Lemma 76.9.2. In Situation 76.7.1. Assume

$f$ is locally of finite presentation,

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

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

$u$ 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.
$\square$

## Comments (0)