A short note on the non-negativity of partial Euler characteristics

Abstract

Let $(A,\mathfrak{m})$ be a Noetherian local ring, $M$ a finite $A$-module and x_1,...,x_n\in \m such that \lambda (M/\x M) is finite. Serre proved that all partial Euler characteristics of $M$ with respect to \x is non-negative. This fact is easy to show when $A$ contains a field. We give an elementary proof of Serre's result when $A$ does not contain a field.Comment: 2 page