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The inverse problem for universal deformation rings and the special linear group


Author: Krzysztof Dorobisz
Journal: Trans. Amer. Math. Soc. 368 (2016), 8597-8613
MSC (2010): Primary 11G99; Secondary 11F80, 20C20
DOI: https://doi.org/10.1090/tran6644
Published electronically: March 1, 2016
MathSciNet review: 3551582
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Abstract: We present a solution to the inverse problem for universal deformation rings of group representations. Namely, we show that every complete noetherian local commutative ring $ R$ with a finite residue field $ k$ can be realized as the universal deformation ring of a continuous linear representation of a profinite group. More specifically, $ R$ is the universal deformation ring of the natural representation of $ \textup {SL}_n(R)$ in $ \textup {SL}_n(k)$, provided that $ n\geq 4$. We also check for which $ R$ an analogous result is true in case $ n=2$ and $ n=3$.


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Additional Information

Krzysztof Dorobisz
Affiliation: Mathematisch Instituut, Universiteit Leiden, P.O. Box 9512, 2300 RA Leiden, The Netherlands
Email: kdorobisz@gmail.com

DOI: https://doi.org/10.1090/tran6644
Keywords: Deformations of group representations, universal deformation rings, inverse problem, special linear group
Received by editor(s): December 5, 2013
Received by editor(s) in revised form: July 18, 2014, and October 24, 2014
Published electronically: March 1, 2016
Article copyright: © Copyright 2016 American Mathematical Society

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