The Stacks project

Lemma 114.11.3. Let $X$ be a Noetherian scheme. Let $\mathcal{P}$ be a property of coherent sheaves on $X$ such that

  1. For any short exact sequence of coherent sheaves if two out of three of them have property $\mathcal{P}$ then so does the third.

  2. If $\mathcal{P}$ holds for a direct sum of coherent sheaves then it holds for both.

  3. For every integral closed subscheme $Z \subset X$ with generic point $\xi $ there exists some coherent sheaf $\mathcal{G}$ such that

    1. $\text{Supp}(\mathcal{G}) = Z$,

    2. $\mathcal{G}_\xi $ is annihilated by $\mathfrak m_\xi $, and

    3. property $\mathcal{P}$ holds for $\mathcal{G}$.

Then property $\mathcal{P}$ holds for every coherent sheaf on $X$.

Proof. This follows from Lemma 114.11.2 in exactly the same way that Cohomology of Schemes, Lemma 30.12.6 follows from Cohomology of Schemes, Lemma 30.12.5. $\square$


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Your email address will not be published.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 01YK. Beware of the difference between the letter 'O' and the digit '0'.