On injective resolutions of local cohomology modules

Let $K$ be a field of characteristic zero and let $R = K[X_1,\ldots,X_n]$. Let $I$ be an ideal in $R$ and let $M = H^i_I(R)$ be the $i^{th}$-local cohomology module of $R$ with respect to $I$. Let $c = \ injdim \ M$. We prove that if $P$ is a prime ideal in $R$ with Bass number $\mu_c(P,M) > 0$ then $P$ is a maximal ideal in $R$

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

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