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. $\square$

Proof. To check $\mathcal{P}\! \mathit{olarized}\to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is quasi-separated we have to show that its diagonal is quasi-compact and quasi-separated. This is immediate from Lemma 108.11.1. To prove that $\mathcal{P}\! \mathit{olarized}\to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is locally of finite presentation, it suffices to show that $\mathcal{P}\! \mathit{olarized}$ is limit preserving, see Limits of Stacks, Proposition 102.3.8. This is Quot, Lemma 99.14.8. $\square$

Proof. Observe that $|\mathcal{P}\! \mathit{olarized}|$ is a locally Noetherian topological space, see Morphisms of Stacks, Lemma 101.8.3 (this also uses that $\mathop{\mathrm{Spec}}(\mathbf{Z})$ is Noetherian and hence $\mathcal{P}\! \mathit{olarized}$ is a locally Noetherian algebraic stack by Lemma 108.11.2 and Morphisms of Stacks, Lemma 101.17.5). Thus any quasi-compact subset of $|\mathcal{P}\! \mathit{olarized}|$ is a Noetherian topological space and any subset of such is also Noetherian, see Topology, Lemmas 5.9.4 and 5.9.2. Thus all we have to do is a find a quasi-compact subset containing $T$.

By Lemma 108.7.7 the algebraic space

is proper over $\mathop{\mathrm{Spec}}(\mathbf{Z})$. By Quot, Lemma 99.9.3¹ the identity morphism of $H$ corresponds to a closed subspace

which is proper, flat, and of finite presentation over $H$ and such that the restriction $\mathcal{N} = \mathcal{O}(1)|_ Z$ is relatively ample on $Z/H$ and has Hilbert polynomial $P$ on the fibres of $Z \to H$. In particular, the pair $(Z \to H, \mathcal{N})$ defines a morphism

which sends a morphism of schemes $U \to H$ to the classifying morphism of the family $(Z_ U \to U, \mathcal{N}_ U)$, see Quot, Lemma 99.14.4. Since $H$ is a Noetherian algebraic space (as it is proper over $\mathbf{Z})$) we see that $|H|$ is Noetherian and hence quasi-compact. The map

is continuous, hence the image is quasi-compact. Thus it suffices to prove $T$ is contained in the image of $|H| \to |\mathcal{P}\! \mathit{olarized}|$. However, assumptions (1) and (2) exactly express the fact that this is the case: any choice of a closed immersion $i : X \to \mathbf{P}^ n_ k$ with $i^*\mathcal{O}_{\mathbf{P}^ n}(1) \cong \mathcal{L}$ we get a $k$-valued point of $H$ by the moduli interpretation of $H$. This finishes the proof of the lemma. $\square$