The Gorenstein property and the Kemper et al. conjecture

Amiram Braun

Abstract. Let G \subset SL(V) be a finite group, where V is a finite dimensional vector space
over a field F with charF = p \ge 0. Let S(V) be the symmetric algebra of V and S(V)^G
the ring of G-invariants. We provide for each p > 0 an example of a group G, where
S(V)^G is Cohen-Macaulay, but is not Gorenstein. This refutes a natural conjecture due
to G. Kemper, E. K\"ording, G. Malle, B. H. Matzat, D. Vogel and G. Wiese. We also prove
a "local-global" criterion for the Gorenstein property to hold in S(V)^G. This is then used
to characterize the Gorenstein property of S(F^2)^G for any finite subgroups G of SL(2,F).