Lemma 108.4.1 (0DLY)—The Stacks project

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Table of contents
Part 8: Topics in Moduli Theory
Chapter 108: Moduli Stacks
Section 108.4: Properties of the stack of coherent sheaves
Lemma 108.4.1 (cite)

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Lemma 108.4.1. The diagonal of $\mathcal{C}\! \mathit{oh}_{X/B}$ over $B$ is affine and of finite presentation.

Proof. The representability of the diagonal by algebraic spaces was shown in Quot, Lemma 99.5.3. From the proof we find that we have to show $\mathit{Isom}(\mathcal{F}, \mathcal{G}) \to T$ is affine and of finite presentation for a pair of finitely presented $\mathcal{O}_{X_ T}$ -modules $\mathcal{F}$ , $\mathcal{G}$ flat over $T$ with support proper over $T$ . This was discussed in Section 108.3. $\square$

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