The Stacks project

Proof. By Lemma 108.10.2 the morphism $\mathit{Mor}_ B(Y, X) \to B$ is separated and locally of finite presentation. Thus it suffices to show that the open and closed subspace $\mathit{Mor}^{P, \mathcal{M}}_ B(Y, X)$ of Remark 108.10.4 is quasi-compact over $B$.

The question is étale local on $B$ (Morphisms of Spaces, Lemma 67.8.8). Thus we may assume $B$ is affine.

Assume $B = \mathop{\mathrm{Spec}}(\Lambda )$. Note that $X$ and $Y$ are schemes and that $\mathcal{L}$ and $\mathcal{N}$ are ample invertible sheaves on $X$ and $Y$ (this follows immediately from the definitions). Write $\Lambda = \mathop{\mathrm{colim}}\nolimits \Lambda _ i$ as the colimit of its finite type $\mathbf{Z}$-subalgebras. Then we can find an $i$ and a system $X_ i, Y_ i, \mathcal{L}_ i, \mathcal{N}_ i$ as in the lemma over $B_ i = \mathop{\mathrm{Spec}}(\Lambda _ i)$ whose base change to $B$ gives $X, Y, \mathcal{L}, \mathcal{N}$. This follows from Limits, Lemmas 32.10.1 (to find $X_ i$, $Y_ i$), 32.10.3 (to find $\mathcal{L}_ i$, $\mathcal{N}_ i$), 32.8.6 (to make $X_ i \to B_ i$ separated), 32.13.1 (to make $Y_ i \to B_ i$ proper), and 32.4.15 (to make $\mathcal{L}_ i$, $\mathcal{N}_ i$ ample). Because

and similarly for $\mathit{Mor}^ P_ B(Y, X)$ we reduce to the case discussed in the next paragraph.

Assume $B$ is a Noetherian affine scheme. By Properties, Lemma 28.26.15 we see that $\mathcal{M}$ is ample. By Lemma 108.7.8 we see that $\mathrm{Hilb}^{P, \mathcal{M}}_{Y \times _ B X/B}$ is of finite presentation over $B$ and hence Noetherian. By construction

is an open subspace of $\mathrm{Hilb}^{P, \mathcal{M}}_{Y \times _ B X/B}$ and hence quasi-compact (as an open of a Noetherian algebraic space is quasi-compact). $\square$