (fg)^{!} ˜ ⟶g^{!}f^{!}

2. If M^{•} and N^{•} are in D^{−}(O_{F,X}) then there exists a canonical isomorphism

f^{!}(M^{•} ⊗^{𝕃}_{O_{X}} N^{•}) ˜ ⟶f^{!}M^{•} ⊗_{O_{Y}} f^{!}N^{•}[−d_{Y/X}]

2.4. Let f : Y → X be a map of smooth k-schemes. We denote by ω_{Y/X} the relative dualising sheaf. This is isomorphic to ω_{Y} ⊗_{O_{Y}} f^{∗}ω_{X}^{−1}, which is a line bundle, since X and Y are smooth. The (f^{−1}O_{F,X}, O_{Y})-bimodule f^{−1}O_{F,X} ⊗_{f^{−1}O_{X}} ω_{Y/X} can be given the structure of a (f^{−1}O_{F,X}, O_{F,Y})-bimodule [EK 1, 3.3.1]. The construction involves the formalism of duality theory for quasi-coherent sheaves, and especially the Cartier operator. Rather than giving it in generality, we describe the bimodule structure explicitly in two special cases.

In the first example, we take O_{Y} = O_{X}/a, where a cuts out the smooth divisor Y of X. In this case we have an exact sequence

0 → O_{Y} · da → f^{∗}Ω_{X}^{1} → Ω_{Y}^{1} → 0

so that ω_{Y/X} = O_{Y} · (da)^{−1}. The required right O_{F,Y} -module structure is then given by [EK 1, 3.3.4]

bF^{n}(da)^{−1} ↦^{·F} ba^{p^{n}(p−1)}F^{n+1}(da)^{−1}

where b is a section of O_{X}.

Next suppose that Y → X is smooth of relative dimension 1. Then locally, Y admits an étale map to the X-scheme, 𝔾_{m} × X. If t denotes the canonical co-ordinate on 𝔾_{m}, then this induces an invertible section t of O_{Y} , and ω_{Y/X} is generated by dt/t. The required right O_{F,Y} -module structure is then given by

aF^{n}t^{i}dt/t ↦^{·F} aF^{n+1}t^{i/p}dt/t

where a is a section of f^{−1}O_{X}, and t^{i/p} = 0 if p ∤ i.

We denote the (f^{−1}O_{F,X}, O_{F,Y})-bimodule f^{−1}O_{F,X} ⊗_{f^{−1}O_{X}} ω_{Y/X} by O_{F,X←Y} . If g : Z → Y is another morphism of smooth k-schemes, then the natural isomorphism ω_{Z/X} ˜ ⟶ω_{Z/Y} ⊗_{O_{Z}} g^{∗}ω_{Y/X} induces an isomorphism of ((fg)^{−1}O_{F,X}, O_{Z})- bimodules

O_{F,X←Z} ˜ ⟶g^{−1}O_{F,X←Y} ⊗_{g^{−1}O_{F,Y}} O_{F,Y ←Z}  (2.4.1)

In fact this map is an isomorphism of (f^{−1}O_{F,X}, O_{F,Z})-bimodules [EK 1, 3.3.3]. The calculation of the general right O_{F,Y} -module structure on O_{F,X←Y} can be reduced to the two special cases considered above, by locally factoring any map Y → X as a composite of a closed embedding, and smooth curve fibrations, and using the isomorphism (2.4.1).

We define a functor

f_{+} : D^{−}(O_{F,Y}) → D^{−}(O_{F,X})

by

f_{+}M^{•} = Rf_{∗}(O_{F,X←Y} ⊗^{𝕃}_{O_{F,Y}} M^{•})

By (1.3.2), and the fact that Rf_{∗} has finite cohomological dimension, f_{+} is well defined (in fact we could have even worked with unbounded complexes). Moreover, it induces a functor

f_{+} : D^{b}(O_{F,Y}) → D^{b}(O_{F,X})

As for f^{!}, we have the following result [EK 1, 3.5.3, 5.8, 6.8.4]

Proposition 2.4.2. Let ∗ be one of qc, u, or lfgu. Then f_{+} restrict to a functor

f_{+} : D_{∗}^{−}(O_{F,Y}) → D_{∗}^{−}(O_{F,X})

We record some other properties of f_{+}

Proposition 2.4.3. If f : Y → X and g : Z → Y are morphisms of smooth k-schemes, then

1. There exists a canonical isomorphism (fg)_{+} ˜ ⟶g_{+}f_{+}.

2. If M^{•} is in D^{−}(O_{F,Y}) and N^{•} is in D_{qc}^{−}(O_{F,X}) then we have the projection formula

f_{+}(M^{•} ⊗^{𝕃}_{O_{Y}} f^{!}N^{•}) ˜ ⟶M^{•}[d_{Y/X}] ⊗^{𝕃}_{O_{X}} f_{+}f^{!}N^{•}

3. If M^{•} is in D^{−}(O_{F,X}) and N^{•} is in D^{b}(O_{F,Y}), and f is an open immersion, then there is a canonical isomorphism

Rf_{∗}\underline{Rhom}_{O_{F,Y}} (f^{!}M^{•}, N^{•}) ˜ ⟶\underline{Rhom}_{O_{F,X}} (M^{•}, f_{+}N^{•})

4. If M^{•} is in D_{qc}^{−}(O_{F,Y}) and N^{•} is in D_{qc}^{b}(O_{F,X}), and f is a proper map, then there is a canonical isomorphism

\underline{Rhom}_{O_{F,X}} (f_{+}M^{•}, N^{•}) ˜ ⟶Rf_{∗}\underline{Rhom}_{O_{F,Y}} (M^{•}, f^{!}N^{•})

Proof. (1) is a formal consequence of the isomorphism (2.4.1). (2) and (3) are formal consequences of respectively, the projection formula and the adjointness between Rf_{∗} and f^{−1} for Zariski sheaves [EK 1, 4.2, 4.3]. The proof of (4) is one of most delicate points in the whole theory, and uses Grothendieck duality, expressed on the level of residual complexes [Ha, Ch VI]. A key point is the construction of the trace map, explained in the following lemma [EK 1, 4.4.9]. ⊓ ⊔

Lemma 2.4.4. Let f : Y → X be a proper map of smooth k-schemes. Then the complex of O_{F,X}-modules f_{+}O_{F,Y} has a natural structure of an object in the derived category of (O_{F,X}, O_{F,X})-bimodules, and there exists a map

tr_{F,f} : f_{+}O_{F,Y} [d_{Y/X}] → O_{F,X}

in this derived category, which on the level of complexes of O_{F,X}-modules becomes the map

f_{+}O_{F,Y} [d_{Y/X}] = O_{F,X} ⊗_{O_{X}} Rf_{∗}ω_{Y/X}[d_{Y/X}] →^{1⊗tr_{f}} O_{F,X}

where tr_{f} : Rf_{∗}ω_{Y/X}[d_{Y/X}] → O_{X} is the trace map of Grothendieck-Serre duality.

2.5. In this section we explain the analogue of Kashiwara’s theorem for unit modules. The following result is proved in [EK 1, 5.10.1, 5.10.3]

Theorem 2.5.1. Let f : Y → X be a closed immersion of smooth k schemes.

1. If M is a unit O_{F,X}-module supported on Y, then \underline{H}^{0}(f^{!}M) ˜ ⟶f^{!}M.

2. If M is a unit O_{F,Y} -module then \underline{H}^{0}(f_{+}M) ˜ ⟶f_{+}M.

3. The functors \underline{H}^{0}(f^{!}M) and \underline{H}^{0}(f_{+}M) induce an equivalence between the category of unit O_{F,Y} modules, and the category of unit O_{F,X}-modules supported on Y.

4. If M is any unit O_{F,X}, then adjunction map f_{+}f^{!}M → M induced by the adjointness of (2.4.3)(4) induces an inclusion f_{+}\underline{H}^{0}(f^{!}M) ⊂ M which identifies f_{+}\underline{H}^{0}(f^{!})M with the subsheaf Γ_{Y} (M) of M, consisting of sections supported on Y.

We also have an analogous result for the derived category [EK 1, 5.11.3, 5.11.5].To explain it, for any closed subset Y ⊂ X, denote by D_{u,Y}^{−} (O_{F,X}) the full (triangulated) subcategory of D_{u}^{−}(O_{F,X}) consisting of complexes whose cohomology sheaves are supported on Y.

Corollary 2.5.2. Let f : Y → X be a closed immersion of smooth k-schemes.

1. The functor

f_{+} : D_{u}^{−}(O_{F,Y}) → D_{u}^{−}(O_{F,X})

induces an equivalence of D_{u}^{−}(O_{F,Y}) with D_{u,Y}^{−} (O_{F,X}). The functor f^{!} provides a quasi-inverse.

2. If M^{•} is in D_{u}^{b}(O_{F,X}), then RΓ_{Y} (M^{•}) has a natural structure of an object of D_{u,Y} (O_{F,X}), and there is an isomorphism

RΓ_{Y} (M^{•}) ˜ ⟶f_{+}f^{!}M^{•})

The following result is often used in [EK 1] to make arguments by induction on the support of M^{•} [EK 1, 5.12.1]

Corollary 2.5.3. Let M^{•} be in D_{u}^{−}(O_{F,X}), and let Y be the support of the cohomology sheaves of M. Then for any smooth open subscheme U ⊂ Y, if f : U → X denotes the natural map, then there is a natural map M^{•} → f_{+}f^{!}M^{•} whose cone is supported on Y \U.

Proof. Let W = X \ (Y \ U). Then U is closed in W and W is open in X. Let i : U → W and j : W → X denote the corresponding immersions, so that f = ji. Then by (2.4.3)(3), there is the natural morphism of adjunction M^{•} → j_{+}j^{!}M^{•}. Since M^{•} is supported on Y, we see that j^{!}M^{•} is supported on Y ∩ W = U. Thus by (2.5.2) we see that

j^{!}M^{•} ˜ ⟶i_{+}i^{!}j^{!}M^{•}  (2.5.4)

Thus we get maps

M^{•} → j_{+}j^{!}M^{•} →^{∼} j_{+}i_{+}i^{!}j^{!}M^{•} ˜ ⟶f_{+}f^{!}M^{•}

Applying j^{!} to the composite of these maps recovers the isomorphism (2.5.4), so this composite map has cone supported on X \ W = Z \ U. ⊓ ⊔

3. The Riemann-Hilbert correspondence

3.1. We now move from working in the Zariski site, to the étale site. If X is a smooth k-scheme, we denote by π_{X} : X_{ét} → X_{Zar} the natural morphism of sites. We set O_{F,X_{ét}} = π_{X}^{∗}O_{F,X}. We call a O_{F,X_{ét}}-module unit (resp. lfgu) if it has this property when restricted to the Zariski site of an étale covering of X. Recall that the exact functors π_{X}^{∗} and π_{X}^{∗} induce quasi-inverse equivalences of categories between quasi-coherent sheaves on X_{ét} and quasi-coherent sheaves on X_{Zar}. Using this one sees easily that every unit (resp. lfgu) O_{F,X_{ét}}-module M_{ét} has the form M_{ét} = π_{X}^{∗}M for some unit (resp. lfgu) O_{F,X}-module.

For • equal to one of ∅, +, −, or b, and ∗ one of ∅, qc, u, or lfgu we denote by D_{∗}^{•}(O_{F,X_{ét}}) the derived category of O_{F,X_{ét}}-modules with the indicated boundedness condition, and whose cohomology sheaves satisfy the condition ∗. If ∗ is one of qc, u or lfgu, then π_{X}^{∗} induces an equivalence of triangulated categories between this category and D_{∗}^{•}(O_{F,X}) [EK 1, 7.3.2]. This essentially follows from descent, and a theorem of Bernstein which implies that any complex in D^{b}(O_{F,X_{ét}}) is quasiisomorphic to a complex of quasi-coherent O_{F,X}-modules.

For a map of smooth k-schemes f : Y → X, we can define the bimodules O_{F,Y →X} and O_{F,X←Y} , and the functors ⊗_{O_{X_{ét}}} , f^{!} and f_{+}. When restricted to bounded complexes with quasi-coherent cohomology, these satisfy

π_{X}^{∗}(– ⊗_{O_{X}} –) ˜ ⟶π_{X}^{∗}(–) ⊗_{O_{X_{ét}}} π_{X}^{∗}(–), π_{Y}^{∗} f^{!} ˜ ⟶f^{!}π_{X}^{∗} and π_{X}^{∗}f_{+} ˜ ⟶π_{Y}^{∗} f_{+}

3.2. For • as above, we denote by D^{•}(X_{ét}, ℤ/pZ) the derived category of étale sheaves of ℤ/pℤ-modules on X with the indicated boundedness condition. We denote by D_{c}^{•}(X_{ét}, ℤ/pℤ) the full triangulated subcategory of D^{•}(X_{ét}) consisting of complexes with constructible cohomology sheaves.

Now define the functor

Sol_{ét}(–) = \underline{Rhom}_{O_{F,X_{ét}}} (–, O_{X_{ét}})[d_{X}] : D_{lfgu}^{−}(O_{F,X_{ét}}) → D^{+}(X_{ét}, ℤ/pℤ)

This functor has the following properties [EK 1, 9.3, 9.7.1, 9.8, 9.9]

Proposition 3.2.1. Let f : Y → X be a morphism of smooth k-schemes.

1. There is a natural isomorphism f^{−1} Sol_{ét} ˜ ⟶ Sol_{ét} f^{!}

2. If f is quasi-projective, there is a natural isomorphism Sol_{ét} f_{+} ˜ ⟶f_{!} Sol_{ét} .

3. There is a natural isomorphism

Sol_{ét}(–) ⊗^{𝕃}_{ℤ/pℤ} Sol_{ét}(–) ˜ ⟶ Sol_{ét}(– ⊗^{𝕃}_{O_{X_{ét}}} –)[d_{X}]

4. Sol_{ét} restricts to a functor

Sol_{ét} : D_{lfgu}^{b}(O_{F,X_{ét}}) → D_{c}^{b}(X_{ét}, ℤ/pℤ)

Sketch of proof. Each of these claims is non-trivial. For (1), the required natural transformation is deduced from the map

\underline{Rhom}_{O_{F,X_{ét}}} (M^{•}, O_{X_{ét}}) → \underline{Rhom}_{O_{F,Y_{ét}}} (f^{!}M^{•}, f^{!}O_{X_{ét}})

and the remark that f^{!}O_{X_{ét}} = O_{Y_{ét}}[d_{Y/X}].

We will explain how to check that this map is an isomorphism in the special case that Y is a smooth divisor, cut out by a section a of O_{X}, and M^{•} is a single lfgu O_{F,X}-module, with a generator of the form μ : O_{X_{ét}}^{n} → O_{X_{ét}}^{n}. The general case can be reduced to this one using various formal arguments, and the technique explained in the proof of (2.3.1).

Using the resolution (1.3.4) to compute Sol_{ét}, one sees that what we have to show is that the natural map

f^{−1}(O_{X_{ét}}^{n} →^{1−μF} O_{X_{ét}}^{n}) → (O_{Y_{ét}}^{n} →^{1−μF} O_{Y_{ét}}^{n})  (3.2.2)

is a quasi-isomorphism. Both sides of (3.2.2) are surjective in the étale topology (because the differential of 1 − μF is the identity), so we have to show (3.2.2) induces an isomorphism on kernels.

Suppose that a section (ax_{i}) of aO_{X_{ét}}^{n} satisfies (1−μF)(ax_{i}) = 0. Then (ax_{i}) = μF (ax_{i}) = a^{p}μF (x_{i}), and

(x_{i}) = a^{p−1}μF (x_{i}) = a^{p−1}μF (a^{p−1}μF (x_{i}))

= a^{(p+1)(p−1)}μF μF (a^{p−1}μF (x_{i})) = …. (3.2.3)

Continuing, we find that the x_{i} are divisible by arbitrarily high powers of a, and so vanish. This shows that (3.2.2) induces an injective map on kernels. To see the surjectivity, suppose that (x̄_{i}) are sections of (O_{X_{ét}}/a)^{n} which satisfy (1 − μF^{r})(x̄_{i}) = 0. Let (x_{i}) be a section of O_{X_{ét}}^{n} lifting (x̄_{i}). Then (1 − μF)(x_{i}) = (ax′_{i}) for some section (x′_{i}) of O_{X_{ét}}. Now étale locally, we may find a section (y_{i}) of O_{X_{ét}}^{n} solving (1 − a^{p−1}μF)(y_{i}) = (x′_{i}). Then x_{i} − ay_{i} = x̄_{i} modulo a, and

(1 − μF)(x_{i} − ay_{i}) = (ax′_{i} − ax′_{i}) = 0

For the proof of (2) one considers separately the case of a proper map, and of an open immersion. The general isomorphism is then constructed by factoring a quasi-projective map as a composite of such maps (one has to work to check that the result is independent of the factorisation). The case of a proper map can be deduced from the adjointness in (2.4.3)(4). For the case of f, an open immersion, we have a natural isomorphism f^{−1} Sol_{ét}(f_{+}M^{•}) ˜ ⟶ Sol_{ét}(M^{•}), because the definition of Sol_{ét} is local on X. By adjointness of f_{!} and f^{−1}, this gives a map Sol_{ét}(f_{+}M^{•}) ˜ ⟶f_{!} Sol_{ét}(M^{•}). To show this is an isomorphism one has to show that the stalks of Sol_{ét}(f_{+}M^{•}) have vanishing cohomology at points of X\Y. This is done (after some reductions) by an explicit calculation, similar to the one we explained in the proof of (1).

The proofs of (3) and (4) proceed by induction on the support of (the cohomology sheaves of) M^{•}, using (1.2.2) and (2.5.3). This argument uses (1) and (2). Note that when M is an F -crystal (i.e. a unit module, which is finite flat over O_{X_{ét}}) of rank n, then computing Sol_{ét}(M) by using the resolution (1.3.4), with the root μ = ϕ_{M}^{−1}, one sees that

Sol_{ét}(M) = (M →^{1−μF} M)[d_{X}]

Then Katz’s theorem, mentioned in the introduction, implies that Sol_{ét}(M) = L[d_{X}] where L is a locally constant étale sheaf of ℤ/pℤ-modules of rank n. ⊓ ⊔ 3.3. We define a functor

M_{ét} = \underline{Rhom}_{ℤ/pℤ}(–, O_{X_{ét}})[d_{X}] : D_{c}^{b}(X_{ét}, ℤ/pℤ) → D^{+}(O_{F,X_{ét}})

We have the following result [EK 1, 10.4]

Proposition 3.3.1. The functor M_{ét} induces a functor

M_{ét} : D_{c}^{b}(X_{ét}, ℤ/pℤ) → D_{lfgu}^{b}(O_{F,X_{ét}})

The proof of (3.3.1) uses excision and induction on the dimension of the support of (the cohomology sheaves) of a complex L^{•} in D_{c}^{b}(X_{ét}, ℤ/pℤ), together with the fact that if L is an étale local system, then M_{ét}(L) is easily seen to be an F -crystal. As in the proof of (3.2.1), in order to carry out this excision argument, one first has to prove that in certain special cases, M_{ét} exchanges f_{+} and f_{!}, and f^{!} and f^{−1}.

3.4. Now using (3.2.1)(4) and (3.3.1) we can define functors

Sol = Sol_{ét} ∘π_{X}^{∗} : D_{lfgu}^{b}(O_{F,X}) → D_{c}^{b}(X_{ét}, ℤ/pℤ)

and

M = Rπ_{X∗} ∘ M_{ét} : D_{c}^{b}(X_{ét}, ℤ/pℤ) → D_{lfgu}^{b}(O_{F,X})

The natural map

M^{•} → \underline{Rhom}_{ℤ/pℤ}(\underline{Rhom}_{O_{F,X_{ét}}} (M^{•}, O_{X_{ét}}), O_{X_{ét}})

induces a natural transformation ζ : id → M ∘ Sol . Similarly, we get a natural transformation η : id → Sol ∘ M . We have the following theorem [EK 1, 11.3]

Theorem 3.4.1. The natural transformations ζ and η are isomorphisms. In particular M and Sol induce quasi-inverse anti-equivalences of triangulated categories.

They exchange the functors ⊗^{𝕃}_{ℤ/pℤ} and ⊗^{𝕃}_{O_{X}} (up to a shift), and f^{−1} and f^{!}. If f is quasi-projective they exchange f_{!} and f_{+}.

Sketch of proof. That Sol takes f^{!}, f_{+} and ⊗^{𝕃}_{O_{X}} into f^{−1} f_{!} and ⊗^{𝕃}_{ℤ/pℤ} follows from (3.2.1). The analogous claim for M follows once we know that ζ and η are isomorphisms. This is established by an excision argument. ⊓ ⊔

4. Perverse Sheaves

4.1. The canonical t-structure on D_{lfgu}^{b}(O_{F,X}) induces via the anti-equivalence of categories in (3.4.1), a t-structure on D_{c}^{b}(X_{ét}, ℤ/pℤ). To describe this exotic t-structure explicitly we recall the notion of a perversity function.

Let X be a k-scheme. For x ∈ X, denote by i_{x} : x → X the natural inclusion. Recall that the “middle perversity” function on X is defined by p(x) = − dim \overline{\{x\}}. Define full sub-categories ^{p}D^{≤0} and ^{p}D^{≥0} of D_{c}^{b}(X_{ét}, Λ) by the conditions:

F^{•} is in ^{p}D^{≤0} if and only if for all x ∈ X, \underline{H}^{i}(i_{x}^{−1}F^{•}) = 0 for i > p(x).

F^{•} is in ^{p}D^{≥0} if and only if for all x ∈ X, \underline{H}^{i}(i_{x}^{!}F^{•}) = 0 for i < p(x).

Gabber has shown that the subcategories ^{p}D^{≤0} and ^{p}D^{≥0} underlie a (necessarily unique) t-structure on D_{c}^{b}(X_{ét}, ℤ/pℤ) [Ga]. (In fact, Gabber establishes this more generally for perversity functions p having the property that p(x) ≤ p(y)+1 whenever x is an immediate specialisation of y.) In [EK 1, 11.5] we show that this t-structure is equal to the one induced on D_{c}^{b}(X_{ét}, ℤ/pℤ) by the anti-equivalence of categories in (3.4.1). In particular, the analysis there gives an independent proof that the categories ^{p}D^{≤0} and ^{p}D^{≥0} underlie a t-structure on D_{c}^{b}(X_{ét}, ℤ/pℤ).

In this section we want to supplement the results of loc. cit. by proving a structure theorem for complexes in the heart of this t-structure – “perverse sheaves”

– and showing that there is a reasonable theory of middle extension. These results were motivated by those of Blickle’s thesis [Bl], which constructs the middle extension in certain cases.

4.2. The next result follows immediately from [Lyu, Thm. 3.2]

Proposition 4.2.1. Let X be a smooth k-scheme. The category of lfgu O_{F,X}- modules is Artinian.

Corollary 4.2.2. Let j : U → X be a locally closed immersion of smooth kschemes. If M is an lfgu O_{F,U}-module, then the set of lfgu O_{F,X}-submodules M̃ ⊂ \underline{H}^{0}(j_{+}M) such that j^{!}M̃ = M has a smallest element.

Proof. If M̃_{1} and M̃_{2} are two such submodules, then we have an exact sequence

0 → M̃_{1} ∩ M̃_{2} → \underline{H}^{0}(j_{+}M) → \underline{H}^{0}(j_{+}M)/M̃_{1} ⊕ \underline{H}^{0}(j_{+}M)/M̃_{2}

By (2.1.1) the right most term is lfgu, so applying (2.1.1) again shows that M̃_{1} ∩ M̃_{2} is an lfgu O_{F,X}-module. Now the corollary follows from (4.2.1). ⊓ ⊔

We denote the minimal extension M̃ in (4.2.2) by j_{!+}M. We call an lfgu O_{F,X}- module simple if it has no non-trivial, proper lfgu O_{F,X}-submodules.

Corollary 4.2.3. Every simple lfgu O_{F,X}-module on X is of the form j_{!+}N for a locally closed immersion j : U → X, and some simple F -crystal N on U.

Proof. Suppose that N is a simple F -crystal on U. Let M ⊂ j_{!+}N be a non-zero submodule. Let Y denote the closure of U in X, set W = X\(Y \U), and let h : W → X and i : U → W denote the natural (open, resp. closed) immersions. First we claim that M is not supported on Y \U. Indeed, the composite M → \underline{H}^{0}(j_{+}N) → j_{+}N ˜ ⟶h_{+}i_{+}N (in which the isomorphism is provided by (2.4.3)(1)) corresponds, by the adjointness of (2.4.3)(3), to a map h^{!}M → i_{+}N . If M were supported on Y \U, then h^{!}M would vanish, hence the map M → h_{+}i_{+}N = j_{+}N would also vanish, and thus M would vanish, contradicting our assumption. Thus j^{!}M is a non-zero submodule of N. Since N is simple, it follows that j^{!}M = N , which implies that M = j_{!+}N as the latter module is minimal.

Conversely, suppose that M is a simple lfgu O_{F,X}-module. Let Y be the support of M and choose a dense open U ⊂ Y such that, if j : U → X denotes the natural map, then j^{!}M is an F -crystal. Now the map M → j_{+}j^{!}M is non-zero, as it becomes an isomorphism after applying j^{!}. Since M is simple, this map must be an injection, whence j_{!+}j^{!}M ⊂ M. Since M is simple, this inclusion must be an isomorphism. Finally if N ⊂ j^{!}M is a non-zero subcrystal, then j_{!+}N ⊂ M, and this must be an isomorphism, as M is simple, so that N = j^{!}M. Thus, j^{!}M is simple. ⊓ ⊔

4.3. It remains to translate the results of the previous section into statements involving D_{c}^{b}(X_{ét}, ℤ/pℤ). We call a complex L^{•} in D_{c}^{b}(X_{ét}, ℤ/pℤ) a perverse sheaf if it is of the form Sol_{ét}(M) for a single lfgu O_{F,X}-module M. We begin by showing that the construction on perverse sheaves, which corresponds to the functor j_{!+}, has a familiar description.

The conditions given in (4.1) to define ^{p}D^{≤0} and ^{p}D^{≥0} may be applied equally well to complexes of sheaves in D^{b}(X, ℤ/pℤ), and it is shown in [Ga] that they determine a t-structure on this category. (Indeed, Gabber’s strategy is to construct this t-structure first, and then to show that it restricts to a t-structure on D_{c}^{b}(X, ℤ/pℤ), by showing that the perverse truncation of a complex with constructible cohomology sheaves again has constructible cohomology sheaves.) We denote the cohomology functors of this t-structure by ^{p}\underline{H}^{i}. If j : U → X is a locally closed immersion, and L^{•} is a perverse sheaf in D_{c}^{b}(U_{ét}, ℤ/pℤ), then we set

j_{!∗}L^{•} = Im(^{p}\underline{H}^{0}(j_{!}L^{•}) → ^{p}\underline{H}^{0}(Rj_{∗}L^{•})).

Lemma 4.3.1. Let j : U → X be a locally closed immersion, and M an lfgu O_{F,U}-module. If L^{•} = Sol(M) is the perverse sheaf corresponding to M then

1. j_{!}L^{•} is in ^{p}D^{≤0} and Rj_{∗}L^{•} is in ^{p}D^{≥0}

2. We have a canonical isomorphism Sol(j_{!+}M) ˜ ⟶j_{!∗}L^{•}

Proof. To prove (1) let i_{x} : x → X be the inclusion of a point of X. We have to check that \underline{H}^{i}(i_{x}^{−1}j_{!}L^{•}) = 0 for i > p(x) and \underline{H}^{i}(i_{x}^{!}Rj_{∗}L^{•}) = 0 for i < p(x). Let Y be the closure of U in X. If x ∉ Y \U then these equalities follow from the analogous facts for the perverse sheaf L^{•}. If x ∈ Y \U, then we have i_{x}^{−1}j_{!}L^{•} = i_{x}^{!}Rj_{∗}L^{•} = 0, so these equalities are trivial.

Next we prove (2). Let L^{′•} ⊂ ^{p}\underline{H}^{0}(j_{!}L^{•}) be a perverse subsheaf. Consider the composite map

L^{′•} → ^{p}\underline{H}^{0}(j_{!}L^{•}) → ^{p}\underline{H}^{0}(Rj_{∗}L^{•})  (4.3.2)

If the map (4.3.2) vanishes then L^{′•} is supported on Y \U because j^{−1}j_{!∗}L^{•} = L^{•}. Conversely, if L^{′•} is supported on Y \U, then we claim that (4.3.2) vanishes. By (1) it suffices to check that the map obtained by composing (4.3.2) with p\underline{H}^{0}(Rj_{∗}L^{•}) → Rj_{∗}L^{•} vanishes. Since L^{′•} is supported on Y \U this last map factors through a map

L^{′•} → i_{∗}i^{!}Rj_{∗}L^{•} = 0,

where i : Y \U → X denotes the inclusion.

It follows that j_{!∗}L^{•} is equal to the quotient of ^{p}\underline{H}^{0}(j_{!}L^{•}) by its maximal perverse subsheaf supported on Y \U. Now (2) follows from (3.4.1) and the definition of j_{!+}M. ⊓ ⊔

Corollary 4.3.3. The category of perverse sheaves is Artinian. A perverse sheaf is irreducible if and only if it is of the form j_{!∗}L[d_{U}], where j : U → X is a locally closed immersion, and L is an irreducible local system of ℤ/pℤ-sheaves on U.

Proof. The first claim follows from (4.2.1). The second claim follows from (4.2.3) and (4.3.1)(2), keeping in mind that M ↦ \underline{H}^{0}(Sol(M)[−d_{U}]) establishes an equivalence between the category of F -crystals on U, and the category of locally constant étale sheaves of finite ℤ/pℤ-modules (cf. the proof of (3.2.1)). ⊓ ⊔

We remark that in the notes [Ga], Gabber gives another proof that the category of perverse sheaves is Artinian.

5. L-functions

5.1. In this section we will define L-functions for lfgu O_{F,X}-modules and explain some of their properties. The theory explained in previous sections has its strongest applications to L-functions if it is first generalised to include “coefficients”. Thus, in this section we let Λ be a Noetherian 𝔽_{p}-algebra, and we set O_{X}^{Λ} = Λ ⊗_{𝔽_{p}} O_{X} and O_{F,X}^{Λ} = Λ ⊗_{𝔽_{p}} O_{F,X}. We define O_{X_{ét}}^{Λ} and O_{F,X_{ét}}^{Λ} similarly.

We say that a O_{F,X}^{Λ} is “unit” (resp. quasi-coherent) if its underlying O_{F,X}- module is unit (resp. quasi-coherent). We say that a O_{F,X}^{Λ}-module is lfgu if it is unit, and if, locally on X, it is finitely generated over O_{F,X}^{Λ}.

For • = ∅, −, +, or b, and ∗ one of qc, u or lfgu we define D_{∗}^{•}(O_{F,X}^{Λ}) to be the full (triangulated) subcategory of D^{•}(O_{F,X}^{Λ}) consisting of complexes whose cohomology sheaves satisfy the condition ∗.

5.2. If f : Y → X is a morphism of smooth k-schemes, we can define functors

f^{!} : D^{−}(O_{F,X}^{Λ}) → D^{−}(O_{F,Y}^{Λ} )

and

f_{+} : D^{−}(O_{F,X}^{Λ}) → D^{−}(O_{F,X}^{Λ})

using the same definitions as in §2. These functors also preserve the conditions qc, u, and lfgu (except for the last case, this is a formal consequence of the results cited in §2). For proofs of these results we refer to the same references in [EK 1] cited in §2. We also have a bifunctor

– ⊗^{𝕃}_{O_{X}^{Λ}} – : D^{−}(O_{F,X}^{Λ}) × D^{−}(O_{F,X}^{Λ}) → D^{−}(O_{F,X}^{Λ})

but we will not need this in what follows.

Having introduced coefficients, we can also define functors corresponding to extension of scalars: Let Λ → Λ′ be a morphism of Noetherian 𝔽_{p}-algebras. Then we have a functor

⊗^{𝕃}_{Λ} Λ′ : D^{−}(O_{F,X}^{Λ}) → D^{−}(O_{F,X}^{Λ′})

We have the following result

Proposition 5.2.1. If ∗ is one of qc, u or lfgu, then ⊗^{𝕃}_{Λ} Λ′ induces a functor

⊗^{𝕃}_{Λ} Λ′ : D_{∗}^{−}(O_{F,X}^{Λ}) → D_{∗}^{−}(O_{F,X}^{Λ′})

Moreover, – ⊗^{𝕃}_{Λ} Λ′ commutes with f^{!} and f_{+}.

Proof. For the first claim see [EK 1, 1.13, 5.6, 6.5], and for the second [EK 1, 2.8, 3.10] ⊓ ⊔

5.3. If Λ is a finite ring we have a version of the Riemann-Hilbert correspondence. To explain it, we denote by D_{lfgu}^{b}(O_{F,X}^{Λ})^{∘} the full subcategory of D_{lfgu}^{b}(O_{F,X}^{Λ}) consisting of complexes which have finite Tor dimension when viewed as complexes of O_{X}^{Λ}-modules. Note that if Λ is regular, then O_{X}^{Λ} is a sheaf of regular rings, and this finiteness condition is automatic.

We denote by D_{ctf}^{b}(X_{ét}, Λ) the full subcategory of D_{c}^{b}(X_{ét}, Λ) consisting of complexes of sheaves of Λ-modules which have finite Tor dimension.

We define functors

Sol(–) = \underline{Rhom}_{O_{F,X_{ét}}^{Λ}} (π_{X}^{∗}–, O_{X_{ét}}^{Λ}) : D_{lfgu}^{b}(O_{F,X}^{Λ})^{∘} → D(X_{ét}, Λ)

M(–) = Rπ_{X∗}\underline{Rhom}_{Λ}(–, O_{X_{ét}}^{Λ}) : D_{ctf}^{b}(X_{ét}, Λ) → D(O_{F,X}^{Λ})

The Riemann-Hilbert correspondence with “Λ-coefficients” then takes the form (with the same references in [EK 1] cited in §3)

Theorem 5.3.1. Suppose Λ is a finite ring. The functors Sol and M induce quasi-inverse anti-equivalences of triangulated categories between D_{lfgu}^{b}(O_{F,X}^{Λ})^{∘}

and D_{ctf}^{b}(X_{ét}, Λ). It exchanges the functors ⊗^{𝕃}_{O_{X}^{Λ}} and ⊗^{𝕃}_{Λ} (up to a shift), and f^{!} and f^{−1}. If f is quasi-projective, it exchanges f_{+} and f_{!}. The equivalence is

compatible with – ⊗^{𝕃}_{Λ} Λ′ for any map of finite 𝔽_{p}-algebras Λ → Λ′.

5.4. We now begin our definition of L-functions. We assume from now on that k is a finite field. Suppose first that Λ is a product of finitely many fields, and that X = x = Spec k(x) where k(x) is a finite extension of k.

Let M be a lfgu O_{F,x}^{Λ}.-module. Then M is actually finite as a Λ⊗_{𝔽_{p}} k′-module. This follows from the Λ-coefficient version of (1.2.1). Alternatively, in this very simple situation, it follows from the fact that M is finitely generated over Λ[F], and is also a module over the commutative ring Λ[F, F^{−1}]. Since Λ is a finite product of fields, M is finite projective over Λ ⊗_{𝔽_{p}} k(x). Let |k(x)| = p^{s} for some integer s. The structural isomorphism ϕ_{M} : F^{∗}M ˜ ⟶M induces a map

ϕ_{M,x}^{−1} : F^{s−1∗}ϕ_{M}^{−1} ∘ F^{s−2∗}ϕ_{M}^{−1} ∘ ⋯ ∘ ϕ_{M}^{−1} : M ˜ ⟶F^{s∗}M → M

We define

L_{u}(x, M) = det_{Λ⊗_{𝔽p} k(x)}(1 − ϕ_{M,x}^{−1}T^{s}|M)^{−1}

A priori L_{u}(x, M) lies in Λ ⊗_{𝔽_{p}} k(x)[[T]] but one can show that it actually lies in Λ[[T]] [EK 1, 12.1.2].

More generally, for M^{•} in D_{lfgu}^{b}(O_{F,x}^{Λ}) we define

L_{u}(x, M^{•}) = ∏_{i} L_{u}(x, \underline{H}^{i}(M^{•}))^{(−1)^{i}}

Next if X is any smooth k-scheme, and M^{•} is in D_{lfgu}^{b}(O_{F,X}^{Λ}) we define

L_{u}(X, M^{•}) = ∏_{x∈|X|} L_{u}(x, i_{x}^{!}M^{•})

where |X| denotes the set of closed points in X, and i_{x} : x → X denotes the inclusion.

Finally, if Λ is any reduced ring, we denote by Q(Λ) its total ring of fractions and for M^{•} in D_{lfgu}^{b}(O_{F,X}^{Λ}), we define

L_{u}(X, M^{•}) = L_{u}(X, M^{•} ⊗^{𝕃}_{Λ} Q(Λ))

5.5. If Λ is finite we have the following result [EK 1, 12.3.1]

Proposition 5.5.1. Suppose Λ is a finite reduced ring (i.e a product of finite fields), let M^{•} be in D_{lfgu}^{b}(O_{F,X}^{Λ}) and set L^{•} = Sol(M^{•}). Then we have

L_{u}(X, M^{•}) = L_{ét}(X, L^{•})  (5.5.2)

where the right hand side is the L-function defined in [De, p. 116].

For the proof one reduces immediately to the case where X is a point, and the result is then a simple computation.

In fact the right hand side of (5.5.2) is defined for more general Λ, so one might ask if (5.5.2) holds for any (i.e not necessarily reduced) finite 𝔽_{p}-algebra Λ. For this one first needs a definition of the left hand side. Such a definition is given in [EK 1, 12.1.5], and (5.5.2) then holds. However, we will not need this below.

One initial draw back of the definition of L_{u}(X, M^{•}) we have given is that it is not a priori clear that L_{u}(X, M^{•}) is compatible with change of rings, or that it lies in Λ[[T]]. The key to overcoming these difficulties is the following result which proves this in a special case

Proposition 5.5.3. Let λ : Λ → Λ′ be a map of reduced Noetherian 𝔽_{p}-algebras, and suppose that Λ is regular. If M^{•} is in D_{lfgu}^{b}(O_{F,X}^{Λ}) then L_{u}(X, M^{•}) lies in Λ[[T]], and we have

L_{u}(X, M^{•} ⊗^{𝕃}_{Λ} Λ′) = λ(L_{u}(X, M^{•}))

The main point of the proof is that, because Λ has finite projective dimension, one can compute L_{u}(X, M^{•}) without passing to Q(Λ).

As a consequence of (5.5.3) one deduces [EK 1, 12.4.5, 12.4.6]

Theorem 5.5.4. Let λ : Λ → Λ′ be a morphism of Noetherian, reduced 𝔽_{p}- algebras, and M in D_{lfgu}^{b}(O_{F,X}^{Λ}). Then

1. If Λ is normal, then L_{u}(X, M^{•}) is in Λ[[T]].

2. If Λ is of finite type over 𝔽_{p}, M^{•} is in D_{lfgu}^{b}(O_{F,X}^{Λ})^{∘}, and L_{u}(X, M^{•}) lies in Λ[[T]] then

L_{u}(X, M^{•} ⊗^{𝕃}_{Λ} Λ′) = λ(L_{u}(X, M^{•}))

Sketch of proof. For (1), note that Λ is a product of normal domains, and it suffices to consider the case where Λ is a normal domain. Then for every height one prime 𝔭 of Λ we have that L_{u}(X, M^{•}) is in Λ_{𝔭}[[T]] by (5.5.3). On the other hand we have Λ = ∩_{𝔭}Λ_{𝔭}.

The proof of (2) is more involved, and uses de Jong’s theorem on resolution of singularities to reduce to the case Λ regular, when one can apply (5.5.3). ⊓ ⊔

It is now easy to deduce the following result [EK 1, 12.4.3]

Theorem 5.5.5. Let f : Y → X be a morphism of smooth k-schemes, Λ a reduced Noetherian k-algebra, and M^{•} in D_{lfgu}^{b}(O_{F,X}^{Λ}). Then we have

L_{u}(X, M^{•}) = L_{u}(X, f_{+}M^{•})  (5.5.6)

Sketch of proof. To prove the theorem one first reduces by a limit argument to the case where Λ is of finite type over 𝔽_{p}, and then to the case where Λ is smooth over 𝔽_{p}. It suffices to show (5.5.6) holds after specialising both sides modulo every maximal ideal of Λ. Thus (5.5.3) implies that we may assume that Λ is a finite field. Then (5.5.6) follows from (5.3.1) and an analogous formula for complexes of constructible étale Λ-sheaves [De, 2.2, p. 116]. ⊓ ⊔

As a consequence of this theorem we have the following result which includes a conjecture of Goss proved by Taguchi-Wan [TW].

Corollary 5.5.7. Let M^{•} be in D_{lfgu}^{b}(O_{F,X}^{Λ}). Then L_{u}(X, M^{•}) is a rational function.

6. 𝒟_{F,X}-modules

6.1. We again let k be any perfect field. In this section we explain how to use arithmetic 𝒟-modules [Be] to generalise the results of §1 - 3 to schemes which are smooth over W_{n}(k) for some positive integer n. Before giving the formal definitions, we explain the basic idea.

Let X be a smooth W_{n}(k) scheme. Suppose first that X admits a local system of co-ordinates x_{1}, …, x_{d}, - i.e an étale map X → 𝔸_{W_{n}(k)}^{d}. Consider the ring 𝒟_{𝔸_{/ℚp}^{d}} of differential operators on d-dimensional affine space 𝔸_{/ℚ_{p}} over ℚ_{p}. This is a “twisted polynomial ring”:

D_{𝔸_{/ℚp}^{d}} = ℚ_{p}[x_{1}, …, x_{d}, ∂_{x_{1}}, …, ∂_{x_{d}}],

with the commutation relations [∂_{x_{i}}, x_{j}] = δ_{ij}. (Here δ_{ij} denotes the usual Kronecker delta.) For any multi-index I = (i_{1}, ⋯, i_{d}), let ∂^{[I]} = \frac{1}{i_{1}!···i_{d}!}∂_{x_{1}}^{i_{1}} ⋯ ∂_{x_{d}}^{i_{d}}. Then the ℤ_{p}-submodule of D_{𝔸_{/ℚp}^{d}} generated by monomials in the elements x_{i} (as

i ranges between 1 and d) and ∂^{[I]} (as I ranges over all multi-indices) forms a ℤ_{p}-subalgebra of D_{𝔸_{/ℚp}^{d}} , which is in fact the ring of differential operators D_{𝔸_{/ℤp}^{d}} on d-dimensional affine space over ℤ_{p}.

Now consider the composite map h : X → 𝔸_{W_{n}(k)}^{d} → 𝔸_{W (k)}^{d} → 𝔸_{ℤ_{p}}^{d} We set 𝒟_{X} = O_{X} ⊗_{h^{−1}O_{𝔸{d \atop ℤp}}} h^{−1}𝒟_{𝔸_{ℤp}^{d}} . This is the full ring of differential operators on X,

and does not depend on the choice of co-ordinates.

Next let F be a lift of Frobenius to X. Such a lift can be constructed by choosing a lift of Frobenius to 𝔸_{W_{n}(k)}^{n} (e.g x_{i} ↦ x_{i}^{p}), which then lifts uniquely to X by étaleness.

We denote by 𝒟_{F,X} the sheaf of rings obtained from 𝒟_{X} by adjoining an indeterminate F which satisfies the commutation relations

F x_{i} = F (x_{i})F, and ∂_{x_{i}}F = ∂_{x_{i}}(F (x_{i}))F ∂_{x_{i}}.

where x is a section of O_{X}. The key observation is that 𝒟_{F,X} does not depend on the choice of the Frobenius lift F : If F′ is a second lift then, F′ can be expressed by Taylor’s formula

F′ = ∑_{I}(F′(x_{i}) − F (x_{i}))^{I}F ∂^{[I]}

6.2. As a result of the independence of 𝒟_{F,X} of Frobenius lift, we can define a sheaf 𝒟_{F,X} for any smooth W_{n}(k)-scheme X, without assuming the existence of a global Frobenius lift.

The construction of 𝒟_{F,X} can be efficiently expressed in terms of Berthelot’s interpretation of Frobenius descent [Be 2, §2.2]. Namely, Berthelot shows that there is always a functor F^{∗} taking 𝒟_{X}-modules to 𝒟_{X}, and which locally on X realises pullback by Frobenius. Thus, we may set 𝒟_{F,X} = ⊕_{i≥0}F^{∗i}𝒟_{X}. By functoriality F^{i∗}𝒟_{X} is a (𝒟_{X}, 𝒟_{X})-bimodule, and the multiplication in 𝒟_{F,X} is defined via the isomorphism.

F^{i∗}𝒟_{X} ⊗_{𝒟_{X}} F^{j∗}𝒟_{X} ˜ ⟶F^{i+j∗}𝒟_{X}

One checks [EK 2, 1.4.5] that this makes 𝒟_{F,X} into an associative sheaf of 𝒟_{X}- algebras.

The following result result describes 𝒟_{F,X}-modules, in terms of 𝒟_{X}-modules, and is useful for transporting information from one theory to the other [EK 2, 1.4.7, 1.6.4]

Proposition 6.2.1. 1. To give a left 𝒟_{F,X}-module is equivalent to giving a left 𝒟_{X}-module M, and a map of left 𝒟_{X}-modules ϕ_{M} : F^{∗}M → M.

2. To give a right 𝒟_{F,X}-module is equivalent to giving a right 𝒟_{X}-module M and a map of right 𝒟_{F,X}-modules M → F^{!}M.

Here F^{!} is the functor of coherent duality theory, and we have used that if M is a right 𝒟_{X}=module, then F^{!}M has a canonical structure of right 𝒟_{X}-module [Be 2, §2.4], [EK 2, 1.5.1].

6.3. We call a (left) 𝒟_{F,X}-module M a unit 𝒟_{F,X}-module if M is quasi-coherent as an O_{X}-module, and the map ϕ_{M} in (6.2.1)(1) is an isomorphism. We call M an lfgu 𝒟_{F,X}-module if it is a unit module, and is locally finitely generated over 𝒟_{F,X}. The connection between these notions and the corresponding ones for O_{F,X}-modules is given by the following result [EK 2, 3.2.3]

Proposition 6.3.1. Let X be a smooth k-scheme.

1. For any unit O_{F,X}-module M there is a natural isomorphism

M →^{1⊗id} 𝒟_{F,X} ⊗^{𝕃}_{O_{F,X}} M

In particular, M has a natural 𝒟_{F,X}-module structure.

2. The functor M ↦ 𝒟_{F,X} ⊗_{O_{F,X}} M induces an equivalence between the category of unit O_{F,X}-modules, and the category of unit 𝒟_{F,X}-modules.

3. If M is a unit 𝒟_{F,X}-module then M is locally finitely generated as a 𝒟_{F,X}- module if and only if it is locally finitely generated as an O_{F,X}-module.

6.4. As usual we let • denote one of ∅, +, −, or b, and ∗ one of ∅, qc, u or lfgu. We denote by D^{•}(𝒟_{F,X}) the derived category of 𝒟_{F,X}-modules with the indicated bounded condition. We denote by D_{∗}^{•}(𝒟_{F,X}) the full subcategory of D^{•}(𝒟_{F,X}) consisting of complexes whose cohomology sheaves satisfy the condition indicated by ∗. As in §2, each of these subcategories is triangulated [EK 2, 3.1.2, 3.3.4].

In addition to these categories, we denote by D_{∗}^{b}(𝒟_{F,X})^{∘} the full (triangulated) subcategory of D_{∗}^{b}(𝒟_{F,X}) consisting of complexes which have finite Tor dimension as complexes of O_{X}-modules.

As in §2, we have a bifunctor

– ⊗^{𝕃}_{O_{X}} – : D^{−}(𝒟_{F,X}) × D^{−}(𝒟_{F,X}) → D^{−}(𝒟_{F,X})

We also define, for a map of smooth W_{n}(k)-schemes f : Y → X, functors f^{!} and f_{+}. As in §2, the key point is to define the appropriate bimodules.

To construct f^{!} we remark that for any 𝒟_{F,X}-module M, f^{∗}M has a natural structure of 𝒟_{F,Y} -module [EK 2, 2.2.2]. In fact, f^{∗}M is natural a 𝒟_{Y} -module [Be, §2.1], and we have a map

F^{∗}f^{∗}M ˜ ⟶f^{∗}F^{∗}M →^{f^{∗}ϕ_{M}} f^{∗}M

which makes f^{∗}M a 𝒟_{F,Y} -module by (5.2.1)(1).

In particular 𝒟_{F,Y →X} := f^{∗}𝒟_{F,X} is a (𝒟_{F,Y} , f^{−1}𝒟_{F,X})-bimodule by functoriality, and we define a functor

f^{!} : D^{−}(𝒟_{F,X}) → D^{−}(𝒟_{F,Y})

by

f^{!}(M^{•}) = 𝒟_{F,Y →X} ⊗^{𝕃}_{f^{−1}𝒟_{F,X}} f^{−1}M^{•}[d_{Y/X}]

The definition of the (f^{−1}𝒟_{F,X}, 𝒟_{F,Y})-bimodule D_{F,X←Y} used to define f_{+} is more delicate [EK 2, 2.3.6]. Let us only mention that its underlying left f^{−1}𝒟_{X}- module is f^{∗}(𝒟_{F,X} ⊗_{O_{X}} ω_{X}^{−1}) ⊗_{O_{Y}} ω_{Y} . The key points in constructing the right 𝒟_{F,Y} -module structure is first to note that this module has a right 𝒟_{Y} -module structure, by the same argument as in [Be, 2.4.1], and then use the fact that there is an isomorphism of 𝒟_{Y} -modules F^{!}ω_{Y} ˜ ⟶ω_{Y} (obtained from the compatibility of “(–)^{!}” with composition), together with (6.2.1)(2). We define the functor

f_{+} : D^{−}(𝒟_{F,Y}) → D^{−}(𝒟_{F,X})

by

f_{+}M^{•} = Rf_{∗}(𝒟_{F,X←Y} ⊗^{𝕃}_{D_{F,Y}} M^{•})

We remark that on underlying 𝒟_{Y} -modules this functor agrees with the one of [Be, §2.4].

Proposition 6.4.1. Let • be one of b or −, and ∗ one of ∅, qc, u, or lfgu. Also if • = b, let × be one of ∅ or ∘, and ∅ otherwise. If f : Y → X is a map of smooth W_{n}(k)-schemes then the functors f^{!} and f_{+} restrict to functors

f^{!} : D_{∗}^{•}(𝒟_{F,X})^{×} → D_{∗}^{•}(𝒟_{F,Y})^{×}.

and

f_{+} : D_{∗}^{•}(𝒟_{F,Y})^{×} → D_{∗}^{•}(𝒟_{F,X})^{×}.

The bifunctor – ⊗^{𝕃}_{O_{X}} – restricts to a functor

– ⊗^{𝕃}_{O_{X}} – : D_{∗}^{•}(𝒟_{F,X})^{×} × D_{∗}^{b}(𝒟_{F,X})^{∘} → D_{∗}^{•}(𝒟_{F,X})^{×}.

Proof. See [EK 2, 2.2.6, 2.3.13, 2.6, 3.5.1]. ⊓ ⊔

The functors ⊗^{𝕃}_{O_{X}} f^{!}, and f_{+} satisfy analogues of the properties listed in §2 for the case of O_{F,X}-modules. In particular there is an analogue of Kashiwara’s theorem, and f_{+} is left adjoint to f^{!} for proper maps. Rather than repeat the statements, we refer the reader to [EK 2, §2, 3].

6.5. Let X be a smooth W_{n}(k)-scheme. As in (3.1), we denote by π_{X} : X_{ét} → X_{Zar} the natural morphism of sites, and we set O_{X_{ét}} = π_{X}^{∗}, and 𝒟_{F,X_{ét}} = π_{X}^{∗}𝒟_{F,X}. We denote by D_{ctf}^{b}(X_{ét}, ℤ/p^{n}ℤ) the full (triangulated) subcategory of the category D^{b}(X_{ét}, ℤ/p^{n}ℤ), consisting of complexes which have constructible cohomology, and finite Tor dimension over ℤ/p^{n}ℤ.

The sheaf O_{X_{ét}} is naturally a sheaf of 𝒟_{F,X_{ét}}-modules, and so we may define the functors

Sol(–) = \underline{Rhom}_{𝒟_{F,X_{ét}}} (π_{X}^{∗}–, O_{X_{ét}}) : D_{lfgu}^{b}(𝒟_{F,X})^{∘} → D(X_{ét}, ℤ/p^{n}ℤ)

and

M(–) = Rπ_{X∗}\underline{Rhom}_{ℤ/p^{n}ℤ}(–, O_{X_{ét}}) : D_{ctf}^{b}(X_{ét}, ℤ/p^{n}ℤ) → D(𝒟_{F,X})

The Riemann-Hilbert correspondence for unit 𝒟_{F,X}-modules is the following result [EK 2, 4.2.8]

Theorem 6.5.1. The functors Sol and M induce quasi-inverse, anti-equivalences of triangulated categories between D_{lfgu}^{b}(𝒟_{F,X})^{∘} and D_{ctf}^{b}(X_{ét}, ℤ/p^{n}ℤ). They ex-

change the functors ⊗^{𝕃}_{ℤ/p^{n}ℤ} and ⊗^{𝕃}_{O_{X}} (up to a shift) and f^{−1} and f^{!}. If f is quasi-projective, they exchange f_{!} and f_{+}.

The proof of the theorem uses a dévissage technique to reduce to the case n = 1, when one can apply the results of [EK 1], thanks to (6.3.1).

References

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gies p-adiques et applications arithemtiques (II), 1–80, Astérisque 270, 2002. [Ber 2] P. Berthelot 𝒟-modules arithmetiques II. Descente par Frobenius, Mémoires

Soc. Math. France 81, 2000.

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[Bo] A. Borel, et al. Algebraic D-modules, Perspectives in Mathematics 2, Birkhäuser-Boston, 1984.

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[EK1] M. Emerton, M. Kisin A Riemann-Hilbert Correspondence for Unit F - Crystals. I, 143 pages, submitted.

[EK2] M. Emerton, M. Kisin A Riemann-Hilbert Correspondence for Unit F - Crystals. II, 69 pages, submitted.

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[Ka] N. Katz p-adic Properties of Modular Forms, Modular Functions of One Variable 69–190, Lecture Notes in Math. 350, Springer, 1973.

[Lyu] G. Lyubeznik F -modules: applications to local cohomology and D-modules in characteristic p > 0, J. Reine Angew. Math. 491, 1997, 65–130.

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Matthew Emerton, Northwestern University, Evanston

Email: emerton@math.northwestern.edu

Mark Kisin, University of Chicago, Chicago

Email: kisin@math.uchicago.edu