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 Grobner basis theory. At the end of this paper, some examples and applications are given.