Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-19T15:56:09.317Z Has data issue: false hasContentIssue false
  • English
  • Français

Wild ramification and the characteristic cycle of an ℓ-adic sheaf

Part of: (Co)homology theory Algebraic number theory: local and $p$-adic fields

Published online by Cambridge University Press:  06 January 2009

Takeshi Saito
Takeshi Saito
Affiliation:
Department of Mathematical Sciences, University of Tokyo, Tokyo 153-8914, Japan ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We propose a geometric method to measure the wild ramification of a smooth étale sheaf along the boundary. Using the method, we study the graded quotients of the logarithmic ramification groups of a local field of characteristic p > 0 with arbitrary residue field. We also define the characteristic cycle of an ℓ-adic sheaf, satisfying certain conditions, as a cycle on the logarithmic cotangent bundle and prove that the intersection with the 0-section computes the characteristic class, and hence the Euler number.

Keywords

ℓ-adic sheafwild ramificationcharacteristic cycle

MSC classification

Secondary: 14F20: Étale and other Grothendieck topologies and (co)homologies 11S15: Ramification and extension theory
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 8 , Issue 4 , October 2009 , pp. 769 - 829
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Abbes, A. and Saito, T., Ramification of local fields with imperfect residue fields, I, Am. J. Math. 124(5) (2002), 879–920.CrossRefGoogle Scholar
2.Abbes, A. and Saito, T., Ramification of local fields with imperfect residue fields, II, Documenta Math. Extra Volume: Kazuya Kato's Fiftieth Birthday (2003), 3–70.Google Scholar
3.Abbes, A. and Saito, T., The characteristic class and ramification of an ℓ-adic étale sheaf, Invent. Math. 168 (2007), 567–612.CrossRefGoogle Scholar
4.Abbes, A. and Saito, T., Analyse micro-locale ℓ-adique en caractéristique p > 0: le cas d'un trait, Publ. RIMS, in press.+0:+le+cas+d'un+trait,+Publ.+RIMS,+in+press.>Google Scholar
5.André, Y., Structure des connexions méromorphes formelles de plusieurs variables et semi-continuité de l'irrégularité, Invent. Math. 170 (2007), 147–198.CrossRefGoogle Scholar
6.Bosch, S., Lütkebohmert, W. and Raynaud, M., Formal and rigid geometry, IV, The reduced fiber theorem, Invent. Math. 119 (1995), 361–398.CrossRefGoogle Scholar
7.Bourbaki, N., Algèbre commutative (Masson-Dunod, Paris, 1985 (réimpression)).Google Scholar
8.Deligne, P., Cohomologie à supports propres, Séminaire de géométrie algébrique 4, tome 3, exposé XII, Lecture Notes in Mathematics, Volume 305, pp. 250–461 (Springer, 1973).Google Scholar
9.Epp, H. P., Eliminating wild ramification, Invent. Math. 19 (1973), 235–249.CrossRefGoogle Scholar
10.Fulton, W., Intersection theory, 2nd edn (Springer, 1998).CrossRefGoogle Scholar
11.Grothendieck, A. and Dieudonné, J., Éléments de géométrie algébriques, IV-1, IV-2, IV-3, IV-4, Publ. Math. IHES 20 (1964), 24 (1965), 28 (1966), 32 (1967).Google Scholar
12.Grothendieck, A., Rédigé par Illusie, L., Formule de Lefschetz, exposé III, Séminaire de géométrie algébrique 5, Lecture Notes in Mathematics, Volume 589, pp. 73–137 (Springer, 1977).Google Scholar
13.Kashiwara, M. and Schapira, P., Sheaves on manifolds, A Series of Comprehensive Studies in Mathematics, Volume 292 (Springer, 1990).CrossRefGoogle Scholar
14.Kato, K., Swan conductors for characters of degree one in the imperfect residue field case, Algebraic K-theory and algebraic number theory, Contemp. Math. 83 (1989), 101–131.CrossRefGoogle Scholar
15.Kato, K., Class field theory, -modules, and ramification of higher dimensional schemes, Part I, Am. J. Math. 116 (1994), 757–784.CrossRefGoogle Scholar
16.Kato, K. and Saito, T., On the conductor formula of Bloch, Publ. Math. IHES 100 (2004), 5–151.CrossRefGoogle Scholar
17.Katz, N. and Laumon, G., Transformation de Fourier et majoration de sommes exponentielles, Publ. Math. IHES 62 (1985), 361–418.CrossRefGoogle Scholar
18.Kuhlmann, F.-V., A correction to Epp's paper ‘Elimination of wild ramification’, Invent. Math. 153 (2003), 679–681.CrossRefGoogle Scholar
19.Oesterlé, J. and dit d'Costa, L. Pharamond, Fermetures intégrales des -algèbres, J. Ramanujan Math. Soc. 12(2) (1997), 147–159.Google Scholar
20.Serre, J.-P., Groupes proalgébriques, Publ. Math. IHES 7 (1960), 5–68.CrossRefGoogle Scholar