The Stacks project

Proof. Recall that $\mathcal{P}\! \mathit{olarized}$ is a limit preserving algebraic stack, see Quot, Lemma 99.14.8. By Limits of Stacks, Lemma 102.3.6 this implies that $\Delta : \mathcal{P}\! \mathit{olarized}\to \mathcal{P}\! \mathit{olarized}\times \mathcal{P}\! \mathit{olarized}$ is limit preserving. Hence $\Delta$ is locally of finite presentation by Limits of Stacks, Proposition 102.3.8.

Let us prove that $\Delta$ is separated. To see this, it suffices to show that given an affine scheme $U$ and two objects $\upsilon = (Y, \mathcal{N})$ and $\chi = (X, \mathcal{L})$ of $\mathcal{P}\! \mathit{olarized}$ over $U$, the algebraic space

is separated. The rule which to an isomorphism $\upsilon _ T \to \chi _ T$ assigns the underlying isomorphism $Y_ T \to X_ T$ defines a morphism

Since we have seen in Lemmas 108.10.2 and 108.10.3 that the target is a separated algebraic space, it suffices to prove that this morphism is separated. Given an isomorphism $f : Y_ T \to X_ T$ over some scheme $T/U$, then clearly

Here $[f] : T \to \mathit{Isom}_ U(Y, X)$ indicates the $T$-valued point corresponding to $f$ and $\mathit{Isom}(\mathcal{N}_ T, f^*\mathcal{L}_ T)$ is the algebraic space discussed in Section 108.3. Since this algebraic space is affine over $U$, the claim implies $\Delta$ is separated.

To finish the proof we show that $\Delta$ is quasi-compact. Since $\Delta$ is representable by algebraic spaces, it suffice to check the base change of $\Delta$ by a surjective smooth morphism $U \to \mathcal{P}\! \mathit{olarized}\times \mathcal{P}\! \mathit{olarized}$ is quasi-compact (see for example Properties of Stacks, Lemma 100.3.3). We can assume $U = \coprod U_ i$ is a disjoint union of affine opens. Since $\mathcal{P}\! \mathit{olarized}$ is limit preserving (see above), we see that $\mathcal{P}\! \mathit{olarized}\to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is locally of finite presentation, hence $U_ i \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is locally of finite presentation (Limits of Stacks, Proposition 102.3.8 and Morphisms of Stacks, Lemmas 101.27.2 and 101.33.5). In particular, $U_ i$ is Noetherian affine. This reduces us to the case discussed in the next paragraph.

In this paragraph, given a Noetherian affine scheme $U$ and two objects $\upsilon = (Y, \mathcal{N})$ and $\chi = (X, \mathcal{L})$ of $\mathcal{P}\! \mathit{olarized}$ over $U$, we show the algebraic space

is quasi-compact. Since the connected components of $U$ are open and closed we may replace $U$ by these. Thus we may and do assume $U$ is connected. Let $u \in U$ be a point. Let $P$ be the Hilbert polynomial $n \mapsto \chi (Y_ u, \mathcal{N}_ u^{\otimes n})$, see Varieties, Lemma 33.45.1. Since $U$ is connected and since the functions $u \mapsto \chi (Y_ u, \mathcal{N}_ u^{\otimes n})$ are locally constant (see Derived Categories of Schemes, Lemma 36.32.2) we see that we get the same Hilbert polynomial in every point of $U$. Set $\mathcal{M} = \text{pr}_1^*\mathcal{N} \otimes _{\mathcal{O}_{Y \times _ U X}} \text{pr}_2^*\mathcal{L}$ on $Y \times _ U X$. Given $(f, \varphi ) \in \mathit{Isom}_{\mathcal{P}\! \mathit{olarized}}(\upsilon , \chi )(T)$ for some scheme $T$ over $U$ then for every $t \in T$ we have

where in the middle equality we use the isomorphism $\varphi : f^*\mathcal{L}_ T \to \mathcal{N}_ T$. Setting $P'(t) = P(2t)$ we find that the morphism

(see earlier) has image contained in the intersection

The intersection is an intersection of open subspaces of $\mathit{Mor}_ U(Y, X)$ (see Lemma 108.10.3 and Remark 108.10.4). Now $\mathit{Mor}^{P', \mathcal{M}}_ U(Y, X)$ is a Noetherian algebraic space as it is of finite presentation over $U$ by Lemma 108.10.5. Thus the intersection is a Noetherian algebraic space too. Since the morphism

is affine (see above) we conclude. $\square$