Generalized Gorensteinness and a homological determinant for preprojective algebras

The study of invariants of group actions on commutative polynomial rings has motivated many developments in commutative algebra and algebraic geometry. It has been of particular interest to understand what conditions on the group result in an invariant ring satisfying useful properties. In particular, Watanabe's Theorem states that the invariant subring of $k[x_1,\ldots,x_n]$ under the natural action of a finite subgroup of $SL_n(k)$ is always Gorenstein. In this paper, we study this question in the more general setting of group actions on noncommutative non-connected algebras $A$. We develop the notion of a homological determinant of an automorphism of $A$, then use the homological determinant to study actions of finite groups $G$ on $A$. We give a sufficient condition so that the invariant ring $A^G$ has finite injective dimension and satisfies the generalized Gorenstein condition. More precisely, let $A$ be a noetherian $\mathbb{N}$-graded generalized Gorenstein algebra with finite global dimension. Suppose all elements of $G$ fix the idempotents of $A$ and act with trivial homological determinant. Then the invariant ring $A^G$ is generalized Gorenstein.