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Poincaré duality and Steinberg's Theorem on rings of coinvariants


Authors: W. G. Dwyer and C. W. Wilkerson
Journal: Proc. Amer. Math. Soc. 138 (2010), 3769-3775
MSC (2010): Primary 57T10, 13A50; Secondary 20F55
DOI: https://doi.org/10.1090/S0002-9939-2010-10429-X
Published electronically: May 26, 2010
MathSciNet review: 2661576
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Abstract: Let $ k$ be a field, $ V$ an $ r$-dimensional $ k$-vector space, and $ W$ a finite subgroup of $ \mathrm{Aut}_k(V )$. Let $ S = S[V^{\char93 }]$ be the symmetric algebra on $ V^\char93 $, the $ k$-dual of $ V$, and $ R = S^W$ the ring of invariants under the natural action of $ W$ on $ S$. Define $ P_*$ to be the quotient algebra $ S\otimes_R k$.

Steinberg has shown that $ R$ is polynomial if $ k$ is the field of complex numbers and the quotient algebra $ P_* = S\tensor_R k$ satisfies Poincaré duality.

In this paper we use elementary methods to prove Steinberg's result for fields of characteristic 0 or of characteristic prime to the order of $ W$. This gives a new proof even in the characteristic zero case.

Theorem 0.1. If the characteristic of $ k$ is zero or prime to the order of $ W$ and $ P_*$ satisfies Poincaré duality, then $ R$ is isomorphic to a polynomial algebra on $ r$ generators.


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Additional Information

W. G. Dwyer
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Email: dwyer.1@nd.edu

C. W. Wilkerson
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907-1395 – and – Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
Email: cwilkers@purdue.edu, cwilkers@math.tamu.edu

DOI: https://doi.org/10.1090/S0002-9939-2010-10429-X
Keywords: Invariants, coinvariants, Gorenstein, Poincar\'e duality, duality, reflection groups
Received by editor(s): May 19, 2006
Received by editor(s) in revised form: January 31, 2010
Published electronically: May 26, 2010
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2010 W. G. Dwyer and C. W. Wilkerson

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