The Stacks project

Lemma 76.9.3. In Situation 76.7.1. Assume

  1. $f$ is locally of finite presentation,

  2. $\mathcal{G}$ is of finite type,

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

Then $F_{surj}$ is an algebraic space and $F_{surj} \to B$ is an open immersion.

Proof. Consider $\mathop{\mathrm{Coker}}(u)$. Observe that $\mathop{\mathrm{Coker}}(u_ T) = \mathop{\mathrm{Coker}}(u)_ T$ for any $T/B$. Note that formation of the support of a finite type quasi-coherent module commutes with pullback (Morphisms of Spaces, Lemma 66.15.1). Hence $F_{surj}$ is representable by the open subspace of $B$ corresponding to the open set

\[ |B| \setminus |f|(\text{Supp}(\mathop{\mathrm{Coker}}(u))) \]

see Properties of Spaces, Lemma 65.4.8. This is an open because $|f|$ is closed on $\text{Supp}(\mathcal{G})$ and $\text{Supp}(\mathop{\mathrm{Coker}}(u))$ is a closed subset of $\text{Supp}(\mathcal{G})$. $\square$


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