A short note on the non-negativity of partial Euler characteristics
Open Access Publications from the University of California

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

• Author(s): Puthenpurakal, Tony J.
• et al.

## Published Web Location

https://arxiv.org/pdf/math/0409059.pdf
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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.

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