Theorem 77.9.1 (0CWD)—The Stacks project

Processing math: 100%

Theorem 77.9.1. In Situation 77.7.1 assume

$f$ is of finite presentation,
$\mathcal{F}$ is of finite presentation, flat over $B$ , and pure relative to $B$ , and
$u$ is surjective.

Then $F_{iso}$ is representable by a closed immersion $Z \to B$ . Moreover $Z \to S$ is of finite presentation if $\mathcal{G}$ is of finite presentation.

Proof. Let $\mathcal{K} = \mathop{\mathrm{Ker}}(u)$ and denote $v : \mathcal{K} \to \mathcal{F}$ the inclusion. By Lemma 77.7.5 we see that $F_{u, iso} = F_{v, zero}$ . By Lemma 77.8.5 applied to $v$ we see that $F_{u, iso} = F_{v, zero}$ is representable by a closed subspace of $B$ . Note that $\mathcal{K}$ is of finite type if $\mathcal{G}$ is of finite presentation, see Modules on Sites, Lemma 18.24.1. Hence we also obtain the final statement of the lemma. $\square$

Comments (0)

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).

Comments (0)

Post a comment