Rings of coinvariants and $p$-subgroups
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Abstract:
Let $\varrho :G\hookrightarrow GL(n, \mathbb {F})$ be a faithful representation of a finite group $G$ over the field $\mathbb {F}$ and let $V \cong \mathbb {F}^n$ be an $\mathbb {F}(G)$-module. It has been shown by L. Smith that if $n=3$ and the order of $G$ is divisible by the positive characteristic $p$ of $\mathbb {F}$, then $\mathbb {F} [V]^G$ is Cohen-Macaulay. Under the condition $n=3$ we prove the following conjecture through this remarkable result: If $\mathbb {F} [V]_G$ is a Poincaré duality algebra, then $\mathbb {F} [V]_{\operatorname {Syl}_{p}(G)}$ is a complete intersection, where $\operatorname {Syl}_{p}(G)$ is a Sylow $p$-subgroup of $G$.References
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Additional Information
- Tzu-Chun Lin
- Affiliation: Department of Applied Mathematics, Feng Chia University, 100 Wenhwa Road, Tai- chung 407, Taiwan, Republic of China
- Email: lintc@fcu.edu.tw
- Received by editor(s): October 19, 2005
- Received by editor(s) in revised form: May 13, 2008, October 20, 2008, and February 26, 2010
- Published electronically: July 1, 2010
- Communicated by: Bernd Ulrich
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 138 (2010), 4243-4247
- MSC (2000): Primary 13A50; Secondary 20F55
- DOI: https://doi.org/10.1090/S0002-9939-2010-10470-7
- MathSciNet review: 2680050