The Stacks project

107.6 Boundedness for Quot

Contrary to what happens classically, we already know the Quot functor is an algebraic space, but we don't know that it is ever represented by a finite type algebraic space.

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 33.35.18. 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 33.35.18 and our choice of $m$ we see that $\mathcal{K}$ is $m$-regular. By Varieties, Lemma 33.35.12 we see that $\mathcal{K}(m)$ is globally generated. By Varieties, Lemma 33.35.10 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. $\square$

Lemma 107.6.2. Let $B$ be an algebraic space. Let $X = B \times \mathbf{P}^ n_\mathbf {Z}$. Let $\mathcal{L}$ be the pullback of $\mathcal{O}_{\mathbf{P}^ n}(1)$ to $X$. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module of finite presentation. The algebraic space $\mathrm{Quot}^ P_{\mathcal{F}/X/B}$ parametrizing quotients of $\mathcal{F}$ having Hilbert polynomial $P$ with respect to $\mathcal{L}$ is proper over $B$.

Proof. The question is étale local over $B$, see Morphisms of Spaces, Lemma 66.40.2. Thus we may assume $B$ is an affine scheme. In this case $\mathcal{L}$ is an ample invertible module on $X$ (by Constructions, Lemma 27.10.6 and the definition of ample invertible modules in Properties, Definition 28.26.1). Thus we can find $r' \geq 0$ and $r \geq 0$ and a surjection

\[ \mathcal{O}_ X^{\oplus r} \longrightarrow \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes r'} \]

by Properties, Proposition 28.26.13. By Lemma 107.5.10 we may replace $\mathcal{F}$ by $\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes r'}$ and $P(t)$ by $P(t + r')$. By Lemma 107.5.8 we obtain a closed immersion

\[ \mathrm{Quot}^ P_{\mathcal{F}/X/B} \longrightarrow \mathrm{Quot}^ P_{\mathcal{O}_ X^{\oplus r}/X/B} \]

Since we've shown that $\mathrm{Quot}^ P_{\mathcal{O}_ X^{\oplus r}/X/B} \to B$ is proper in Lemma 107.6.1 we conclude. $\square$

Lemma 107.6.3. Let $f : X \to B$ be a proper morphism of finite presentation of algebraic spaces. Let $\mathcal{F}$ be a finitely presented $\mathcal{O}_ X$-module. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module ample on $X/B$, see Divisors on Spaces, Definition 70.14.1. The algebraic space $\mathrm{Quot}^ P_{\mathcal{F}/X/B}$ parametrizing quotients of $\mathcal{F}$ having Hilbert polynomial $P$ with respect to $\mathcal{L}$ is proper over $B$.

Proof. The question is étale local over $B$, see Morphisms of Spaces, Lemma 66.40.2. Thus we may assume $B$ is an affine scheme. Then we can find a closed immersion $i : X \to \mathbf{P}^ n_ B$ such that $i^*\mathcal{O}_{\mathbf{P}^ n_ B}(1) \cong \mathcal{L}^{\otimes d}$ for some $d \geq 1$. See Morphisms, Lemma 29.39.3. Changing $\mathcal{L}$ into $\mathcal{L}^{\otimes d}$ and the numerical polynomial $P(t)$ into $P(dt)$ leaves $\mathrm{Quot}^ P_{\mathcal{F}/X/B}$ unaffected; some details omitted. Hence we may assume $\mathcal{L} = i^*\mathcal{O}_{\mathbf{P}^ n_ B}(1)$. Then the isomorphism $\mathrm{Quot}_{\mathcal{F}/X/B} \to \mathrm{Quot}_{i_*\mathcal{F}/\mathbf{P}^ n_ B/B}$ of Lemma 107.5.7 induces an isomorphism $\mathrm{Quot}^ P_{\mathcal{F}/X/B} \cong \mathrm{Quot}^ P_{i_*\mathcal{F}/\mathbf{P}^ n_ B/B}$. Since $\mathrm{Quot}^ P_{i_*\mathcal{F}/\mathbf{P}^ n_ B/B}$ is proper over $B$ by Lemma 107.6.2 we conclude. $\square$

Lemma 107.6.4. Let $f : X \to B$ be a separated morphism of finite presentation of algebraic spaces. Let $\mathcal{F}$ be a finitely presented $\mathcal{O}_ X$-module. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module ample on $X/B$, see Divisors on Spaces, Definition 70.14.1. The algebraic space $\mathrm{Quot}^ P_{\mathcal{F}/X/B}$ parametrizing quotients of $\mathcal{F}$ having Hilbert polynomial $P$ with respect to $\mathcal{L}$ is separated of finite presentation over $B$.

Proof. We have already seen that $\mathrm{Quot}_{\mathcal{F}/X/B} \to B$ is separated and locally of finite presentation, see Lemma 107.5.2. Thus it suffices to show that the open subspace $\mathrm{Quot}^ P_{\mathcal{F}/X/B}$ of Remark 107.5.9 is quasi-compact over $B$.

The question is étale local on $B$ (Morphisms of Spaces, Lemma 66.8.8). Thus we may assume $B$ is affine.

Assume $B = \mathop{\mathrm{Spec}}(\Lambda )$. 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 $X_ i, \mathcal{F}_ i, \mathcal{L}_ i$ as in the lemma over $B_ i = \mathop{\mathrm{Spec}}(\Lambda _ i)$ whose base change to $B$ gives $X, \mathcal{F}, \mathcal{L}$. This follows from Limits of Spaces, Lemmas 69.7.1 (to find $X_ i$), 69.7.2 (to find $\mathcal{F}_ i$), 69.7.3 (to find $\mathcal{L}_ i$), and 69.5.9 (to make $X_ i$ separated). Because

\[ \mathrm{Quot}_{\mathcal{F}/X/B} = B \times _{B_ i} \mathrm{Quot}_{\mathcal{F}_ i/X_ i/B_ i} \]

and similarly for $\mathrm{Quot}^ P_{\mathcal{F}/X/B}$ we reduce to the case discussed in the next paragraph.

Assume $B$ is affine and Noetherian. We may replace $\mathcal{L}$ by a positive power, see Lemma 107.5.11. Thus we may assume there exists an immersion $i : X \to \mathbf{P}^ n_ B$ such that $i^*\mathcal{O}_{\mathbf{P}^ n}(1) = \mathcal{L}$. By Morphisms, Lemma 29.7.7 there exists a closed subscheme $X' \subset \mathbf{P}^ n_ B$ such that $i$ factors through an open immersion $j : X \to X'$. By Properties, Lemma 28.22.5 there exists a finitely presented $\mathcal{O}_{X'}$-module $\mathcal{G}$ such that $j^*\mathcal{G} = \mathcal{F}$. Thus we obtain an open immersion

\[ \mathrm{Quot}_{\mathcal{F}/X/B} \longrightarrow \mathrm{Quot}_{\mathcal{G}/X'/B} \]

by Lemma 107.5.6. Clearly this open immersion sends $\mathrm{Quot}^ P_{\mathcal{F}/X/B}$ into $\mathrm{Quot}^ P_{\mathcal{G}/X'/B}$. Now $\mathrm{Quot}^ P_{\mathcal{G}/X'/B}$ is proper over $B$ by Lemma 107.6.3. Therefore it is Noetherian and since any open of a Noetherian algebraic space is quasi-compact we win. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0DP9. Beware of the difference between the letter 'O' and the digit '0'.