Poincaré duality and Steinberg’s Theorem on rings of coinvariants

Abstract:

Let $k$ be a field, $V$ an $r$-dimensional $k$-vector space, and $W$ a finite subgroup of $\mathrm {Aut}_k(V )$. Let $S = S[V^{\#}]$ be the symmetric algebra on $V^\#$, 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 $P_* = S\otimes _R k$ 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.