The Stacks project

Lemma 108.4.1. The diagonal of $\mathcal{C}\! \mathit{oh}_{X/B}$ over $B$ is affine and of finite presentation.

Proof. The representability of the diagonal by algebraic spaces was shown in Quot, Lemma 99.5.3. From the proof we find that we have to show $\mathit{Isom}(\mathcal{F}, \mathcal{G}) \to T$ is affine and of finite presentation for a pair of finitely presented $\mathcal{O}_{X_ T}$-modules $\mathcal{F}$, $\mathcal{G}$ flat over $T$ with support proper over $T$. This was discussed in Section 108.3. $\square$

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Your email address will not be published.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



View Lemma 108.4.1 as pdf