ON THE TRACE OF GRADED AUTOMORPHISMS

Let A = +(d greater than or equal to 0) A(d) be a connected algebra wi th a graded algebra endomorphism sigma. The trace of sigma is defined to be Tr(sigma, t) = Sigma(d greater than or equal to 0) tr(sigma\A(d) )t(d). We prove that Tr(sigma, t) is a rational function if A is eithe r finitely generated commutative or right noetherian with finite globa l dimension or regular. A version of Molien's theorem follows in these three cases. If A is a regular algebra or a Frobenius algebra we prov e a reciprocity for the trace. We also partially generalize a theorem of Watanabe on the Gorenstein property to the noncommutative case. (C) 1997 Academic Press.