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Riemann–Hilbert Correspondence for Unit F-Crystals On R AN IE N R A U L E O S F D T E U L T I ’ I T N S ANNALES DE L’INSTITUT FOURIER Sachio OHKAWA Riemann–Hilbert correspondence for unit F -crystals on embeddable algebraic varieties Tome 68, no 3 (2018), p. 1077-1120. <http://aif.cedram.org/item?id=AIF_2018__68_3_1077_0> © Association des Annales de l’institut Fourier, 2018, Certains droits réservés. Cet article est mis à disposition selon les termes de la licence CREATIVE COMMONS ATTRIBUTION – PAS DE MODIFICATION 3.0 FRANCE. http://creativecommons.org/licenses/by-nd/3.0/fr/ L’accès aux articles de la revue « Annales de l’institut Fourier » (http://aif.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://aif.cedram.org/legal/). cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Ann. Inst. Fourier, Grenoble 68, 3 (2018) 1077-1120 RIEMANN–HILBERT CORRESPONDENCE FOR UNIT F -CRYSTALS ON EMBEDDABLE ALGEBRAIC VARIETIES by Sachio OHKAWA Abstract. — For a separated scheme X of finite type over a perfect field k of characteristic p > 0 which admits an immersion into a proper smooth scheme over the truncated Witt ring Wn, we define the bounded derived category of lo- cally finitely generated unit F -crystals with finite Tor-dimension on X over Wn, independently of the choice of the immersion. Then we prove the anti-equivalence of this category with the bounded derived category of constructible étale sheaves n of Z/p Z-modules with finite Tor-dimension. We also discuss the relationship of t-structures on these derived categories when n = 1. Résumé. — Soit X un schéma séparé de type fini sur un corps parfait k de caractéristique p > 0 et qui admet une immersion vers un schéma propre et lisse sur l’anneau des vecteurs de Witt tronqué Wn. Nous définissons la catégorie dé- rivée bornée des F -cristaux unités localement finiment engendrés sur Wn avec Tor-dimension finie sur X, indépendemment de l’immersion. Nous provons ensuite que cette catégorie est anti-équivalente à la catégorie dérivée bornée des faisceaux n constructibles étales de Z/p Z-modules avec Tor-dimension finie. Nous étudions aussi les relations de t-structures sur ces catégories dérivées lorsque n = 1. 1. Introduction For a complex manifold X, Kashiwara [9] and Mebkhout [12] indepen- dently established an anti-equivalence, which is called the Riemann–Hilbert b correspondence, between the triangulated category Drh(DX) of DX -modules b with regular holonomic cohomologies and that Dc(X, C) of sheaves of C- vector spaces on X with constructible cohomologies. There is a significant property from the point of view of relative cohomology theories that this ! ∗ anti-equivalence respects Grothendieck’s six operations f , f!, f , f∗, RHom L b b and ⊗ defined on Drh(DX ) and Dc(X, C). Keywords: D-modules, Frobenius structure, étale sheaves. 2010 Mathematics Subject Classification: 14F30, 14F10. 1078 Sachio OHKAWA In [4], Emerton and Kisin studied a positive characteristic analogue of the Riemann–Hilbert correspondence. Let k be a perfect field of charac- teristic p > 0. We denote by Wn := Wn(k) the ring of Witt vectors of length n. For a smooth scheme X over Wn, Emerton and Kisin defined the sheaf DF,X of OX -algebras by adjoining to OX the differential op- erators of all orders on X and a “local lift of Frobenius”. By using DF,X , b they introduced the triangulated category Dlfgu(DF,X ) of DX -modules with Frobenius structures with locally finitely generated unit cohomologies and proved the anti-equivalence b ◦ ∼= b n Dlfgu(DF,X ) −→ Dctf (X´et, Z/p Z) b ◦ b between the subcategory Dlfgu(DF,X ) of Dlfgu(DF,X ) consisting of com- plexes of finite Tor dimension over OX and the triangulated category b n n Dctf (X´et, Z/p Z) of étale sheaves of Z/p Z-modules with constructible co- homologies and of finite Tor dimension over Z/pnZ, which they call the Riemann–Hilbert correspondence for unit F -crystals. They also introduced three of Grothendieck’s six operations, which are the direct image f+, the ! L b inverse image f and the tensor product ⊗ on Dlfgu(DF,X ), and proved −1 that their Riemann–Hilbert correspondence exchanges these to f!, f and L b n ⊗ on Dctf (X´et, Z/p Z). Emerton and Kisin established the Riemann–Hilbert correspondence for unit F -crystals only for smooth schemes X over Wn. Since the triangulated b n category Dctf (X´et, Z/p Z) depends only on the mod p reduction of X, it is natural to expect that there exists a definition of the triangulated category b ◦ Dlfgu(DF,X ) and the Riemann–Hilbert correspondence depending only on the mod p reduction of X. Also, there should be the Riemann–Hilbert correspondence for algebraic varieties over k which are not smoothly liftable to Wn. The purpose of this article is to generalize the Emerton–Kisin theory to the case of Wn-embeddable algebraic varieties over k. Here we say a separated k-scheme X of finite type is Wn-embeddable if there exists a proper smooth Wn-scheme P and an immersion X,→ P such that the diagram X / P (1.1) Spec k / Spec Wn is commutative. A quasi projective variety over k is a typical example of Wn-embeddable variety and thus Wn-embeddable varieties form a suffi- ciently wide class of algebraic varieties in some sense. ANNALES DE L’INSTITUT FOURIER RIEMANN–HILBERT CORRESPONDENCE 1079 The first problem is to define a reasonable D-module category for Wn- embeddable algebraic varieties over k. Our construction is based on Kashi- wara’s theorem which roughly asserts that, for any closed immersion X,→ P of smooth algebraic varieties, the category of D-modules on P supported on X is naturally equivalent to the category of D-modules on X. Using the characteristic p > 0 analogue of Kashiwara’s theorem due to Emerton– Kisin [4, Prop. 15.5.3], we show that, when we are given the diagram (1.1), b ◦ the full triangulated subcategory of Dlfgu(DF,P ) consisting of complexes supported on X does not depend on the choice of immersion X,→ P b ◦ (Corollary 4.6). We denote this full subcategory by Dlfgu(X/Wn) . Then we show the Riemann–Hilbert correspondence b ◦ ∼= b n Dlfgu(X/Wn) −→ Dctf (X´et, Z/p Z) for any Wn-embeddable k-scheme X (Theorem 4.12). As in the case of [4], we can naturally introduce three of Grothendieck’s six operations, that is, direct and inverse images and tensor products. We then prove that the Riemann–Hilbert correspondence respects these operations (Theorem 4.14). A striking consequence of the Riemann–Hilbert correspondence over com- plex numbers is that one can introduce an exotic t-structure on the topo- logical side called the perverse t-structure, which corresponds to the stan- dard t-structure on the D-module side. For an algebraic variety X over b k, Gabber introduced in [5] the perverse t-structure on Dc(X´et, Z/pZ), which we call Gabber’s perverse t-structure. In the case when X is smooth over k, Emerton and Kisin showed that the standard t-structure on the D-module side corresponds to Gabber’s perverse t-structure. In this paper, we generalize it to the case of k-embeddable k-schemes. In the complex situation, conversely, a t-structure on the D-module side corresponding to the standard t-structure on the topological side is explicitly described by Kashiwara in [10]. In this paper, we construct the analogue of Kashiwara’s b t-structure on Dlfgu(X/k) and discuss the relationship of it and the stan- b dard t-structure on Dc(X´et, Z/pZ). The content of each section is as follows: In the second section, we re- call several notions, terminologies and cohomological operations on DF,P - modules from [4] which we often use in this paper. We also recall the statement of the Riemann–Hilbert correspondence for unit F -crystals of Emerton–Kisin (Theorem 2.3). In the third section, we define the local co- homology functor RΓZ for DF,P -modules and prove compatibilities with RΓZ and other operations for DF,P -modules, which are essential tools to b ◦ define and study the triangulated category Dlfgu(X/Wn) for any Wn- embeddable k-scheme X. In subsection 4.1, we introduce the category TOME 68 (2018), FASCICULE 3 1080 Sachio OHKAWA b ◦ Dlfgu(X/Wn) for any Wn-embeddable k-scheme X and in subsection 4.2, b ◦ we construct three of Grothendieck’s six operations on Dlfgu(X/Wn) . Our arguments in these subsections are heavily inspired by that of Caro in [2]. In subsection 4.3, we prove the Riemann–Hilbert correspondence for unit F -crystals on Wn-embeddable k-schemes, which is our main result. In the b fifth section, we discuss several properties on Dlfgu(X/k) (in the case n = 1) related to t-structures. In subsection 5.1, we introduce the standard t- b structure on Dlfgu(X/k) depending on the choice of the immersion X,→ P . We prove that the standard t-structure corresponds to Gabber’s perverse t-structure via the Riemann–Hilbert correspondence. As a consequence, we know that the definition of the standard t-structure is independent of the choice of X,→ P (Theorem 5.5). In subsection 5.2, we define the abelian b category µlfgu,X as the heart of the standard t-structure on Dlfgu(X/k), b b and prove that the natural functor D (µlfgu,X ) → Dlfgu(X/k) is an equiv- alence of triangulated categories (Theorem 5.6), which can be regarded as an analogue of Beilinson’s theorem.
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