On the conductor formula of Bloch
Publications Mathématiques de l'IHÉS, Volume 100  (2004), p. 5-151

In [6], S. Bloch conjectures a formula for the Artin conductor of the ℓ-adic etale cohomology of a regular model of a variety over a local field and proves it for a curve. The formula, which we call the conductor formula of Bloch, enables us to compute the conductor that measures the wild ramification by using the sheaf of differential 1-forms. In this paper, we prove the formula in arbitrary dimension under the assumption that the reduced closed fiber has normal crossings.

@article{PMIHES_2004__100__5_0,
     author = {Kato, Kazuya and Saito, Takeshi},
     title = {On the conductor formula of Bloch},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     publisher = {Springer},
     volume = {100},
     year = {2004},
     pages = {5-151},
     doi = {10.1007/s10240-004-0026-6},
     zbl = {1099.14009},
     mrnumber = {2102698},
     language = {en},
     url = {http://www.numdam.org/item/PMIHES_2004__100__5_0}
}
Kato, Kazuya; Saito, Takeshi. On the conductor formula of Bloch. Publications Mathématiques de l'IHÉS, Volume 100 (2004) , pp. 5-151. doi : 10.1007/s10240-004-0026-6. http://www.numdam.org/item/PMIHES_2004__100__5_0/

1. A. Abbes, Cycles on arithmetic surfaces, Compos. Math., 122 (2000), no. 1, 23-111. | MR 1771449 | Zbl 0986.14014

2. A. Abbes, The Whitney sum formula for localized Chern classes, to appear in J. Théor. Nombres Bordx.

3. A. Abbes and T. Saito, Ramification groups of local fields with imperfect residue fields II, Doc. Math., Extra Volume Kato (2003), 3-70. | MR 2000773 | Zbl 1127.11349

4. K. Arai, Conductor formula of Bloch, in tame case (in Japanese), Master thesis at University of Tokyo, 2000.

5. A. Blanco, J. Majadas, and A. Rodicio, Projective exterior Koszul homology and decomposition of the Tor functor, Invent. Math., 123 (1996), 123-140. | MR 1376249 | Zbl 0856.13009

6. S. Bloch, Cycles on arithmetic schemes and Euler characteristics of curves, Algebraic geometry, Bowdoin, 1985, 421-450, Proc. Symp. Pure Math. 46, Part 2, Am. Math. Soc., Providence, RI (1987). | MR 927991 | Zbl 0654.14004

7. T. Chinburg, G. Pappas, and M. Taylor, ε-constants and Arakelov Euler characteristics, Math. Res. Lett., 7 (2000), no. 4, 433-446. | Zbl 1097.14501

8. A. J. De Jong, Smoothness, semi-stability and alterations, Publ. Math., Inst. Hautes Étud. Sci., 83 (1996), 51-93. | Numdam | MR 1423020 | Zbl 0916.14005

9. P. Deligne, Équations différentielles à points singuliers réguliers, Lect. Notes Math. 163, Springer, Berlin-New York (1970). | MR 417174 | Zbl 0244.14004

10. P. Deligne and N. Katz, Groupes de monodromie en géométrie algébrique, (SGA 7 II), Lect. Notes Math. 340, Springer, Berlin-New York (1973). | MR 354657 | Zbl 0258.00005

11. A. Dold and D. Puppe, Homologie nicht-additiver Funktoren, Anwendungen, Ann. Inst. Fourier, 11 (1961), 201-312. | Numdam | MR 150183 | Zbl 0098.36005

12. K. Fujiwara and K. Kato, Logarithmic etale topology theory, preprint.

13. W. Fulton, Intersection theory, 2nd ed. Ergeb. Math. Grenzgeb., 3. Folge. 2, Springer, Berlin (1998). | MR 1644323 | Zbl 0541.14005

14. W. Fulton and S. Lang, Riemann-Roch algebra, Grundlehren Math. Wiss. 277, Springer, Berlin-New York (1985). | MR 801033 | Zbl 0579.14011

15. A. Grothendieck with J. Dieudonné, Eléments de géométrie algèbrique IV, Publ. Math., Inst. Hautes Étud. Sci., 20, 24, 28, 32 (1964-1967). | Numdam | MR 173675 | Zbl 0136.15901

16. A. Grothendieck et. al., Théorie des topos et cohomologie étale des schemas, (SGA 4), tome 3, Lect. Notes Math. 305, Springer, Berlin-New York (1973). | MR 463174 | Zbl 0245.00002

17. A. Grothendieck et. al., Théorie des intersections et théorème de Riemann-Roch, (SGA 6), Lect. Notes Math. 225, Springer, Berlin-New York (1971). | MR 354655

18. R. Hartshorne, Residues and Duality, Lect. Notes Math. 20, Springer, Berlin-New York (1966). | MR 222093 | Zbl 0212.26101

19. L. Illusie, Complexe cotangent et déformations I, Lect. Notes Math. 239, Springer, Berlin-New York (1971). | MR 491680 | Zbl 0224.13014

20. L. Illusie, An overview of the work of K. Fujiwara, K. Kato, and C. Nakayama on logarithmic etale cohomology, Cohomologies p-adiques et applications arithmétiques, II. Astérisque, 279 (2002), 271-322. | MR 1922832 | Zbl 1052.14005

21. L. Illusie, Champs toriques et log lissité, preprint (2000).

22. B. Iversen, Critical points of an algebraic function, Invent. Math. 12 (1971), 210-224. | MR 342512 | Zbl 0223.14003

23. K. Kato, Logarithmic structures of Fontaine-Illusie, Algebraic analysis, geometry, and number theory (J.-I. Igusa ed.), Johns Hopkins UP, Baltimore (1989), 191-224. | MR 1463703 | Zbl 0776.14004

24. K. Kato, Class field theory, 𝒟-modules, and ramification on higher dimensional schemes, preprint, unpublished version. | Zbl 0864.11057

25. K. Kato, Toric singularities, Am. J. Math., 116 (1994), 1073-1099. | MR 1296725 | Zbl 0832.14002

26. K. Kato, S. Saito, and T. Saito, Artin characters for algebraic surfaces, Am. J. Math., 110 (1988), no. 1, 49-75. | MR 926738 | Zbl 0673.14020

27. M. Nagata, A generalization of the imbedding problem of an abstract variety in a complete variety, J. Math. Kyoto Univ., 3 (1963), 89-102. | MR 158892 | Zbl 0223.14011

28. C. Nakayama, Logarithmic étale cohomology, Math. Ann., 308 (1997), 365-404. | MR 1457738 | Zbl 0877.14016

29. C. Nakayama, Nearby cycles for log smooth families, Compos. Math., 112 (1998), 45-75. | MR 1622751 | Zbl 0926.14006

30. T. Ochiai, l-independence of the trace of monodromy, Math. Ann., 315 (1999), no. 2, 321-340. | MR 1715253 | Zbl 0980.14014

31. A. Ogg, Elliptic curves and wild ramification, Am. J. Math. 89 (1967), 1-21. | MR 207694 | Zbl 0147.39803

32. M. Olsson, Logarithmic geometry and algebraic stacks, Ann. Sci. Éc. Norm. Supér., 36 (2003), 747-791. | Numdam | MR 2032986 | Zbl 1069.14022

33. F. Orgogozo, Conjecture de Bloch et nombres de Milnor, Ann. Inst. Fourier, 53 (2003), 1739-1754. | Numdam | MR 2038779 | Zbl 1065.14005

34. D. Quillen, Notes on the homology of commutative rings, Mimeographed Notes, MIT (1968). | Zbl 0234.18010

35. D. Quillen, On the (co-) homology of commutative rings, in Applications of Categorical Algebra (Proc. Sympos. Pure Math., XVII, New York, 1968, 65-87. Am. Math. Soc., Providence, R.I. | MR 257068 | Zbl 0234.18010

36. J.-P. Serre, Corps locaux, 3rd ed., Hermann, Paris (1968). | MR 354618 | Zbl 0137.02601

37. J.-P. Serre, Représentations linéaires des groupes finis, 3rd ed., Hermann, Paris (1978). | MR 543841 | Zbl 0407.20003

38. T. Saito, Conductor, discriminant, and the Noether formula for arithmetic surfaces, Duke Math. J., 57 (1988), no. 1, 151-173. | MR 952229 | Zbl 0657.14017

39. T. Saito, Self-intersection 0-cycles and coherent sheaves on arithmetic schemes, Duke Math. J., 57 (1988), no. 2, 555-578. | MR 962520 | Zbl 0687.14004

40. T. Saito, Parity in Bloch's conductor formula in even dimension, to appear in J. Théor. Nombres Bordx. | Numdam | Zbl 1093.14030

41. T. Saito, Weight spectral sequences and independence of ℓ, J. de l'Institut Math. de Jussieu, 2 (2003), 1-52. | Zbl 1084.14027

42. C. Weibel, An introduction to homological algebra, Cambr. Stud. Adv. Math., 38, Cambridge UP, Cambridge (1994). | MR 1269324 | Zbl 0797.18001