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.