The Stacks project

Lemma 76.9.2. In Situation 76.7.1. Assume

  1. $f$ is locally of finite presentation,

  2. $\mathcal{F}$ is locally of finite presentation and flat over $B$,

  3. the support of $\mathcal{F}$ is proper over $B$, and

  4. $u$ is surjective.

Then the functor $F_{iso}$ is an algebraic space and $F_{iso} \to B$ is a closed immersion. If $\mathcal{G}$ is of finite presentation, then $F_{iso} \to B$ is of finite presentation.

Proof. Let $\mathcal{K} = \mathop{\mathrm{Ker}}(u)$ and denote $v : \mathcal{K} \to \mathcal{F}$ the inclusion. By Lemma 76.7.5 we see that $F_{u, iso} = F_{v, zero}$. By Lemma 76.8.6 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. $\square$

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