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
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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
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- Takeshi Saito, The characteristic cycle and the singular support of a constructible sheaf, Invent. Math. (online), DOI 10.1007/S00222-016-0675-3.
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References
- Ahmed Abbes and Takeshi Saito, Ramification of local fields with imperfect residue fields, Amer. J. Math. 124 (2002), no. 5, 879â920. MR 1925338 (2003m:11196)
- Ahmed Abbes and Takeshi Saito, Ramification of local fields with imperfect residue fields. II, Doc. Math. Extra Vol. (2003), 5â72 (electronic). Kazuya Katoâs fiftieth birthday. MR 2046594 (2005g:11231)
- Ahmed Abbes and Takeshi Saito, The characteristic class and ramification of an $l$-adic Ă©tale sheaf, Invent. Math. 168 (2007), no. 3, 567â612. MR 2299562 (2008k:14045), 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 dâun trait, Publ. Res. Inst. Math. Sci. 45 (2009), no. 1, 25â74 (French, with English and French summaries). MR 2512777 (2009m:11197), DOI https://doi.org/10.2977/prims/1234361154
- Ahmed Abbes and Takeshi Saito, Ramification and cleanliness, Tohoku Math. J. (2) 63 (2011), no. 4, 775â853. MR 2872965, DOI https://doi.org/10.2748/tmj/1325886290
- M. Artin, Morphismes acycliques, Exp. XV, ThĂ©orie des Topos et Cohomologie Ătale 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
- P. Berthelot, Immersions rĂ©guliĂšres et calcul du K$^\bullet$ dâun schĂ©ma Ă©clatĂ©, Exp. VII, ThĂ©orie des intersections et thĂ©orĂšme de Riemann-Roch (SGA6), Lecture Notes in Math. 225, 1971, 416-465. MR 0354655 (50 \#7133)
- J.-E. Bertin, Généralités sur les schémas en groupes, Exp. VI$_\textrm {B}$, Schémas en groupes (SGA3) Tome I, SMF Edition recomposé 2011.
- Pierre Deligne, ThĂ©orie de Hodge. III, Inst. Hautes Ătudes Sci. Publ. Math. 44 (1974), 5â77 (French). MR 0498552 (58 \#16653b)
- Pierre Deligne, La formule de Milnor, Groupes de Monodromie en GĂ©omĂ©trie AlgĂ©brique, Lecture Notes in Math. 340, Springer, Berlin, 1973, pp. 197â211.
- Pierre Deligne, ThĂ©orĂšme de finitude en cohomologie $\ell$-adique, Cohomologie Ă©tale SGA 4$\frac 12$, Springer Lecture Notes in Math. 569 (1977), 233â251. MR 0463174
- Pierre Deligne, Notes sur Euler-Poincaré: brouillon project, unpublished notes dated 8/2/2011.
- A. Grothendieck, Sous-groupes de Cartan, éléments réguliers. Groupes algébriques affines de dimension 1, Séminaire C. Chevalley, 1956-1958, ENS, Exposé 7.
- A. Grothendieck and J. A. Dieudonné, Eléments de géométrie algébrique. (PremiÚre Partie), Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 166, Springer-Verlag, Berlin, 1971 (French). MR 3075000
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- Luc Illusie, Complexe cotangent et déformations. II, Lecture Notes in Mathematics, Vol. 283, Springer-Verlag, Berlin-New York, 1972 (French). MR 0491681 (58 \#10886b)
- 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), 252â261.
- Masaki Kashiwara and Pierre Schapira, Sheaves on manifolds, with a chapter in French by Christian Houzel. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 292, Springer-Verlag, Berlin, 1990. MR 1074006 (92a:58132)
- 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. 101â131. MR 991978 (90g:11164), DOI https://doi.org/10.1090/conm/083/991978
- Kazuya Kato, Class field theory, ${\mathcal {D}}$-modules, and ramification on higher-dimensional schemes. I, Amer. J. Math. 116 (1994), no. 4, 757â784. MR 1287939 (96f:11087), DOI https://doi.org/10.2307/2375001
- Kazuya Kato and Takeshi Saito, On the conductor formula of Bloch, Publ. Math. Inst. Hautes Ătudes Sci. 100 (2004), 5â151. MR 2102698 (2006a:14038), 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 Ătudes Sci. 107 (2008), 109â168. MR 2434692 (2009f:14003a), DOI https://doi.org/10.1007/s10240-008-0011-6
- G. Laumon, Semi-continuitĂ© du conducteur de Swan (dâaprĂšs P. Deligne), CaractĂ©ristique dâEuler-PoincarĂ©, AstĂ©risque, vol. 83, Soc. Math. France, Paris, 1981, pp. 173â219 (French). MR 629128 (83g:14007)
- GĂ©rard Laumon, CaractĂ©ristique dâEuler-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. 193â207. MR 737931 (85m:14032)
- Gérard Laumon and Laurent Moret-Bailly, Champs algébriques, 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. MR 1771927 (2001f:14006)
- Takeshi Saito, Wild ramification and the characteristic cycle of an $l$-adic sheaf, J. Inst. Math. Jussieu 8 (2009), no. 4, 769â829. MR 2540880 (2011e:14039), 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, DeuxiĂšme Ă©dition; Publications de lâUniversitĂ© de Nancago, No. VIII. Hermann, Paris, 1968. MR 0354618 (50 \#7096)
- Liang Xiao, On ramification filtrations and $p$-adic differential modules, I: the equal characteristic case, Algebra Number Theory 4 (2010), no. 8, 969â1027. 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.
