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AROUND THE NEARBY CYCLE FUNCTOR FOR ARITHMETIC $\mathscr{D}$-MODULES

Part of: (Co)homology theory

Published online by Cambridge University Press:  28 August 2019

TOMOYUKI ABE
TOMOYUKI ABE*
Affiliation:
Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba, 277-8583, Japan email [email protected]
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Abstract

We will establish a nearby and vanishing cycle formalism for the arithmetic $\mathscr{D}$-module theory following Beilinson’s philosophy. As an application, we define smooth objects in the framework of arithmetic $\mathscr{D}$-modules whose category is equivalent to the category of overconvergent isocrystals.

MSC classification

Primary: 14F10: Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 14F30: $p$-adic cohomology, crystalline cohomology
Type
Article
Information
Nagoya Mathematical Journal , Volume 236: Celebrating the 60th Birthday of Shuji Saito , December 2019 , pp. 1 - 28
Copyright
© 2019 Foundation Nagoya Mathematical Journal  

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References

Abe, T., Explicit calculation of Frobenius isomorphisms and Poincaré duality in the theory of arithmetic 𝒟-modules , Rend. Sem. Math. Univ. Padova 131 (2014), 89–149.10.4171/RSMUP/131-7Google Scholar
Abe, T., Langlands correspondence for isocrystals and existence of crystalline companion for curves , J. Amer. Math. Soc. 31 (2018), 921–1057.10.1090/jams/898Google Scholar
Abe, T. and Caro, D., Theory of weights in p-adic cohomology , Amer. J. Math. 140(4) (2018), 879–975.10.1353/ajm.2018.0021Google Scholar
Abe, T. and Caro, D., On Beilinson’s equivalence for p-adic cohomology , Selecta Math. 24 (2018.), 591–608.10.1007/s00029-017-0370-2Google Scholar
Abe, T. and Marmora, A., On p-adic product formula for epsilon factors , J. Inst. Math. Jussieu 14 (2015), 275–377.10.1017/S1474748014000024Google Scholar
Beilinson, A., How to Glue Perverse Sheaves, Lecture Notes in Math. 1289 , 42–51. Springer, Berlin, 1987.Google Scholar
Beilinson, A., Bernstein, J. and Deligne, P., Faisceaux pervers , Astérisque 100 (1982).Google Scholar
Berthelot, P., Géométrie rigide et cohomologie des variétés algébriques de caractéristique p , Mém. Soc. Math. Fr. 23 (1986), 7–32.Google Scholar
Caro, D., Pleine fidélité sans structure de Frobenius et isocristaux partiellement surconvergents , Math. Ann. 349 (2011), 747–805.10.1007/s00208-010-0539-xGoogle Scholar
Caro, D., Sur la préservation de la surconvergence par l’image directe d’un morphisme propre et lisse , Ann. E.N.S. 48 (2015), 131–169.Google Scholar
Caro, D., Sur la préservation de la cohérence par image inverse extraordinaire d’une immersion fermée , Nagoya J. Math. https://doi.org/10.1017/nmj.2019.16.Google Scholar
Caro, D. and Vauclair, D., Logarithmic $p$ -bases and arithmetical differential modules, preprint, 2015, arXiv:1511.07797.Google Scholar
Christol, G. and Mebkhout, Z., Sur le théorème de l’indice des équations différentielles p-adiques IV , Invent. Math. 143 (2001), 629–672.10.1007/s002220000116Google Scholar
Crew, R., Arithmetic 𝓓-modules on the unit disk , Compos. Math. 148 (2012), 227–268.10.1112/S0010437X11005471Google Scholar
Crew, R., Arithmetic ${\mathcal{D}}$ -modules on adic formal schemes, preprint, 2017, arXiv:1701.01324.Google Scholar
Deligne, P., “ Théorèmes de finitude en cohomologie ℓ-adique ”, Cohomologie Etale, Lecture Notes in Mathematics 569 , Springer, Berlin, Heidelberg, 1977.10.1007/BFb0091516Google Scholar
Ekedahl, T., “ On the adic formalism ”, in The Grothendieck Festschrift, Modern Birkhäuser Classics, (eds. Cartier, P., Katz, N. M., Manin, Y. I., Illusie, L., Laumon, G. and Ribet, K. A.) Birkhäuser, Boston, 2007, 197–218.10.1007/978-0-8176-4575-5_4Google Scholar
Kedlaya, K. S., Semistable reduction for overconvergent F-isocrystals, I: Unipotence and logarithmic extensions , Compos. Math. 143 (2007), 1164–1212.10.1112/S0010437X07002886Google Scholar
Kedlaya, K. S., Semistable reduction for overconvergent F-isocrystals, IV: local semistable reduction at nonmonomial valuations , Compos. Math. 147 (2011), 467–523.10.1112/S0010437X10005142Google Scholar
Lazda, C. and Pal, A., Rigid Cohomology over Laurent Series Fields, Algebra and Applications 21 , Springer, 2016.10.1007/978-3-319-30951-4Google Scholar
Shiho, A., Cut-by-curves criterion for the log extendability of overconvergent isocrystals , Math. Z. 269 (2011), 59–82.10.1007/s00209-010-0716-3Google Scholar
Stacks project, https://stacks.math.columbia.edu/.Google Scholar