On a generalization of a result of Peskine and Szpiro

Abstract

Let $(R,\mathfrak{m})$ be a regular local ring containing a field $K$. Let $I$ be a Cohen–Macaulay ideal of height $g$. If $\mathrm{char}K=p>0$ then by a result of Peskine and Szpiro the local cohomology modules $H_I^i(R)$ vanish for $i>g$. This result is not true if $\mathrm{char}K=0$. However, we prove that the Bass numbers of the local cohomology module $H_I^g(R)$ completely determine whether $H_I^i(R)$ vanish for $i>g$. The result of this paper has been proved more generally for Gorenstein local rings by Hellus and Schenzel (2008) (Theorem 3.2). However, our result is elementary to prove. In particular, we do not use spectral sequences in our proof.