The Stacks project

Lemma 33.32.3. Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\mathcal{F}$ be a coherent sheaf with $\dim (\text{Supp}(\mathcal{F})) \leq 0$. Then

  1. $\mathcal{F}$ is generated by global sections,

  2. $H^ i(X, \mathcal{F}) = 0$ for $i > 0$,

  3. $\chi (X, \mathcal{F}) = \dim _ k\Gamma (X, \mathcal{F})$, and

  4. $\chi (X, \mathcal{F} \otimes \mathcal{E}) = n\chi (X, \mathcal{F})$ for every locally free module $\mathcal{E}$ of rank $n$.

Proof. By Cohomology of Schemes, Lemma 30.9.7 we see that $\mathcal{F} = i_*\mathcal{G}$ where $i : Z \to X$ is the inclusion of the scheme theoretic support of $\mathcal{F}$ and where $\mathcal{G}$ is a coherent $\mathcal{O}_ Z$-module. Since the dimension of $Z$ is $0$, we see $Z$ is affine (Properties, Lemma 28.10.5). Hence $\mathcal{G}$ is globally generated and the higher cohomology groups of $\mathcal{G}$ are zero (Cohomology of Schemes, Lemma 30.2.2). Hence $\mathcal{F} = i_*\mathcal{G}$ is globally generated. Since the cohomologies of $\mathcal{F}$ and $\mathcal{G}$ agree (Cohomology of Schemes, Lemma 30.2.4) we conclude that the higher cohomology groups of $\mathcal{F}$ are zero which gives the first formula. By the projection formula (Cohomology, Lemma 20.49.2) we have

\[ i_*(\mathcal{G} \otimes i^*\mathcal{E}) = \mathcal{F} \otimes \mathcal{E}. \]

Since $Z$ has dimension $0$ the locally free sheaf $i^*\mathcal{E}$ is isomorphic to $\mathcal{O}_ Z^{\oplus n}$ and arguing as above we see that the second formula holds. $\square$

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