Cohomological and projective dimensions. (July 2013) - Varbaro, Matteo
Abstract Let $\mathfrak{a}$ be a homogeneous ideal of a polynomial ring $R$ in $n$ variables over a field $\mathbb{k}$ . Assume that $\mathrm{depth} (R/ \mathfrak{a})\geq t$ , where $t$ is some number in $\{ 0, \ldots, n\} $ . A result of Peskine and Szpiro says that if $\mathrm{char} (\mathbb{k})\gt 0$ , then the local cohomology modules ${ H}_{\mathfrak{a}}^{i} (M)$ vanish for all $i\gt n- t$ and all $R$ -modules $M$ . In characteristic $0$ , there are counterexamples to this for all $t\geq 4$ . On the other hand, when $t\leq 2$ , by exploiting classical results of Grothendieck, Lichtenbaum, Hartshorne and Ogus it is not difficult to extend the result to any characteristic. In this paper we settle the remaining case; specifically, we show that if $\mathrm{depth} (R/ \mathfrak{a})\geq 3$ , then the local cohomology modules ${ H}_{\mathfrak{a}}^{i} (M)$ vanish for all $i\gt n- 3$ and all $R$ -modules $M$ , whatever the characteristic of $\mathbb{k}$ is.