Lemma 53.6.5. In Situation 53.6.2. Given an exact sequence
of coherent $\mathcal{O}_ X$-modules with $\dim (\text{Supp}(\mathcal{Q})) = 0$ and $\dim _ k H^0(X, \mathcal{Q}) \geq 2$ and such that there is no nonzero submodule $\mathcal{Q}' \subset \mathcal{F}$ such that $\mathcal{Q}' \to \mathcal{Q}$ is injective. Then the submodule of $\mathcal{F}$ generated by global sections surjects onto $\mathcal{Q}$.
Proof. Let $\mathcal{F}' \subset \mathcal{F}$ be the submodule generated by global sections and the image of $\omega _ X \to \mathcal{F}$. Since $\dim _ k H^0(X, \mathcal{Q}) \geq 2$ and $\dim _ k H^1(X, \omega _ X) = \dim _ k H^0(X, \mathcal{O}_ X) = 1$, we see that $\mathcal{F}' \to \mathcal{Q}$ is not zero and $\omega _ X \to \mathcal{F}'$ is not an isomorphism. Hence $H^1(X, \mathcal{F}') = 0$ by Lemma 53.6.3 and our assumption on $\mathcal{F}$. Consider the short exact sequence
If the quotient on the right is nonzero, then we obtain a contradiction because then $H^0(X, \mathcal{F})$ is bigger than $H^0(X, \mathcal{F}')$. $\square$
Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)