Ramification theory for varieties over a local field
Publications Mathématiques de l'IHÉS, Volume 117 (2013), pp. 1-178.

We define generalizations of classical invariants of wild ramification for coverings on a variety of arbitrary dimension over a local field. For an -adic sheaf, we define its Swan class as a 0-cycle class supported on the wild ramification locus. We prove a formula of Riemann-Roch type for the Swan conductor of cohomology together with its relative version, assuming that the local field is of mixed characteristic.

We also prove the integrality of the Swan class for curves over a local field as a generalization of the Hasse-Arf theorem. We derive a proof of a conjecture of Serre on the Artin character for a group action with an isolated fixed point on a regular local ring, assuming the dimension is 2.

DOI: 10.1007/s10240-013-0048-z
Kato, Kazuya 1; Saito, Takeshi 2

1 Department of Mathematics, University of ChicagoChicago, IL, 60637USA
2 Department of Mathematical Sciences, University of TokyoTokyo, 153-8914Japan
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Kato, Kazuya; Saito, Takeshi. Ramification theory for varieties over a local field. Publications Mathématiques de l'IHÉS, Volume 117 (2013), pp. 1-178. doi : 10.1007/s10240-013-0048-z. http://archive.numdam.org/articles/10.1007/s10240-013-0048-z/

[1.] Abbes, A. The Grothendieck-Ogg-Shafarevich formula for arithmetic surfaces, J. Algebr. Geom., Volume 9 (2000), pp. 529-576 | MR | Zbl

[2.] Abbes, A. Cycles on arithmetic surfaces, Compos. Math., Volume 122 (2000), pp. 23-111 | DOI | MR | Zbl

[3.] Bloch, S. Cycles on arithmetic schemes and Euler characteristics of curves, Algebraic Geometry (Proc. Symp. Pure Math.) (1987), pp. 421-450 (Part 2) | Zbl

[4.] Berthelot, P. Immersions réguliérès et calcul du K d’un schema éclaté, Exp. VII, SGA 6 (Lect. Notes Math.) (1971), pp. 416-465 | Zbl

[5.] Bourbaki, N. Algèbre commutative, Hermann, Paris, 1964 | Zbl

[6.] Jong, A. J. Smoothness, semi-stability and alterations, Publ. Math. IHÉS, Volume 83 (1996), pp. 51-93 | Zbl

[7.] Deligne, P. Lemme de Gabber, Séminaire sur les pinceaux arithmétiques: la conjecture de Mordell, Astérisque, Volume 127 (1985), pp. 131-150 | Numdam | MR | Zbl

[8.] Deligne, P.; Mumford, D. The irreducibility of the space of curves of given genus, Publ. Math. IHÉS, Volume 36 (1969), pp. 75-109 | MR | Zbl

[9.] Fujiwara, K.; Kato, F. Rigid geometry and applications, Adv. Stud. Pure Math., Volume 45 (2006), pp. 327-386 | MR | Zbl

[10.] Fulton, W. Intersection Theory, Ergeb. Math. Ihrer Grenzgeb, Springer, Berlin, 1998 (Folge 2) | DOI | Zbl

[11.] A. Grothendieck, avec la collaboration de J. Dieudonné, Éléments de géométrie algébrique III (Seconde partie), Publ. Math. IHÉS, 17 (1963).

[12.] A. Grothendieck, avec la collaboration de J. Dieudonné, Éléments de géométrie algébrique IV (Premiére partie), Publ. Math. IHÉS, 24 (1964).

[13.] A. Grothendieck, avec la collaboration de J. Dieudonné, Éléments de géométrie algébrique IV (Quatrième partie), Publ. Math. IHÉS, 32 (1967). | Numdam | Zbl

[14.] Illusie, L. Complexe cotangent et déformations I, Lect. Notes Math., Springer, Berlin, 1971 | DOI | Zbl

[15.] Illusie, L. Existence de résolutions globales, Exp. II, SGA 6 (Lect. Notes Math.) (1971), pp. 160-221 | Zbl

[16.] Illusie, L. Conditions de finitude relatives, Exp. III, SGA 6 (Lect. Notes Math.) (1971), pp. 222-273 | Zbl

[17.] Illusie, L. Appendice à “P. Deligne, Théorème de finitude en cohomologie -adique”, Cohomologie étale, SGA $4\frac{1}{2}$ (Lect. Notes Math.) (1977), pp. 252-261 | Zbl

[18.] Illusie, L. Théorie de Brauer et caractéristique d’Euler-Poincaré, Astérisque, Volume 82–83 (1981), pp. 161-172 | Numdam | MR | Zbl

[19.] Illusie, L. An overview of the work of K. Fujiwara, K. Kato, and C. Nakayama on logarithmic étale cohomology, Astérisque, Volume 279 (2002), pp. 271-322 | Numdam | Zbl

[20.] Kato, K. Residue homomorphisms in Milnor K-theory, Galois groups and their representations, Adv. Stud. Pure Math., Volume 2 (1983), pp. 153-172 | Zbl

[21.] K. Kato, Swan conductors for characters of degree one in the imperfect residue field case, in Contemporary Mathematics, vol. 83, pp. 101–131, 1989. | Zbl

[22.] Kato, K. Logarithmic structures of Fontaine-Illusie, Algebraic Analysis, Geometry, and Number Theory (1989), pp. 191-224 | Zbl

[23.] Kato, K. Generalized class field theory, Proceedings of ICM, Vol. I (1991), pp. 419-428 | Zbl

[24.] Kato, K. Class field theory, ${\mathcal{D}}$-modules, and ramification of higher dimensional schemes, Part I, Am. J. Math., Volume 116 (1994), pp. 757-784 | DOI | Zbl

[25.] Kato, K. Toric singularities, Am. J. Math., Volume 116 (1994), pp. 1073-1099 | DOI | Zbl

[26.] Kato, K.; Saito, T. On the conductor formula of Bloch, Publ. Math. IHÉS, Volume 100 (2004), pp. 5-151 | Numdam | Zbl

[27.] Kato, K.; Saito, T. Ramification theory for varieties over a perfect field, Ann. Math., Volume 168 (2008), pp. 33-96 | DOI | MR | Zbl

[28.] Kato, K.; Saito, S.; Saito, T. Artin characters for algebraic surfaces, Am. J. Math., Volume 109 (1987), pp. 49-76 | MR | Zbl

[29.] Knudsen, F. The projectivity of the moduli space of stable curves. II. The stacks M g,n , Math. Scand., Volume 52 (1983), pp. 161-199 | MR | Zbl

[30.] Mumford, D.; Fogarty, J.; Kirwan, F. Geometric Invariant Theory, Ergeb. Math. Ihrer Grenzgeb, Springer, Berlin, 1994 | DOI | Zbl

[31.] Nagata, M. A generalization of the imbedding problem of an abstract variety in a complete variety, J. Math. Kyoto Univ., Volume 3 (1963), pp. 89-102 | MR | Zbl

[32.] Nakayama, C. Logarithmic etale cohomology, Math. Ann., Volume 308 (1997), pp. 365-404 | DOI | MR | Zbl

[33.] Nakayama, C. Nearby cycles for log smooth families, Compos. Math., Volume 112 (1998), pp. 45-75 | DOI | MR | Zbl

[34.] Oda, T. Convex Bodies and Algebraic Geometry, Ergeb. Math. Ihrer Grenzgeb, Springer, Berlin, 1988 | Zbl

[35.] Rapoport-T. Zink, M. Über die lokale Zetafunktion von Shimuravarietäten, Monodromiefiltration und verschwindende Zyklen in ungleicher Characteristik, Invent. Math., Volume 68 (1982), pp. 21-201 | DOI | MR | Zbl

[36.] Raynaud, M.; Gruson, L. Critères de platitude et de projectivité, Techniques de “platification” d’un module, Invent. Math., Volume 13 (1971), pp. 1-89 | DOI | MR | Zbl

[37.] Mme M. Raynaud (d’après notes inédites de A. Grothendieck), Propreté cohomologique des faisceaux d’ensembles et des faisceaux de groupes non commutatifs, exposé XIII, in SGA 1, Lect. Notes Math., vol. 224, Springer, Berlin, 1971. Édition recomposée SMF (2003).

[38.] Saito, T. The Euler numbers of -adic sheaves of rank 1 in positive characteristic, ICM90 Satellite Conference Proceedings, Arithmetic and Algebraic Geometry (1991), pp. 165-181 | Zbl

[39.] Serre, J.-P. Corps Locaux, Hermann, Paris, 1968 | Zbl

[40.] Serre, J.-P. Représentations linéaires des groupes finis, Hermann, Paris, 1968 | Zbl

[41.] Serre, J.-P. Sur la rationalité des représentations d’Artin, Ann. Math., Volume 72 (1960), pp. 406-420 | DOI | Zbl

[42.] Tsushima, T. On localizations of the characteristic classes of -adic sheaves and conductor formula in characteristic p>0, Math. Z., Volume 269 (2011), pp. 411-447 | DOI | MR | Zbl

[43.] Vidal, I. Theorie de Brauer et conducteur de Swan, J. Algebr. Geom., Volume 13 (2004), pp. 349-391 | DOI | MR | Zbl

[44.] Zariski, O.; Samuel, P. Commutative Algebra II, Grad. Texts Math., Springer, Berlin, 1975 | Zbl

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