The inverse problem for universal deformation rings and the special linear group
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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 $\mathrm {SL}_n(R)$ in $\mathrm {SL}_n(k)$, provided that $n\geq 4$. We also check for which $R$ an analogous result is true in case $n=2$ and $n=3$.References
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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
- 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
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 8597-8613
- MSC (2010): Primary 11G99; Secondary 11F80, 20C20
- DOI: https://doi.org/10.1090/tran6644
- MathSciNet review: 3551582