On a theorem of R. Steinberg on rings of coinvariants

Let :G ,!GL(n, F) be a representation of a finite group G over the field F. G acts via on the ring of polynomial functions F[V ], V = Fn. LetH(G) F[V ] be the Hilbert ideal, i.e., the ideal in F[V ] generated by all the homogeneous G-invariant forms of strictly positive degree. R. Steinberg [Trans. Amer. Math. Soc. 112 (1964), 392–400; MR0167535 (29 #4807)] proved that for F of characteristic zero, the algebra of coinvariants F[V ]G = F[V ]/H(G) satisfies Poincar´e duality if and only if G is a pseudoreflection group (formulation of R. M. Kane [Canad. Math. Bull. 37 (1994), no. 1, 82–88; MR1261561 (96e:51016)]). In the paper under review, the author explores the situation for fields of non-zero characteristic. He proves that an analogue of Steinberg’s theorem holds for the case n = 2 and, for n = 3, it holds in a weak sense for representations of p-groups over a field F of characteristic p. He gives a counterexample in the case n = 4, using the vector invariants of Z/2 over F2: F2[V ]Z/2 is a Poincar´e duality algebra, although F2[V ]Z/2 is not a polynomial algebra and Z/2 contains no pseudoreflections. He also shows the following interesting property: F2[V ]Z/2 is isomorphic to the algebra of coinvariants obtained from a degree 4 faithful representation of the group Z/2× Z/2, which is generated by reflections and has polynomial invariants.