Journal article
Universal deformation rings, endo-trivial modules, and semidihedral and generalized quaternion 2-groups
Journal of Pure and Applied Algebra, Vol.223(5), pp.1897-1912
12/12/2016
DOI: 10.1016/j.jpaa.2018.08.006
Abstract
J. Pure Appl. Algebra 223 (2019), no. 3, 1897-1912 Let $k$ be a field of characteristic $p>0$, and let $W$ be a complete discrete valuation ring of characteristic $0$ that has $k$ as its residue field. Suppose $G$ is a finite group and $G^{\mathrm{ab},p}$ is its maximal abelian $p$-quotient group. We prove that every endo-trivial $kG$-module $V$ has a universal deformation ring that is isomorphic to the group ring $WG^{\mathrm{ab},p}$. In particular, this gives a positive answer to a question raised by Bleher and Chinburg for all endo-trivial modules. Moreover, we show that the universal deformation of $V$ over $WG^{\mathrm{ab},p}$ is uniquely determined by any lift of $V$ over $W$. In the case when $p=2$ and $G=\mathrm{D}$ is a $2$-group that is either semidihedral or generalized quaternion, we give an explicit description of the universal deformation of every indecomposable endo-trivial $k\mathrm{D}$-module $V$.
Details
- Title: Subtitle
- Universal deformation rings, endo-trivial modules, and semidihedral and generalized quaternion 2-groups
- Creators
- Frauke M BleherTed ChinburgRoberto C Soto
- Resource Type
- Journal article
- Publication Details
- Journal of Pure and Applied Algebra, Vol.223(5), pp.1897-1912
- DOI
- 10.1016/j.jpaa.2018.08.006
- ISSN
- 0022-4049
- eISSN
- 1873-1376
- Grant note
- name: NSF FRG, award: DMS-1360621, DMS-1360767, DMS-1265290; name: NSF SaTC, award: CNS-1701785, CNS-1513671; DOI: 10.13039/100000893, name: Simons Foundation, award: 338379
- Language
- English
- Date published
- 12/12/2016
- Academic Unit
- Mathematics
- Record Identifier
- 9983985967902771
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