The Stacks project

Proof. By the Grothendieck's vanishing result in Cohomology, Proposition 20.20.7 we conclude that $\dim (U) \geq d$. Assume $\dim (U) = d$. Choose a compactification $U \to X$ such that $U$ is dense in $X$. (This is possible by More on Flatness, Theorem 38.33.8 and Lemma 38.32.2.) After replacing $X$ by its reduction we find that $X$ is a proper variety of dimension $d$ and we see that $U$ is proper if and only if $U = X$. Set $Z = X \setminus U$. We will show that $H^ d(U, \mathcal{F})$ is zero if $Z$ is nonempty.

Choose a coherent $\mathcal{O}_ X$-module $\mathcal{G}$ whose restriction to $U$ is $\mathcal{F}$, see Properties, Lemma 28.22.5. Let $\omega _ X^\bullet $ denote the dualizing complex of $X$ as in Section 48.27. Set $\omega _ U^\bullet = \omega _ X^\bullet |_ U$. Then $H^ d(U, \mathcal{F})$ is dual to

by Lemma 48.33.1. By Lemma 48.27.1 we see that the cohomology sheaves of $\omega _ X^\bullet $ vanish in degrees $< -d$ and $H^{-d}(\omega _ X^\bullet ) = \omega _ X$ is a coherent $\mathcal{O}_ X$-module which is $(S_2)$ and whose support is $X$. In particular, $\omega _ X$ is torsion free, see Divisors, Lemma 31.11.10. Thus we see that the cohomology sheaf

is torsion free, see Divisors, Lemma 31.11.12. Consequently this sheaf has no nonzero sections vanishing on any nonempty open of $X$ (those would be torsion sections). Thus it follows from Lemma 48.33.3 that $H^{-d}_ c(U, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ U}(\mathcal{F}, \omega _ U^\bullet ))$ is zero, and hence $H^ d(U, \mathcal{F})$ is zero as desired. $\square$