The Stacks project

Lemma 36.36.3. Let $f : X \to Y$ be a morphism of schemes. Assume

  1. $Y$ is quasi-compact and quasi-separated and has the resolution property,

  2. there exists an $f$-ample invertible module on $X$.

Then $X$ has the resolution property.

Proof. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module. Let $\mathcal{L}$ be an $f$-ample invertible module. Choose an affine open covering $Y = V_1 \cup \ldots \cup V_ m$. Set $U_ j = f^{-1}(V_ j)$. By Properties, Proposition 28.26.13 for each $j$ we know there exists finitely many maps $s_{j, i} : \mathcal{L}^{\otimes n_{j, i}}|_{U_ j} \to \mathcal{F}|_{U_ j}$ which are jointly surjective. Consider the quasi-coherent $\mathcal{O}_ Y$-modules

\[ \mathcal{H}_ n = f_*(\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes n}) \]

We may think of $s_{j, i}$ as a section over $V_ j$ of the sheaf $\mathcal{H}_{-n_{j, i}}$. Suppose we can find finite locally free $\mathcal{O}_ Y$-modules $\mathcal{E}_{i, j}$ and maps $\mathcal{E}_{i, j} \to \mathcal{H}_{-n_{j, i}}$ such that $s_{j, i}$ is in the image. Then the corresponding maps

\[ f^*\mathcal{E}_{i, j} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes n_{i, j}} \longrightarrow \mathcal{F} \]

are going to be jointly surjective and the lemma is proved. By Properties, Lemma 28.22.3 for each $i, j$ we can find a finite type quasi-coherent submodule $\mathcal{H}'_{i, j} \subset \mathcal{H}_{-n_{j, i}}$ which contains the section $s_{i, j}$ over $V_ j$. Thus the resolution property of $Y$ produces surjections $\mathcal{E}_{i, j} \to \mathcal{H}'_{j, i}$ and we conclude. $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 36.36: The resolution property

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Your email address will not be published.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 0FDD. Beware of the difference between the letter 'O' and the digit '0'.