Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Vol. 46, No. 2, pp. 559560 (2005) 

A short note on the nonnegativity of partial Euler characteristicsTony. J. PuthenpurakalDepartment of Mathematics, IIT Bombay, Powai, Mumbai 400 076, email{tputhen@math.iitb.ac.inAbstract: Let $(A,\mathfrak{m})$ be a Noetherian local ring, $M$ a finite $A$module and $x_1,\ldots,x_n\in \m$ such that $\lambda (M/{\mathbf x} M)$ is finite. Serre ([S, Appendix 2]) proved that all partial Euler characteristics of $M$ with respect to $\mathbf x$ is nonnegative. This fact is easy to show when $A$ contains a field ([BH, 4.7.12]). We give an elementary proof of Serre's result when $A$ does not contain a field. [BH] Bruns,W.; Herzog, J.: CohenMacaulay rings. Cambridge Studies in Advanced Mathematics {\bf 39}, Cambridge 1993. [S] Serre, J. P.: Local Algebra. Springer Monographs in Mathematics, SpringerVerlag, Berlin 2000. Full text of the article:
Electronic version published on: 18 Oct 2005. This page was last modified: 29 Dec 2008.
© 2005 Heldermann Verlag
