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Stability of the holonomy on quasi-projective varieties. (Stabilité de l’holonomie sur les variétés quasi-projectives.) (French. English summary) Zbl 1263.14024

Let \(p\) be a prime number, \(V\) a mixed characteristic complete discrete valuation ring with perfect residue field \(k\), of characteristic \(p\). Let \({\mathcal P}\) a smooth formal scheme over \(V\) and \(P\) its special fiber. In this context, P. Berthelot [Astérisque 279, 1–80 (2002; Zbl 1098.14010)] constructed a category of arithmetic holonomic \({\mathcal D}_{{\mathcal P}}\)-modules with Frobenius structure, and proved that this category is stable by the following cohomological operations: extraordinary inverse image by a smooth morphism, external tensor product, dual functor. He conjectured that this category is stable by Grothendieck \(6\) operations. It turns out that this category is attached to the \(k\)-variety \(P\).
On the other hand, Caro constructed a category of overholonomic arithmetic \({\mathcal D}\)-modules with Frobenius structure, and could prove the stability by Grothendieck \(6\) operations of this category [D. Caro and N. Tsuzuki, Ann. Math. (2) 176, No. 2, 747–813 (2012; Zbl 1276.14031)]. Moreover, the overholonomicity implies the holonomicity in the sense of Berthelot. Thus, to prove Berthelot’s conjectures, it is enough to prove the converse statement: and that holonomicity in the sense of Berthelot implies overholonomicity. Caro does this in this paper in the case of quasi-projective varieties. By general arguments, he reduces the statement to the projective case. The restriction to the quasi-projective case is technical and involves another construction of arithmetic \({\mathcal D}\)-modules due to Z. Mebkhout and L. Narvaez-Macarro [Lect. Notes Math. 1454, 267–308 (1990; Zbl 0727.14011)] over weak formal schemes. More precisely, consider \({\mathcal P}\) a smooth projective formal scheme, \(H_0\) a divisor of the special fiber \(P\) of \({\mathcal P}\), and \({\mathcal E}\) a coherent arithmetic \({\mathcal D}\)-module with overconvergent singularities along \(H_0\), then Caro proves that, after enlarging \(H_0\), \({\mathcal E}\) comes from an overconvergent isocrystal. This allows him to prove that, holonomicity with Frobenius structure, in the sense of Berthelot, implies overholonomicity, in the case of quasi-projective formal schemes (Théorème 2.1.5).
As an important corollary of his comparison result, Caro proves Berthelot’s conjectures for holonomic \({\mathcal D}\)-modules for quasi-projective \(k\)-varieties. Indeed, Caro defines the category of holonomic \({\mathcal D}\)-modules (arithmetic \({\mathcal D}\)-modules, with Frobenius structures) over any quasi-projective \(k\)-varieties and proves that this category is stable by Grothendieck \(6\)-operations. This comes of course of the fact that overholonomic \({\mathcal D}\)-modules are stable the Grothendieck \(6\)-operations.

MSC:

14F30 \(p\)-adic cohomology, crystalline cohomology
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
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References:

[1] doi:10.1090/S1056-3911-04-00381-9 · Zbl 1065.14020 · doi:10.1090/S1056-3911-04-00381-9
[4] doi:10.1007/s00208-010-0539-x · Zbl 1307.14025 · doi:10.1007/s00208-010-0539-x
[5] doi:10.1112/S0010437X05001880 · Zbl 1167.14012 · doi:10.1112/S0010437X05001880
[7] doi:10.1007/s00222-007-0070-1 · Zbl 1203.14025 · doi:10.1007/s00222-007-0070-1
[11] doi:10.1090/S1056-3911-02-00296-5 · Zbl 1053.14015 · doi:10.1090/S1056-3911-02-00296-5
[13] doi:10.1023/A:1000124232370 · Zbl 0956.14010 · doi:10.1023/A:1000124232370
[15] doi:10.1017/CBO9780511543128 · doi:10.1017/CBO9780511543128
[16] doi:10.1007/s00222-006-0517-9 · Zbl 1114.14011 · doi:10.1007/s00222-006-0517-9
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