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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

   

 

Inverse problems for deformation rings


Authors: Frauke M. Bleher, Ted Chinburg and Bart de Smit
Journal: Trans. Amer. Math. Soc. 365 (2013), 6149-6165
MSC (2010): Primary 11F80; Secondary 11R32, 20C20
Published electronically: May 14, 2013
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Abstract: Let $ W$ be a complete Noetherian local commutative ring with residue field $ k$ of positive characteristic $ p$. We study the inverse problem for the universal deformation rings $ R_{W}(\Gamma ,V)$ relative to $ W$ of finite dimensional representations $ V$ of a profinite group $ \Gamma $ over $ k$. We show that for all $ p$ and $ n \ge 1$, the ring $ W[[t]]/(p^n t,t^2)$ arises as a universal deformation ring. This ring is not a complete intersection if $ p^nW\neq \{0\}$, so we obtain an answer to a question of M. Flach in all characteristics. We also study the `inverse inverse problem' for the ring $ W[[t]]/(p^n t,t^2)$; this is to determine all pairs $ (\Gamma , V)$ such that $ R_{W}(\Gamma ,V)$ is isomorphic to this ring.


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Additional Information

Frauke M. Bleher
Affiliation: Department of Mathematics, University of Iowa, Iowa City, Iowa 52242-1419
Email: frauke-bleher@uiowa.edu

Ted Chinburg
Affiliation: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6395
Email: ted@math.upenn.edu

Bart de Smit
Affiliation: Mathematisch Instituut, University of Leiden, P.O. Box 9512, 2300 RA Leiden, The Netherlands
Email: desmit@math.leidenuniv.nl

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05848-5
PII: S 0002-9947(2013)05848-5
Keywords: Universal deformation rings, complete intersections, inverse problems
Received by editor(s): February 24, 2012
Received by editor(s) in revised form: April 5, 2012
Published electronically: May 14, 2013
Additional Notes: The first author was supported in part by NSF Grant DMS0651332 and NSA Grant H98230-11-1-0131. The second author was supported in part by NSF Grants DMS0801030 and DMS1100355. The third author was funded in part by the European Commission under contract MRTN-CT-2006-035495.
Article copyright: © Copyright 2013 Frauke M. Bleher, Ted Chinburg, and Bart de Smit