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

  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 20.51.2) 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. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0AYT. Beware of the difference between the letter 'O' and the digit '0'.