The Stacks project

108.3 Properties of Hom and Isom

Let $f : X \to B$ be a morphism of algebraic spaces which is of finite presentation. Assume $\mathcal{F}$ and $\mathcal{G}$ are quasi-coherent $\mathcal{O}_ X$-modules. If $\mathcal{G}$ is of finite presentation, flat over $B$ with support proper over $B$, then the functor $\mathit{Hom}(\mathcal{F}, \mathcal{G})$ defined by

is an algebraic space affine over $B$. If $\mathcal{F}$ is of finite presentation, then $\mathit{Hom}(\mathcal{F}, \mathcal{G}) \to B$ is of finite presentation. See Quot, Proposition 99.3.10.

If both $\mathcal{F}$ and $\mathcal{G}$ are of finite presentation, flat over $B$ with support proper over $B$, then the subfunctor

is an algebraic space affine of finite presentation over $B$. See Quot, Proposition 99.4.3.