On Lefschetz series

Abstract

Let R={\oplus }_{i=\mathrm{0}}^{\infty }{R}^i be a connected graded commutative algebra over the field \mathbb{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 \mathbb{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.

Keywords

Lefschetz series / Gröbner basis / endomorphism of graded algebra

References

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2014 Higher Education Press and Springer-Verlag Berlin Heidelberg