The Stacks project

Proposition 98.4.3. In Situation 98.3.1 assume that

  1. $f$ is of finite presentation, and

  2. $\mathcal{F}$ and $\mathcal{G}$ are finitely presented $\mathcal{O}_ X$-modules, flat over $B$, with support proper over $B$.

Then the functor $\mathit{Isom}(\mathcal{F}, \mathcal{G})$ is an algebraic space affine of finite presentation over $B$.

Proof. We will use the abbreviations $H = \mathit{Hom}(\mathcal{F}, \mathcal{G})$, $I = \mathit{Hom}(\mathcal{F}, \mathcal{F})$, $H' = \mathit{Hom}(\mathcal{G}, \mathcal{F})$, and $I' = \mathit{Hom}(\mathcal{G}, \mathcal{G})$. By Proposition 98.3.10 the functors $H$, $I$, $H'$, $I'$ are algebraic spaces and the morphisms $H \to B$, $I \to B$, $H' \to B$, and $I' \to B$ are affine and of finite presentation. The composition of maps gives a morphism

\[ c : H' \times _ B H \longrightarrow I \times _ B I',\quad (u', u) \longmapsto (u \circ u', u' \circ u) \]

of algebraic spaces over $B$. Since $I \times _ B I' \to B$ is separated, the section $\sigma : B \to I \times _ B I'$ corresponding to $(\text{id}_\mathcal {F}, \text{id}_\mathcal {G})$ is a closed immersion (Morphisms of Spaces, Lemma 66.4.7). Moreover, $\sigma $ is of finite presentation (Morphisms of Spaces, Lemma 66.28.9). Hence

\[ \mathit{Isom}(\mathcal{F}, \mathcal{G}) = (H' \times _ B H) \times _{c, I \times _ B I', \sigma } B \]

is an algebraic space affine of finite presentation over $B$ as well. Some details omitted. $\square$


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