Beitr\ EMIS ELibM Electronic Journals Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Vol. 46, No. 2, pp. 559-560 (2005)

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

Tony. J. Puthenpurakal

Department of Mathematics, IIT Bombay, Powai, Mumbai 400 076, e-mail{

Abstract: 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 non-negative. 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.: Cohen-Macaulay rings. Cambridge Studies in Advanced Mathematics {\bf 39}, Cambridge 1993. [S] Serre, J. P.: Local Algebra. Springer Monographs in Mathematics, Springer-Verlag, Berlin 2000.

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Electronic version published on: 18 Oct 2005. This page was last modified: 29 Dec 2008.

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