The Stacks project

Lemma 74.8.6. In Situation 74.7.1. Assume

  1. $f$ is locally of finite presentation,

  2. $\mathcal{G}$ is an $\mathcal{O}_ X$-module of finite presentation flat over $B$,

  3. the support of $\mathcal{G}$ is proper over $B$.

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

Proof. If $f$ is of finite presentation, then this follows immediately from Lemmas 74.8.5 and 74.3.6. This is the only case of interest and we urge the reader to skip the rest of the proof, which deals with the possibility (allowed by the assumptions in this lemma) that $f$ is not quasi-separated or quasi-compact.

Let $i : Z \to X$ be the closed subspace cut out by the zeroth fitting ideal of $\mathcal{G}$ (Divisors on Spaces, Section 68.5). Then $Z \to B$ is proper by assumption (see Derived Categories of Spaces, Section 72.7). On the other hand $i$ is of finite presentation (Divisors on Spaces, Lemma 68.5.2 and Morphisms of Spaces, Lemma 64.28.12). There exists a quasi-coherent $\mathcal{O}_ Z$-module $\mathcal{H}$ of finite type with $i_*\mathcal{H} = \mathcal{G}$ (Divisors on Spaces, Lemma 68.5.3). In fact $\mathcal{H}$ is of finite presentation as an $\mathcal{O}_ Z$-module by Algebra, Lemma 10.6.4 (details omitted). Then $F_{zero}$ is the same as the functor $F_{zero}$ for the map $i^*\mathcal{F} \to \mathcal{H}$ adjoint to $u$, see Lemma 74.7.6. The sheaf $\mathcal{H}$ is flat relative to $B$ because the same is true for $\mathcal{G}$ (check on stalks; details omitted). Moreover, note that if $\mathcal{F}$ is of finite type, then $i^*\mathcal{F}$ is of finite type. Hence we have reduced the lemma to the case discussed in the first paragraph of the proof. $\square$


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