Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

Wild ramification and the cotangent bundle


Author: Takeshi Saito
Journal: J. Algebraic Geom. 26 (2017), 399-473
DOI: https://doi.org/10.1090/jag/681
Published electronically: September 19, 2016
MathSciNet review: 3647790
Full-text PDF

Abstract | References | Additional Information

Abstract:

We define the characteristic cycle of a locally constant Ă©tale sheaf on a smooth variety in positive characteristic ramified along the boundary as a cycle in the cotangent bundle of the variety, at least on a neighborhood of the generic point of the divisor on the boundary. The crucial ingredient in the definition is the commutative group structure on the boundary induced by the groupoid structure of multiple self-products.

We prove a compatibility with pull-back and local acyclicity in non-characteristic situations. We also give a relation with the cohomological characteristic class under a certain condition and a concrete example where the intersection with the $0$-section computes the Euler-Poincaré characteristic.


References [Enhancements On Off] (What's this?)

References
  • Ahmed Abbes and Takeshi Saito, Ramification of local fields with imperfect residue fields, Amer. J. Math. 124 (2002), no. 5, 879–920. MR 1925338
  • Ahmed Abbes and Takeshi Saito, Ramification of local fields with imperfect residue fields. II, Doc. Math. Extra Vol. (2003), 5–72. Kazuya Kato’s fiftieth birthday. MR 2046594
  • Ahmed Abbes and Takeshi Saito, The characteristic class and ramification of an $l$-adic Ă©tale sheaf, Invent. Math. 168 (2007), no. 3, 567–612. MR 2299562, DOI https://doi.org/10.1007/s00222-007-0040-7
  • Ahmed Abbes and Takeshi Saito, Analyse micro-locale $l$-adique en caractĂ©ristique $p>0$: le cas d’un trait, Publ. Res. Inst. Math. Sci. 45 (2009), no. 1, 25–74 (French, with English and French summaries). MR 2512777, DOI https://doi.org/10.2977/prims/1234361154
  • Ahmed Abbes and Takeshi Saito, Ramification and cleanliness, Tohoku Math. J. (2) 63 (2011), no. 4, 775–853. MR 2872965, DOI https://doi.org/10.2748/tmj/1325886290
  • M. Artin, Morphismes acycliques, Exp. XV, ThĂ©orie des Topos et Cohomologie Étale des SchĂ©mas (SGA4), Lecture Notes in Math. 305, Springer, Berlin-New York, 1973, pp. 168-205.
  • A. Beilinson, Constructible sheaves are holonomic, to appear in Selecta Math., arXiv:1505.06768
  • ThĂ©orie des intersections et thĂ©orĂšme de Riemann-Roch, Lecture Notes in Mathematics, Vol. 225, Springer-Verlag, Berlin-New York, 1971 (French). SĂ©minaire de GĂ©omĂ©trie AlgĂ©brique du Bois-Marie 1966–1967 (SGA 6); DirigĂ© par P. Berthelot, A. Grothendieck et L. Illusie. Avec la collaboration de D. Ferrand, J. P. Jouanolou, O. Jussila, S. Kleiman, M. Raynaud et J. P. Serre. MR 0354655
  • J.-E. Bertin, GĂ©nĂ©ralitĂ©s sur les schĂ©mas en groupes, Exp. VI$_\textrm {B}$, SchĂ©mas en groupes (SGA3) Tome I, SMF Edition recomposĂ© 2011.
  • Pierre Deligne, ThĂ©orie de Hodge. III, Inst. Hautes Études Sci. Publ. Math. 44 (1974), 5–77 (French). MR 498552
  • Pierre Deligne, La formule de Milnor, Groupes de Monodromie en GĂ©omĂ©trie AlgĂ©brique, Lecture Notes in Math. 340, Springer, Berlin, 1973, pp. 197–211.
  • P. Deligne, Cohomologie Ă©tale, Lecture Notes in Mathematics, vol. 569, Springer-Verlag, Berlin, 1977 (French). SĂ©minaire de gĂ©omĂ©trie algĂ©brique du Bois-Marie SGA $4\frac {1}{2}$. MR 463174
  • Pierre Deligne, Notes sur Euler-PoincarĂ©: brouillon project, unpublished notes dated 8/2/2011.
  • A. Grothendieck, Sous-groupes de Cartan, Ă©lĂ©ments rĂ©guliers. Groupes algĂ©briques affines de dimension 1, SĂ©minaire C. Chevalley, 1956-1958, ENS, ExposĂ© 7.
  • A. Grothendieck and J. A. DieudonnĂ©, ÉlĂ©ments de gĂ©omĂ©trie algĂ©brique. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 166, Springer-Verlag, Berlin, 1971 (French). MR 3075000
  • A. Grothendieck, ÉlĂ©ments de gĂ©omĂ©trie algĂ©brique. IV. Étude locale des schĂ©mas et des morphismes de schĂ©mas. I, Inst. Hautes Études Sci. Publ. Math. 20 (1964), 259 (French). MR 173675
  • A. Grothendieck, ÉlĂ©ments de gĂ©omĂ©trie algĂ©brique. IV. Étude locale des schĂ©mas et des morphismes de schĂ©mas IV, Inst. Hautes Études Sci. Publ. Math. 32 (1967), 361 (French). MR 238860
  • Luc Illusie, Complexe cotangent et dĂ©formations. II, Lecture Notes in Mathematics, Vol. 283, Springer-Verlag, Berlin-New York, 1972 (French). MR 0491681
  • Luc Illusie, Appendice Ă  ThĂ©orĂšme de finitude en cohomologie $\ell$-adique, Cohomologie Ă©tale SGA 4$\frac 12$, Springer Lecture Notes in Math. 569 (1977), 252–261.
  • Masaki Kashiwara and Pierre Schapira, Sheaves on manifolds, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 292, Springer-Verlag, Berlin, 1990. With a chapter in French by Christian Houzel. MR 1074006
  • Kazuya Kato, Swan conductors for characters of degree one in the imperfect residue field case, Algebraic $K$-theory and algebraic number theory (Honolulu, HI, 1987) Contemp. Math., vol. 83, Amer. Math. Soc., Providence, RI, 1989, pp. 101–131. MR 991978, DOI https://doi.org/10.1090/conm/083/991978
  • Kazuya Kato, Class field theory, ${\scr D}$-modules, and ramification on higher-dimensional schemes. I, Amer. J. Math. 116 (1994), no. 4, 757–784. MR 1287939, DOI https://doi.org/10.2307/2375001
  • Kazuya Kato and Takeshi Saito, On the conductor formula of Bloch, Publ. Math. Inst. Hautes Études Sci. 100 (2004), 5–151. MR 2102698, DOI https://doi.org/10.1007/s10240-004-0026-6
  • Yves Laszlo and Martin Olsson, The six operations for sheaves on Artin stacks. I. Finite coefficients, Publ. Math. Inst. Hautes Études Sci. 107 (2008), 109–168. MR 2434692, DOI https://doi.org/10.1007/s10240-008-0011-6
  • G. Laumon, Semi-continuitĂ© du conducteur de Swan (d’aprĂšs P. Deligne), The Euler-PoincarĂ© characteristic (French), AstĂ©risque, vol. 83, Soc. Math. France, Paris, 1981, pp. 173–219 (French). MR 629128
  • GĂ©rard Laumon, CaractĂ©ristique d’Euler-PoincarĂ© des faisceaux constructibles sur une surface, Analysis and topology on singular spaces, II, III (Luminy, 1981) AstĂ©risque, vol. 101, Soc. Math. France, Paris, 1983, pp. 193–207 (French). MR 737931
  • GĂ©rard Laumon and Laurent Moret-Bailly, Champs algĂ©briques, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 39, Springer-Verlag, Berlin, 2000 (French). MR 1771927
  • Takeshi Saito, Wild ramification and the characteristic cycle of an $l$-adic sheaf, J. Inst. Math. Jussieu 8 (2009), no. 4, 769–829. MR 2540880, DOI https://doi.org/10.1017/S1474748008000364
  • Takeshi Saito, The characteristic cycle and the singular support of a constructible sheaf, Invent. Math. (online), DOI 10.1007/S00222-016-0675-3.
  • Jean-Pierre Serre, Corps locaux, Hermann, Paris, 1968 (French). DeuxiĂšme Ă©dition; Publications de l’UniversitĂ© de Nancago, No. VIII. MR 0354618
  • Liang Xiao, On ramification filtrations and $p$-adic differential modules, I: the equal characteristic case, Algebra Number Theory 4 (2010), no. 8, 969–1027. MR 2832631, DOI https://doi.org/10.2140/ant.2010.4.969


Additional Information

Takeshi Saito
Affiliation: School of Mathematical Sciences, University of Tokyo, Tokyo 153-8914, Japan
MR Author ID: 236565
Email: t-saito@ms.u-tokyo.ac.jp

Received by editor(s): April 10, 2014
Received by editor(s) in revised form: July 24, 2015, and August 26, 2015
Published electronically: September 19, 2016
Article copyright: © Copyright 2016 University Press, Inc.