The Stacks project

Lemma 107.6.1. Let $n \geq 0$, $r \geq 1$, $P \in \mathbf{Q}[t]$. The algebraic space

\[ X = \mathrm{Quot}^ P_{\mathcal{O}^{\oplus r}_{\mathbf{P}^ n_\mathbf {Z}}/ \mathbf{P}^ n_\mathbf {Z}/\mathbf{Z}} \]

parametrizing quotients of $\mathcal{O}_{\mathbf{P}^ n_\mathbf {Z}}^{\oplus r}$ with Hilbert polynomial $P$ is proper over $\mathop{\mathrm{Spec}}(\mathbf{Z})$.

Proof. We already know that $X \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is separated and locally of finite presentation (Lemma 107.5.2). We also know that $X \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ satisfies the existence part of the valuative criterion, see Lemma 107.5.3. By the valuative criterion for properness, it suffices to prove our Quot space is quasi-compact, see Morphisms of Spaces, Lemma 66.44.1. Thus it suffices to find a quasi-compact scheme $T$ and a surjective morphism $T \to X$. Let $m$ be the integer found in Varieties, Lemma 33.34.18. Let

\[ N = r{m + n \choose n} - P(m) \]

We will write $\mathbf{P}^ n$ for $\mathbf{P}^ n_\mathbf {Z} = \text{Proj}(\mathbf{Z}[T_0, \ldots , T_ n])$ and unadorned products will mean products over $\mathop{\mathrm{Spec}}(\mathbf{Z})$. The idea of the proof is to construct a “universal” map

\[ \Psi : \mathcal{O}_{T \times \mathbf{P}^ n}(-m)^{\oplus N} \longrightarrow \mathcal{O}_{T \times \mathbf{P}^ n}^{\oplus r} \]

over an affine scheme $T$ and show that every point of $X$ corresponds to a cokernel of this in some point of $T$.

Definition of $T$ and $\Psi $. We take $T = \mathop{\mathrm{Spec}}(A)$ where

\[ A = \mathbf{Z}[a_{i, j, E}] \]

where $i \in \{ 1, \ldots , r\} $, $j \in \{ 1, \ldots , N\} $ and $E = (e_0, \ldots , e_ n)$ runs through the multi-indices of total degree $|E| = \sum _{k = 0, \ldots n} e_ k = m$. Then we define $\Psi $ to be the map whose $(i, j)$ matrix entry is the map

\[ \sum \nolimits _{E = (e_0, \ldots , e_ n)} a_{i, j, E} T_0^{e_0} \ldots T_ n^{e_ n} : \mathcal{O}_{T \times \mathbf{P}^ n}(-m) \longrightarrow \mathcal{O}_{T \times \mathbf{P}^ n} \]

where the sum is over $E$ as above (but $i$ and $j$ are fixed of course).

Consider the quotient $\mathcal{Q} = \mathop{\mathrm{Coker}}(\Psi )$ on $T \times \mathbf{P}^ n$. By More on Flatness, Lemma 38.21.4 there exists a $t \geq 0$ and closed subschemes

\[ T = T_0 \supset T_1 \supset \ldots \supset T_ t = \emptyset \]

such that the pullback $\mathcal{Q}_ p$ of $\mathcal{Q}$ to $(T_ p \setminus T_{p + 1}) \times \mathbf{P}^ n$ is flat over $T_ p \setminus T_{p + 1}$. Observe that we have an exact sequence

\[ \mathcal{O}_{(T_ p \setminus T_{p + 1}) \times \mathbf{P}^ n}(-m)^{\oplus N} \to \mathcal{O}_{(T_ p \setminus T_{p + 1}) \times \mathbf{P}^ n}^{\oplus r} \to \mathcal{Q}_ p \to 0 \]

by pulling back the exact sequence defining $\mathcal{Q} = \mathop{\mathrm{Coker}}(\Psi )$. Therefore we obtain a morphism

\[ \coprod (T_ p \setminus T_{p + 1}) \longrightarrow \mathrm{Quot}_{\mathcal{O}^{\oplus r}/\mathbf{P}/\mathbf{Z}} \supset \mathrm{Quot}^ P_{\mathcal{O}^{\oplus r}/\mathbf{P}/\mathbf{Z}} = X \]

Since the left hand side is a Noetherian scheme and the inclusion on the right hand side is open, it suffices to show that any point of $X$ is in the image of this morphism.

Let $k$ be a field and let $x \in X(k)$. Then $x$ corresponds to a surjection $\mathcal{O}_{\mathbf{P}^ n_ k}^{\oplus r} \to \mathcal{F}$ of coherent $\mathcal{O}_{\mathbf{P}^ n_ k}$-modules such that the Hilbert polynomial of $\mathcal{F}$ is $P$. Consider the short exact sequence

\[ 0 \to \mathcal{K} \to \mathcal{O}_{\mathbf{P}^ n_ k}^{\oplus r} \to \mathcal{F} \to 0 \]

By Varieties, Lemma 33.34.18 and our choice of $m$ we see that $\mathcal{K}$ is $m$-regular. By Varieties, Lemma 33.34.12 we see that $\mathcal{K}(m)$ is globally generated. By Varieties, Lemma 33.34.10 and the definition of $m$-regularity we see that $H^ i(\mathbf{P}^ n_ k, \mathcal{K}(m)) = 0$ for $i > 0$. Hence we see that

\[ \dim _ k H^0(\mathbf{P}^ n_ k, \mathcal{K}(m)) = \chi (\mathcal{K}(m)) = \chi (\mathcal{O}_{\mathbf{P}^ n_ k}(m)^{\oplus r}) - \chi (\mathcal{F}(m)) = N \]

by our choice of $N$. This gives a surjection

\[ \mathcal{O}_{\mathbf{P}^ n_ k}^{\oplus N} \longrightarrow \mathcal{K}(m) \]

Twisting back down and using the short exact sequence above we see that $\mathcal{F}$ is the cokernel of a map

\[ \Psi _ x : \mathcal{O}_{\mathbf{P}^ n_ k}(-m)^{\oplus N} \to \mathcal{O}_{\mathbf{P}^ n_ k}^{\oplus r} \]

There is a unique ring map $\tau : A \to k$ such that the base change of $\Psi $ by the corresponding morphism $t = \mathop{\mathrm{Spec}}(\tau ) : \mathop{\mathrm{Spec}}(k) \to T$ is $\Psi _ x$. This is true because the entries of the $N \times r$ matrix defining $\Psi _ x$ are homogeneous polynomials $\sum \lambda _{i, j, E} T_0^{e_0} \ldots T_ n^{e_ n}$ of degree $m$ in $T_0, \ldots , T_ n$ with coefficients $\lambda _{i, j, E} \in k$ and we can set $\tau (a_{i, j, E}) = \lambda _{i, j, E}$. Then $t \in T_ p \setminus T_{p + 1}$ for some $p$ and the image of $t$ under the morphism above is $x$ as desired. $\square$


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