The Stacks project

Lemma 48.34.1. Let $U$ be a variety. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ U$-module. If $H^ d(U, \mathcal{F})$ is nonzero, then $\dim (U) \geq d$ and if equality holds, then $U$ is proper.

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

\[ H^{-d}_ c(U, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ U}(\mathcal{F}, \omega _ U^\bullet )) \]

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

\[ H^{-d}(R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{G}, \omega _ X^\bullet )) = \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{G}, \omega _ X) \]

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. $\square$

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