2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/72366Let $(A,\mathfrak{m})$ be a Noetherian local ring, $M$ a finite $A$-module and $x_1,...,x_n\in \m$ such that $λ(M/\x M)$ 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.2 pagesCommutative Algebra13D40text