Naihuan Jing and James J. Zhang.
On the trace of graded automorphisms
Preprint, 12/30/95.

\abstract
Let $A=\oplus_{n\geq 0} A_n$ be a connected graded algebra with a
graded algebra automorphism $\sigma$. The trace of $\sigma$ is
defined to be a power series $Tr(\sigma,t)=\sum_{n\geq 0}tr
(\sigma|_{A_n})t^n$. In particular, the trace of the identity map
of $A$ is the Hilbert series of $A$. We prove that $Tr(\sigma,t)$
is a rational function if either

1. $A$ is a finitely generated commutative algebra, or

2. $A$ is a right noetherian algebra with finite global dimension, or

3. $A$ is a regular (but not necessarily noetherian) algebra with finite
global dimension.

\noindent
Using this we show that the Hilbert series of subrings of invariants are
rational functions in these three cases. Some basic properties of the
trace are discussed and some applications are given. We also study the
Gorenstein property for some special subrings of invariants.
\endabstract