On Lefschetz series

Abstract

Let R = ⊕ ∞ i=0 R i be a connected graded commutative algebra over the field Q of rational numbers, and let f be a graded endomorphism of R. In this paper, we show that the Lefschetz series of f can be computed directly from the induced linear map Q(f ) on the Q vector space of indecomposables of R. We give an explicit algorithm to compute the Lefschetz series of f from Q(f ). The main tool we used is the graded algebra version of Gröbner basis theory. At the end of this paper, some examples and applications are given.