On the Trace of Graded Automorphisms Original Research Article

LetA = circled plusd ≥ 0 Adbe a connected algebra with a graded algebra endomorphism σ. The trace of σ is defined to be Tr(σ, t) = ∑d ≥ 0 tr(σAd)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 Molienʹs 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.