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Universal deformation rings need not be complete intersections

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Abstract

We answer a question of M. Flach by showing that there is a linear representation of a profinite group whose (unrestricted) 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. Finally, we discuss bounds on the singularities of universal deformation rings of representations of finite groups in terms of the nilpotency of the associated defect groups.

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Authors and Affiliations

  1. Department of Mathematics, University of Iowa, Iowa City, IA, 52242-1419, USA

    Frauke M. Bleher

  2. Department of Mathematics, University of Pennsylvania, Philadelphia, PA, 19104-6395, USA

    Ted Chinburg

Authors
  1. Frauke M. Bleher
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  2. Ted Chinburg
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Corresponding author

Correspondence to Ted Chinburg.

Additional information

The first author was supported in part by NSF Grant DMS01-39737 and NSA Grant H98230-06-1-0021. The second author was supported in part by NSF Grants DMS00-70433 and DMS05-00106.

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Bleher, F.M., Chinburg, T. Universal deformation rings need not be complete intersections. Math. Ann. 337, 739–767 (2007). https://doi.org/10.1007/s00208-006-0054-2

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  • DOI: https://doi.org/10.1007/s00208-006-0054-2

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