Inverse problems for deformation rings
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- by Frauke M. Bleher, Ted Chinburg and Bart de Smit PDF
- Trans. Amer. Math. Soc. 365 (2013), 6149-6165
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.References
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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
- 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.
- © Copyright 2013 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
- DOI: https://doi.org/10.1090/S0002-9947-2013-05848-5
- MathSciNet review: 3091278