Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-26T13:29:02.208Z Has data issue: false hasContentIssue false
  • English
  • Français

On nearby cycles and 𝒟-modules of log schemes in characteristic p>0

Part of: (Co)homology theory

Published online by Cambridge University Press:  16 June 2010

Takeshi Tsuji
Takeshi Tsuji*
Affiliation:
Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1, Komaba, Meguro, Tokyo 153-8914, Japan (email: t-tsuji@ms.u-tokyo.ac.jp)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let K be a complete discrete valuation field of mixed characteristic (0,p) with a perfect residue field k. For a semi-stable scheme over the ring of integers OK of K or, more generally, for a log smooth scheme of semi-stable type over k, we define nearby cycles as a single 𝒟-module endowed with a monodromy ∂logt, whose cohomology should give the log crystalline cohomology. We also explicitly describe the monodromy filtration of the 𝒟-module with respect to the endomorphism ∂logt, and construct a weight spectral sequence for the cohomology of the nearby cycles.

Keywords

nearby cycles𝒟-modulelogarithmic scheme

MSC classification

Secondary: 14F30: $p$-adic cohomology, crystalline cohomology 14F10: Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 6 , November 2010 , pp. 1552 - 1616
Copyright
Copyright © Foundation Compositio Mathematica 2010

References

[1]Berthelot, P., Cohomologie rigide et théorie des 𝒟-modules, in p-adic analysis (Trento, 1989), Lecture Notes in Mathematics, vol. 1454 (Springer, Berlin, 1990), 80–124.CrossRefGoogle Scholar
[2]Berthelot, P., 𝒟-modules arithmétiques I. Opérateurs différentiels de niveau fini, Ann. Sci. École Norm. Sup. (4) 29 (1996), 185–272.CrossRefGoogle Scholar
[3]Berthelot, P., 𝒟-modules arithmétiques II. Descente par Frobenius, Mém. Soc. Math. Fr. (N.S.) 81 (2000).Google Scholar
[4]Berthelot, P., Introduction à la théorie arithmétique des 𝒟-modules. Cohomologies p-adiques et applications arithmétiques (II), Astérisque 279 (2002), 1–80.Google Scholar
[5]Gros, M., Sur le 𝒟-module associé au complexe des cycles proches et ses variantes p-adiques, Rend. Sem. Mat. Univ. Padova 112 (2004), 77–95.Google Scholar
[6]Gros, M. and Narváez-Macarro, L., Cohomologie évanescente p-adique: calculs locaux, Rend. Sem. Mat. Univ. Padova 104 (2000), 71–90.Google Scholar
[7]Hartshorne, R., Residues and duality, Lecture Notes in Mathematics, vol. 20 (Springer, Berlin, 1966).CrossRefGoogle Scholar
[8]Hochster, M., Rings of invariants of tori, Cohen–Macaulay rings generated by monomials, and polytopes, Ann. of Math. (2) 96 (1972), 318–337.CrossRefGoogle Scholar
[9]Hyodo, O., On the de Rham–Witt complex attached to a semi-stable family, Compositio Math. 78 (1991), 241–260.Google Scholar
[10]Hyodo, O. and Kato, K., Semi-stable reduction and crystalline cohomology with logarithmic poles, in Périodes p-adiques (Bures-sur-Yvette, 1988), Astérisque 223 (1994), 221–268.Google Scholar
[11]Illusie, L., Généralités sur les conditions de finitude dans les catégories derivées (Exposé I), in Séminaire de géométrie algébrique du Bois-Marie 1966–1967 (SGA 6), Lecture Notes in Mathematics, vol. 225 (Springer, Berlin, 1971), 78–159.Google Scholar
[12]Kashiwara, M., Vanishing cycle sheaves and holonomic systems of differential equations, in Algebraic geometry (Tokyo/Kyoto, 1982), Lecture Notes in Mathematics, vol. 1016 (Springer, Berlin, 1983), 134–142.CrossRefGoogle Scholar
[13]Kato, K., Logarithmic structures of Fontaine–Illusie, in Algebraic analysis, geometry, and number theory (Baltimore, 1988) (Johns Hopkins University Press, Baltimore, 1989), 191–224.Google Scholar
[14]Malgrange, B., Polynômes de Bernstein–Sato et cohomologie évanescente, in Analysis and topology on singular spaces, II, III (Luminy, 1981), Astérisque 101 (1983), 243–267.Google Scholar
[15]Matsumura, H., Commutative algebra, Mathematics Lecture Note Series, vol. 56, second edition (Benjamin/Cummings, Reading, MA, 1980).Google Scholar
[16]Mebkhout, Z. and Sabbah, C., 𝒟X-modules et cycles évanescents, in Le formalisme des six opérations de Grothendieck pour les 𝒟X-modules cohérents, Travaux en Cours, vol. 35 (Hermann, Paris, 1989), 201–239.Google Scholar
[17]Mokrane, A., La suite spectrale des poids en cohomologie de Hyodo–Kato, Duke Math. J. 72 (1993), 301–337.CrossRefGoogle Scholar
[18]Montagnon, C., Généralisation de la théorie arithmétique des𝒟-modules à la géométrie logarithmique, Thesis, Université de Rennes I (2002).Google Scholar
[19]Nakkajima, Y., p-adic weight spectral sequences of log varieties, J. Math. Sci. Univ. Tokyo 12 (2005), 513–661.Google Scholar
[20]Sabbah, C., D-modules et cycles évanescents (d’après B. Malgrange et M. Kashiwara), in Géométrie algébrique et applications, III (La Rábida, 1984), Travaux en Cours, vol. 24 (Hermann, Paris, 1987), 53–98.Google Scholar
[21]Saito, M., Modules de Hodge polarisables, Publ. Res. Inst. Math. Sci. 24 (1988), 849–995.CrossRefGoogle Scholar
[22]Saito, T., Weight spectral sequences and independence of l, J. Inst. Math. Jussieu 2 (2003), 583–634.CrossRefGoogle Scholar
[23]Tsuji, T., Poincaré duality for logarithmic crystalline cohomology, Compositio Math. 118 (1999), 11–41.CrossRefGoogle Scholar
[24]Tsuji, T., On p-adic nearby cycles of log smooth families, Bull. Soc. Math. Fr. 128 (2000), 529–575.CrossRefGoogle Scholar