The Stacks project

Lemma 36.36.9. Let $X = \mathop{\mathrm{lim}}\nolimits X_ i$ be a limit of a direct system of quasi-compact and quasi-separated schemes with affine transition morphisms. Then $X$ has the resolution property if and only if $X_ i$ has the resolution properties for some $i$.

Proof. If $X_ i$ has the resolution property, then $X$ does by Lemma 36.36.4. Assume $X$ has the resolution property. Choose $i \in I$. Choose a finite affine open covering $X_ i = U_{i, 1} \cup \ldots \cup U_{i, m}$. For each $j$ choose a finite type quasi-coherent sheaf of ideals $\mathcal{I}_{i, j} \subset \mathcal{O}_{X_ i}$ such that $X_ i \setminus V(\mathcal{I}_{i, j}) = U_{i, j}$, see Properties, Lemma 28.24.1. Denote $U_ j \subset X$ the inverse image of $U_{i, j}$ and denote $\mathcal{I}_ j \subset \mathcal{O}_ X$ the pullback of $\mathcal{I}_{i, j}$. Since $X$ has the resolution property, we may choose finite locally free $\mathcal{O}_ X$-modules $\mathcal{E}_ j$ and surjections $\mathcal{E}_ j \to \mathcal{I}_ j$. By Limits, Lemmas 32.10.3 and 32.10.2 after increasing $i$ we can find finite locally free $\mathcal{O}_{X_ i}$-modules $\mathcal{E}_{i, j}$ and maps $\mathcal{E}_{i, j} \to \mathcal{O}_{X_ i}$ whose base changes to $X$ recover the compositions $\mathcal{E}_ j \to \mathcal{I}_ j \to \mathcal{O}_ X$, $j = 1, \ldots , m$. The pullbacks of the finitely presented $\mathcal{O}_{X_ i}$-modules $\mathop{\mathrm{Coker}}(\mathcal{E}_{i, j} \to \mathcal{O}_{X_ i})$ and $\mathcal{O}_{X_ i}/\mathcal{I}_{i, j}$ to $X$ agree as quotients of $\mathcal{O}_ X$. Hence by Limits, Lemma 32.10.2 we may assume that these agree, in other words that the image of $\mathcal{E}_{i, j} \to \mathcal{O}_{X_ i}$ is equal to $\mathcal{I}_{i, j}$. Then we conclude that $X_ i$ has the resolution property by Lemma 36.36.6. $\square$

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Your email address will not be published.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 0F8B. Beware of the difference between the letter 'O' and the digit '0'.