POINCAR'E DUALITY AND STEINBERG'S THEOREM ON RINGS OF COINVARIANTS
W. G. DWYER AND C. W. WILKERSON
In this note 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.
1.1. Theorem. Let k be a field, V an r-dimensional k-vector space,
and W a finite subgroup of 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 of
under the natural
action of W on S. Define P* to be the quotient algebra S i\tensor_R k. If the
characteristic of k is zero or prime to the order of W and P* satisfies
Poincar'e duality, then R is isomorphic to a polynomial algebra on r
generators.
Steinberg [9] has shown that R is polynomial if k is the field
of complex numbers and the quotient algebra P* = S\tensor_R k satisfies
Poincar'e duality (1.3). Steinberg's result was extended by Kane [3, 4]
to other fields of characteristic zero, and by T.-C. Lin [5] to the case in
which k is a finite field of characteristic prime to the order of W .
The current proof is independent of previous methods.
(Revised Jan. 25, 2010 to correct typos and to incorporate some remarks by R. J. Shank .)