Explicit calculation of Frobenius isomorphisms and Poincaré duality in the theory of arithmetic \(\mathcal D\)-modules. (English) Zbl 1327.14101

Abe, Tomoyuki

Rend. Semin. Mat. Univ. Padova 131, 89-149 (2014).

Summary: The aim of this paper is to compute the Frobenius structures of some cohomological operators of arithmetic \({\mathcal D}\)-modules. To do this, we calculate explicitly an isomorphism between canonical sheaves defined abstractly. Using this calculation, we establish the relative Poincaré duality in the style of [Théorie des topos et cohomologie étale des schémas (SGA 4). Un séminaire dirigé par M. Artin, A. Grothendieck, J. L. Verdier. Avec la collaboration de P. Deligne, B. Saint-Donat. Tome 3. Exposés IX à XIX. Berlin-Heidelberg-New York: Springer-Verlag (1973; Zbl 0245.00002)]. As another application, we compare the push-forward as arithmetic \({\mathcal D}\)-modules and the rigid cohomologies taking Frobenius into account. These theorems will be used to prove “\(p\)-adic Weil II” and a product formula for \(p\)-adic epsilon factors.

MSC:

14F30	\(p\)-adic cohomology, crystalline cohomology
14F10	Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
12H25	\(p\)-adic differential equations

Keywords:

\(p\)-adic cohomology; arithmetic \(\mathcal D\)-module; rigid cohomology

Citations:

Zbl 0245.00002

References:

[1]	ABE T., Coherence of certain overconvergent isocrystals without Frobe- nius structures on curves, Math. Ann. 350, pp. 577-609 (2011). · Zbl 1264.12005 · doi:10.1007/s00208-010-0573-8
[2]	ABE T., Langlands correspondence for isocrystals and existence of crystal- line companion for curves, available at arxiv.org/abs/1310.0528.
[3]	ABE T. - CARO D., Theory of weights in p-adic cohomology, available at arxiv.org/abs/1303.0662.
[4]	ABE T. - MARMORA A., On p-adic product formula for epsilon factors, To appear in J. Inst. Math. Jussieu. · Zbl 1319.14025 · doi:10.1017/S1474748014000024
[5]	BALDASSARRI F. - BERTHELOT P., On Dwork cohomology for singular hypersurfaces, Geometric aspects of Dwork theory, vol. I, pp. 177-244, Walter de Gruyter (2004). · Zbl 1117.14022
[6]	BERTHELOT P., Cohomologie rigide et cohomologie rigide aÁ supports propres. PremieÁre partie, preprint.
[7]	BERTHELOT P., D -modules arithmeÂtiques I. OpeÂrateurs diffeÂrentiels de niveau fini, Ann. Sci. EÂcole Norm. Sup. 4eÁme seÂrie, 29 no. 2, pp. 185-272 (1996).
[8]	BERTHELOT P., D -modules arithmeÂtiques II. Descente par Frobenius, MeÂm. Soc. Math. Fr. 81 (2000). · Zbl 0948.14017
[9]	BERTHELOT P., Introduction aÁ la theÂorie arithmeÂtique des D -modules, Asterisque 279, pp. 1-80 (2002).
[10]	CARO D., D -modules arithmeÂtiques surcoheÂrents. Application aux fonc- tions L, Ann. Inst. Fourier 54, pp. 1943-1996 (2005).
[11]	CARO D., Fonctions L associeÂes aux h-modulles arithmeÂtiques. Cas des courbes, Compos. Math. 142, pp. 169-206 (2006).
[12]	CARO D., Comparaison des foncteurs duaux des isocristaux surconver- gents, Rend. Sem. Mat. Univ. Padova 114, pp. 131-211 (2006).
[13]	CARO D., DeÂvissages des F-complexes de D -modules arithmeÂtiques en F-isocristauxsurconvergents, Invent. Math. 166 no. 2, pp. 397-456 (2006).
[14]	CARO D., h-modules arithmeÂtiques associeÂs aux isocristaux survonver- gents. Cas Lisse, Bull. Soc. Math. Fr. 137 no. 4, pp. 453-543 (2009).
[15]	CARO D., h-modules arithmeÂtiques surholonomes, Ann. Sci. EÂcole Norm. Sup. 4eÁme seÂrie, 42 no. 1, pp. 141-192 (2009).
[16]	CARO D., Sur la compatibilite aÁ Frobenius de l’isomorphisme de dualite relative, Rend. Sem. Mat. Univ. Padova 122, pp. 235-286 (2009). · Zbl 1310.13009
[17]	CONRAD, B., Grothendieck Duality and Base Change, Lecture Notes in Math. 1750, Springer (2000). · Zbl 0992.14001 · doi:10.1007/b75857
[18]	CREW R., Finiteness theorems for the cohomology of an overconvergent isocrystal on a curve, Ann. Sci. EÂcole Norm. Sup. 4eÁme seÂrie, 31 no. 6, pp. 717-763 (1998). · Zbl 0943.14008 · doi:10.1016/S0012-9593(99)80001-9
[19]	GARNIER L., Descente par Frobenius explicite pour les D y-modules, J. Algebra 205, pp. 542-577 (1998). · Zbl 0926.12006 · doi:10.1006/jabr.1997.7373
[20]	GROSS B. - KOBLITZ N., Gauss sums and the p-adic G-function, Ann. Math. 109, pp. 569-581 (1979). · Zbl 0406.12010 · doi:10.2307/1971226
[21]	HARTSHORNE R., Residues and Duality, Lecture Notes in Math. 20, Springer (1966).
[22]	NOOT-HUYGHE C., Transformation de Fourier des h-modules arithmeÂ- tiques I, Geometric aspects of Dwork theory. vol. II, pp. 857-907, Walter de Gruyter (2004). · Zbl 1126.14016
[23]	NOOT-HUYGHE C., Finitude de la dimension homologique d’algeÁbres d’opeÂrateurs diffeÂrentiels faiblement compleÁtes et aÁ coefficients surcon- vergents, J. Algebra 307, pp. 499-540 (2007). · Zbl 1111.14006
[24]	KASHIWARA M., Senkei-henbibun-houteishiki-kei no daisuuteki kenkyuu, Master thesis, Univ. of Tokyo (1970).
[25]	KEDLAYA K.S., Finiteness of rigid cohomology with coefficients, Duke Math. J. 134, pp. 15-97 (2006). · Zbl 1133.14019 · doi:10.1215/S0012-7094-06-13412-9
[26]	KLEIMAN S.L., Relative duality for quasi-coherent sheaves, Compos. Math. 41, pp. 39-60 (1980). · Zbl 0403.14003
[27]	LE STUM B., Rigid cohomology, Cambridge Tracts in Math. 172, Cam- bridge University Press (2007). · Zbl 1131.14001
[28]	VIRRION A., Dualite locale et holonomie pour les D -modules arithmeÂ- tiques, Bull. Soc. Math. Fr. 128, pp. 1-68 (2000). · Zbl 0955.14015
[29]	VIRRION A., Trace et dualite relative pour les D -modules arithmeÂtiques, Geometric aspects of Dwork theory. vol. II, pp. 1039-1112, Walter de Gruyter (2004). · Zbl 1083.14017