The Stacks project

Lemma 33.33.3. 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. $H^0(X, \mathcal{F}) = \bigoplus _{x \in \text{Supp}(\mathcal{F})} \mathcal{F}_ x$,

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

  4. $\chi (X, \mathcal{F}) = \dim _ k H^0(X, \mathcal{F})$, and

  5. $\chi (X, \mathcal{F} \otimes \mathcal{E}) = n\chi (X, \mathcal{F})$ 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. By definition of the scheme theoretic support the underlying topological space of $Z$ is $\text{Supp}(\mathcal{F})$. 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). In fact, by Lemma 33.20.2 the scheme $Z$ is a finite disjoint union of spectra of local Artinian rings. Thus correspondingly $H^0(Z, \mathcal{G}) = \bigoplus _{z \in Z} \mathcal{G}_ z$. The cohomologies of $\mathcal{F}$ and $\mathcal{G}$ agree by Cohomology of Schemes, Lemma 30.2.4. Thus $H^ i(X, \mathcal{F}) = 0$ for $i > 0$ and $H^0(X, \mathcal{F}) = H^0(Z, \mathcal{G})$. In particular we have (3) is true. For $z \in Z$ corresponding to $x \in \text{Supp}(\mathcal{F})$ we have $\mathcal{G}_ z = (i_*\mathcal{G})_ x = \mathcal{F}_ x$. We conclude that (2) holds. Of course (2) implies (1). We have (4) by definition of the Euler characteristic $\chi (X, \mathcal{F})$ and (3). By the projection formula (Cohomology, Lemma 20.52.2) 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 (5) holds. $\square$

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