On the trace of graded automorphisms

Abstract

LetA=⊕d≥0Adbe a connected algebra with a graded algebra endomorphism σ. The trace of σ is defined to be Tr(σ,t)=∑d≥0tr(σ|Ad)td. We prove that Tr(σ,t) is a rational function ifAis either finitely generated commutative or right noetherian with finite global dimension or regular. A version of Molien's theorem follows in these three cases. IfAis a regular algebra or a Frobenius algebra we prove a reciprocity for the trace. We also partially generalize a theorem of Watanabe on the Gorenstein property to the noncommutative case. © 1997 Academic Press.