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Let be a field, an -dimensional -vector space, and a finite subgroup of . Let be the symmetric algebra on , the -dual of , and the ring of invariants under the natural action of on . Define to be the quotient algebra . Steinberg has shown that is polynomial if is the field of complex numbers and the quotient algebra satisfies Poincare 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 . This gives a new proof even in the characteristic zero case. Theorem 0.1. If the characteristic of is zero or prime to the order of and satisfies Poincare duality, then is isomorphic to a polynomial algebra on generators.

Poincaré duality and Steinberg’s Theorem on rings of coinvariants