Universal deformation rings need not be complete intersections

Frauke M. Bleher; Ted Chinburg

doi:10.1016/j.crma.2005.12.006

Algebra

Universal deformation rings need not be complete intersections
[Les anneaux de déformation universelle ne sont pas nécessairement d'intersection complète]

Présenté par : Jean-Pierre Serre

Frauke M. Bleher ¹ ; Ted Chinburg ²

¹ Department of Mathematics, University of Iowa, Iowa City, IA 52242-1419, USA
² Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395, USA

Comptes Rendus. Mathématique, Volume 342 (2006) no. 4, pp. 229-232.

Résumé
Abstract

Nous répondons à une question de M. Flach en démontrant qu'il existe une représentation linéaire d'un groupe profini dont l'anneau de déformation universelle n'est pas un anneau d'intersection complète. Nous montrons que l'arithmétique fournit de tels exemples dans les situations suivantes. Il existe une infinité de corps quadratiques réels F tels qu'il existe une représentation du groupe de Galois de l'extension maximale non-ramifiée de F sur un corps de caractéristique 2 dont l'anneau de déformation universelle n'est pas un anneau d'intersection complète.

We answer a question of M. Flach by showing that there is a linear representation of a profinite group whose universal deformation ring is not a complete intersection. We show that such examples arise in arithmetic in the following way. There are infinitely many real quadratic fields F for which there is a mod 2 representation of the Galois group of the maximal unramified extension of F whose universal deformation ring is not a complete intersection.

Reçu le : 2005-10-04
Accepté le : 2005-12-06
Publié le : 2006-01-04

DOI : 10.1016/j.crma.2005.12.006

Affiliations des auteurs :

Frauke M. Bleher ¹ ; Ted Chinburg ²

¹ Department of Mathematics, University of Iowa, Iowa City, IA 52242-1419, USA
² Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395, USA

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Frauke M. Bleher; Ted Chinburg. Universal deformation rings need not be complete intersections. Comptes Rendus. Mathématique, Volume 342 (2006) no. 4, pp. 229-232. doi : 10.1016/j.crma.2005.12.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.12.006/

Bibliographie
Cité par

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[3] T. Chinburg, Can deformation rings of group representations not be local complete intersections?, in: Problems from the Workshop on Automorphisms of Curves in Leiden in August 2004, in press

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Timothy Eardley; Jayanta Manoharmayum The inverse deformation problem, Compositio Mathematica, Volume 152 (2016) no. 8, p. 1725 | DOI:10.1112/s0010437x16007582
Krzysztof Dorobisz The inverse problem for universal deformation rings and the special linear group, Transactions of the American Mathematical Society, Volume 368 (2016) no. 12, p. 8597 | DOI:10.1090/tran6644
Frauke Bleher Brauer’s generalized decomposition numbers and universal deformation rings, Transactions of the American Mathematical Society, Volume 366 (2014) no. 12, p. 6507 | DOI:10.1090/s0002-9947-2014-06120-5
Frauke Bleher; Ted Chinburg; Bart de Smit Inverse problems for deformation rings, Transactions of the American Mathematical Society, Volume 365 (2013) no. 11, p. 6149 | DOI:10.1090/s0002-9947-2013-05848-5
Frauke M. Bleher; José A. Vélez-Marulanda Universal deformation rings of modules over Frobenius algebras, Journal of Algebra, Volume 367 (2012), p. 176 | DOI:10.1016/j.jalgebra.2012.06.008
Frauke M. Bleher; Giovanna Llosent; Jennifer B. Schaefer Universal deformation rings and dihedral blocks with two simple modules, Journal of Algebra, Volume 345 (2011) no. 1, p. 49 | DOI:10.1016/j.jalgebra.2011.08.010
Frauke M. Bleher Universal deformation rings and generalized quaternion defect groups, Advances in Mathematics, Volume 225 (2010) no. 3, p. 1499 | DOI:10.1016/j.aim.2010.04.004
Frauke M. Bleher; Giovanna Llosent Universal Deformation Rings for the Symmetric Group S 4, Algebras and Representation Theory, Volume 13 (2010) no. 3, p. 255 | DOI:10.1007/s10468-008-9120-7
Frauke M. Bleher; Ted Chinburg Universal deformation rings need not be complete intersections, Mathematische Annalen, Volume 337 (2007) no. 4, p. 739 | DOI:10.1007/s00208-006-0054-2

Cité par 9 documents. Sources : Crossref

^⁎ Supported (respectively) by NSF Grant DMS01-39737 and NSF Grant DMS00-70433.

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