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Bertini theorem for normality on local rings in mixed characteristic (applications to characteristic ideals)

Part of: Local rings and semilocal rings Differential algebra

Published online by Cambridge University Press:  11 January 2016

Tadashi Ochiai  and
Kazuma Shimomoto
Tadashi Ochiai
Affiliation:
Graduate school of Science, Osaka University 1-1, Machikaneyama Toyonaka, Osaka 560-0043, Japan, ochiai@math.sci.osaka-u.ac.jp
Kazuma Shimomoto
Affiliation:
Department of Mathematics, School of Science and Technology, Meiji University 1-1-1, Higashimita Tama-Ku Kawasaki 214-8571, Japan, shimomotokazuma@gmail.com
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Abstract

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In this article, we prove a strong version of the local Bertini theorem for normality on local rings in mixed characteristic. The main result asserts that a generic hyperplane section of a normal, Cohen–Macaulay, and complete local domain of dimension at least 3 is normal. Applications include the study of characteristic ideals attached to torsion modules over normal domains, which is fundamental in the study of Euler system theory, Iwasawa's main conjectures, and the deformation theory of Galois representations.

MSC classification

Secondary: 13H10: Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) 13N05: Modules of differentials
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 218 , June 2015 , pp. 125 - 173
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2015

References

[1] Bruns, W. and Herzog, J., Cohen–Macaulay Rings, Cambridge Stud. Adv. Math. 39, Cambridge University Press, Cambridge, 1993. MR 1251956.Google Scholar
[2] Eisenbud, D. and Evans, E. G. Jr., Generating modules efficiently: Theorems from algebraic K-theory, J. Algebra 27 (1973), 278–305. MR 0327742.CrossRefGoogle Scholar
[3] Flach, M., The equivariant Tamagawa number conjectures: A survey, with an appendix by Greither, C., Contemp. Math. 358 (2004), 79–125. MR 2088713. DOI 10.1090/conm/358/06537.CrossRefGoogle Scholar
[4] Flenner, H., Die Sätze von Bertini für lokale Ringe, Math. Ann. 229 (1977), 97–111. MR 0460317. CrossRefGoogle Scholar
[5] Fresnel, J. and van der Put, M., Rigid Analytic Geometry and Its Applications, Progr. Math. 218, Birkhäuser, Boston, 2004. MR 2014891. DOI 10.1007/978-1-4612-0041-3.Google Scholar
[6] Greco, S., Two theorems on excellent rings, Nagoya Math. J. 60 (1976), 139–149. MR 0409452.Google Scholar
[7] Grothendieck, A., Élements de géométrie algébrique, IV: Étude locale des schémaset des morphismes de schémas, II, Publ. Math. Inst. Hautes Études Sci. 24 (1965). MR 0199181.Google Scholar
[8] Grothendieck, A., Cohomologie locale des faisceaux coherénts et théorèmes des Lefschetz locauxet globaux, Séminaire de Géométrie Algébrique du Bois-Marie (SGA 2), North-Holland, Amsterdam, 1968. MR 0476737.Google Scholar
[9] Hartshorne, R., Stable reflexive sheaves, Math. Ann. 254 (1980), 121–176. MR 0597077. DOI 10.1007/BF01467074.CrossRefGoogle Scholar
[10] Kunz, E., Kähler Differentials, Adv. Lectures Math., Friedr. Vieweg, Braunschweig, 1986. MR 0864975. DOI 10.1007/978-3-663-14074-0.Google Scholar
[11] Matsumura, H., Commutative Ring Theory, Cambridge Stud. Adv. Math. 8, Cambridge University Press, Cambridge, 1986. MR 0879273.Google Scholar
[12] Neukirch, J., Schmidt, A., and Wingberg, K., Cohomology of Number Fields, Grundlehren Math. Wiss. 323, Springer, Berlin, 2000. MR 1737196.Google Scholar
[13] Northcott, D. G., Finite Free Resolutions, Cambridge Tracts in Math. 71, Cambridge University Press, Cambridge, 1976. MR 0460383.Google Scholar
[14] Ochiai, T., Euler system for Galois deformations, Ann. Inst. Fourier (Grenoble) 55 (2005), 113–146. MR 2141691.Google Scholar
[15] Ochiai, T. and Shimomoto, K., Specialization method in Krull dimension two and Euler system theory over normal deformation rings, in preparation.Google Scholar
[16] Serre, J.-P., Local Fields, Grad. Texts in Math. 67, Springer, New York, 1979. MR 0554237.Google Scholar
[17] Skinner, C. and Urban, E., The Iwasawa main conjectures for GL(2), Invent. Math. 195 (2014), 1–277. MR 3148103. DOI 10.1007/s00222-013-0448-1.CrossRefGoogle Scholar
[18] Swan, R. G., The number of generators of a module, Math. Z. 102 (1967), 318–322. MR 0218347.Google Scholar
[19] Trivedi, V., A local Bertini theorem in mixed characteristic, Comm. Algebra 22 (1994), 823–827. MR 1261007. DOI 10.1080/00927879408824878.Google Scholar