On Lefschetz series (Q644542)

Let \(\bigoplus_{i=0}^\infty R^i\) be a graded commutative algebra over \(Q\) and \(R^+=\bigoplus_{i=1}^\infty R^i\). An element \(x\in R\) is decomposable if and only if there exist homogeneous elements \(y_i, z_i\in R\) such that one can write \(x=\sum y_iz_i\) as a finite sum. Let \(D(R)\) denote all decomposable elements of \(R\) and \(Q(R)=R^+/D(R)\) denote the set of indecomposable elements of \(R\). It is easy to see that \(Q(R)\) and \(D(R)\) are graded vector spaces. If \(f: R_1\mapsto R_2\) is a graded homomorphism, then \(f(Q(R_1))\subset Q(R_2)\). Hence \(Q(f): Q(R_1)\mapsto Q(R_2)\) is a graded homomorphism with \(Q(f)=\bigoplus_{i=1}^\infty Q^i(f)\). In the paper under review the authors show that the Lefschetz series of \(f\) can be computed from \(Q(f)\). As a main result they show that the Lefschetz series \(L_f(t)\) is determined by \(R\) and \(Q(f)\). In fact, they show that the coefficients of \(L_f(t)\) can be written as homogeneous polynomials of the eigenvalues of \(Q(f)\). An explicit algorithm to compute \(L_f(t)\) from \(Q(f)\) is given. At the end some examples and applications are given.