Semistable reduction for multi-filtered vector spaces

TL;DR

This paper establishes a semistable reduction theorem specifically for multi-filtered vector spaces, extending the understanding of their structural properties in algebraic geometry.

Contribution

It introduces a semistable reduction theorem tailored for multi-filtered vector spaces, a novel result in the study of multi-weighted vector spaces.

Findings

Proves a semistable reduction theorem for multi-filtered vector spaces

Extends classical semistable reduction concepts to multi-filtered contexts

Provides foundational results for future research in multi-weighted vector spaces

Abstract

In this paper, we prove a semistable reduction type theorem for multi-filtered vector spaces (or known as multi-weighted vector spaces).

Equations

$$V={\mathrm{Fil}}_i^0\supseteq {\mathrm{Fil}}_i^1\supseteq \cdots \supseteq {\mathrm{Fil}}_i^{l_i}=0.$$

$$w(V,{\mathrm{Fil}}_i^{\bullet })=\sum _i\sum _{j}j\cdot \dim ({\mathrm{Fil}}_i^j/{\mathrm{Fil}}_i^{j+1}).$$

$$\mu (V,{\mathrm{Fil}}_i^{\bullet })=\frac{w(V,{\mathrm{Fil}}_i^{\bullet })}{\dim (V)}.$$

$$0\to \overline{\mathrm{B}}\to \overline{\mathrm{M}}\to \overline{\mathrm{G}}\to 0$$

$$0\to \tilde{\mathrm{B}}\to \tilde{\mathrm{M}}=\mathrm{M}/(\pi \mathrm{M})\to \tilde{\mathrm{G}}\to 0$$

$${\mathrm{Hom}}_{\mathrm{Fi}{\mathrm{l}}_1^{\bullet },\dots ,\mathrm{Fi}{\mathrm{l}}_{\mathrm{n}}^{\bullet }}(\tilde{\mathrm{B}},\tilde{\mathrm{G}})={\mathrm{Hom}}_{\mathrm{Fi}{\mathrm{l}}_1^{\bullet },\dots ,\mathrm{Fi}{\mathrm{l}}_{\mathrm{n}-1}^{\bullet }}(\tilde{\mathrm{B}},\tilde{\mathrm{G}})\cap {\mathrm{Hom}}_{\mathrm{Fi}{\mathrm{l}}_{\mathrm{n}}^{\bullet }}(\tilde{\mathrm{B}},\tilde{\mathrm{G}})$$

$$H/(H_1\cap \mathfrak{m}H+\mathfrak{m}{H}_2)=⨁\mathcal{O}/{\pi }_j\oplus ⨁\mathcal{O}/{\pi }_i^{\prime }\mathfrak{m}⨁{(\mathcal{O}/\mathfrak{m})}^{\oplus s},$$

$$0\to \frac{\pi \mathrm{M}}{\pi {\mathrm{M}}^{\prime }}=\frac{\mathrm{M}}{{\mathrm{M}}^{\prime }}=\tilde{\mathrm{G}}\to \frac{{\mathrm{M}}^{\prime }}{\pi {\mathrm{M}}^{\prime }}\to \frac{{\mathrm{M}}^{\prime }}{\pi \mathrm{M}}=\tilde{\mathrm{B}}\to 0$$

$$0\to \overline{\mathrm{G}}\to \overline{{\mathrm{M}}^{\prime }}\to \overline{\mathrm{B}}\to 0$$

$$\mu (\overline{B^{\prime }})\le \frac{w(\overline{B^{\prime }}\cap \overline{G})+w(\mathrm{Im}(\overline{B^{\prime }}))}{\dim (\overline{B^{\prime }})}\le \frac{\dim (\overline{B^{\prime }}\cap \overline{G})\times \mu (\overline{B})+\dim (\mathrm{Im}(\overline{B^{\prime }}))\times \mu (\overline{B})}{\dim (\overline{B^{\prime }})}=\mu (\overline{B})$$

Full text

Semistable reduction for multi-filtered vector spaces

Shizhang Li

1. Introduction

In this paper, $\mathcal{O}$ will be a valuation ring. We denote $\operatorname{Frac}(\mathcal{O})$ by $K$ and $\mathcal{O}/\mathfrak{m}$ by $k$. For an $\mathcal{O}$ module $\mathrm{M}$, we denote $\mathrm{M}_{K}=\mathrm{M}\otimes_{\mathcal{O}}K$ and $\overline{\mathrm{M}}=\mathrm{M}\otimes_{\mathcal{O}}k$. Also for an $\mathcal{O}/\pi$ module $\widetilde{\mathrm{M}}$ we still denote $\overline{\mathrm{M}}=\widetilde{\mathrm{M}}\otimes_{\mathcal{O}/\pi}k$ for some $\pi\in\mathfrak{m}$. Unless stated otherwise, all the modules considered in this paper will be finitely presented over corresponding ring.

Definition 1.1**.**

A multi-filtered vector space $(V,\operatorname{Fil}^{\bullet}_{i})$ over a field $F$ is a finite dimensional $F$-vector space $V$ together with finitely many filtrations $\operatorname{Fil}^{\bullet}_{i},1\leq i\leq n$, such that for all $i$ the filtration $\operatorname{Fil}^{\bullet}_{i}$ is

1. (1)

decreasing, 2. (2)

indexed by natural numbers, 3. (3)

exhaustive and 4. (4)

separated:

[TABLE]

For any subspace (resp. quotient) $W$ of $V$, we give the obvious induced multi-filtration on them.

Definition 1.2**.**

• •

The weight of a multi-filtered vector space is

[TABLE]

• •

The slope of a multi-filtered vector space is defined as

[TABLE]

• •

A multi-filtered vector space $(V,\operatorname{Fil}^{\bullet}_{i})$ is called semistable if for any nonzero subspace $W\subseteq V$, we have $\mu(W)\leq\mu(V)$.

If $\mathrm{M}$ is an $\mathcal{O}$-lattice of a multi-filtered $K$-vector space $(\mathrm{M}\otimes_{\mathcal{O}}K,\operatorname{Fil}^{\bullet}_{i})$, we will still denote the induced filtration $\operatorname{Fil}^{\bullet}_{i}\cap\mathrm{M}$ on $\mathrm{M}$ by $\operatorname{Fil}^{\bullet}_{i}$. And we use the notation $\overline{\operatorname{Fil}^{\bullet}_{i}}$ (resp. $\widetilde{\operatorname{Fil}^{\bullet}_{i}}$) to denote the induced filtration on $\overline{\mathrm{M}}$ (resp. $\widetilde{\mathrm{M}}=\mathrm{M}\otimes_{\mathcal{O}}\mathcal{O}/\pi$ for some $\pi\in\mathfrak{m}$). We call such a $\mathrm{M}$ an integral model of $(\mathrm{M}\otimes_{\mathcal{O}}K,\operatorname{Fil}^{\bullet}_{i})$.

The main theorem of this paper is the following analogue of Langton’s theorem in the setting of multi-filtered vector spaces.

Theorem 1.3** (Main Theorem).**

Let $K$ be a valued field. For any semistable multi-filtered vector space $(V,\mathrm{Fil}^{\bullet}_{i})$, there exits an integral model $(\mathrm{M},\mathrm{Fil}^{\bullet}_{i})$ such that its reduction $\overline{\mathrm{M}}$ with its induced filtration is again semistable.

2. Proof of the Main Theorem

For technical reason, let us make the following definition.

Definition 2.1**.**

A submodule of $\mathcal{O}^{n}$ (resp. ${({\mathcal{O}/\pi})}^{n}$) is said to be saturated if the quotient is flat over $\mathcal{O}$ (resp. $\mathcal{O}/\pi$).

Definition 2.2**.**

Let $\mathrm{M}$ be an $\mathcal{O}$-lattice of a multi-filtered $K$-vector space $(\mathrm{M}\otimes_{\mathcal{O}}K,\operatorname{Fil}^{\bullet}_{i})$ with its induced filtrations. Given a short exact sequence of $k$-vector spaces

[TABLE]

we say it is liftable modulo $\pi\in\mathfrak{m}$ if there exists a short exact sequence of $\mathcal{O}/\pi$-modules.

[TABLE]

such that

1. (1)

$\widetilde{\mathrm{G}}$ is flat over $\mathcal{O}/\pi$ 2. (2)

this sequence reduces to the above sequence and 3. (3)

$\widetilde{\operatorname{Fil}^{j}_{i}}\cap\widetilde{\mathrm{B}}$ surjects onto $\overline{\operatorname{Fil}^{j}_{i}}\cap\overline{\mathrm{B}}$.

Lemma 2.3**.**

If there is only one filtration $\mathrm{F}^{\bullet}$ on $\mathrm{M}$, then any short exact sequence $0\to\overline{\mathrm{B}}\to\overline{\mathrm{M}}\to\overline{\mathrm{G}}\to 0$ is liftable, i.e., there exists short exact sequence of $\mathcal{O}$-modules $0\to\mathrm{B}\to\mathrm{M}\to\mathrm{G}\to 0$ such that

1. (1)

$\mathrm{G}$* is flat over $\mathcal{O}$* 2. (2)

this sequence reduces to the above sequence 3. (3)

$\mathrm{F}^{\bullet}\cap\mathrm{B}$* surjects onto $\overline{\mathrm{F}^{\bullet}}\cap\overline{\mathrm{B}}$.*

Moreover, fix one lifting and splitting $\mathrm{M}=\mathrm{B}\oplus\mathrm{G}$, then all different liftings are graphs of morphisms in $\mathfrak{m}\mathrm{Hom}_{\mathrm{F}^{\bullet}}(\mathrm{B},\mathrm{G})$.

Proof.

Choose a basis of $\overline{\mathrm{F}^{i}}\cap\overline{\mathrm{B}}$ and lift them inside $\mathrm{F}^{i}$. Then we lift the rest of a basis of $\overline{\mathrm{B}}$ arbitrarily. Let $\mathrm{B}$ be the sum of respective liftings. By construction $\mathrm{B}$ satisfies (2) and (3). Note that $\mathrm{B}$ is a saturated $\mathcal{O}$-submodule in $\mathrm{M}$, hence (1) is also satisfied.

If one has another lifting $0\to\mathrm{B^{\prime}}\to\mathrm{M}\to\mathrm{G^{\prime}}\to 0$, then the projection from $\mathrm{B^{\prime}}$ to $\mathrm{B}$ must be an isomorphism as guaranteed by Nakayama’s lemma. Then this lifting $\mathrm{B^{\prime}}$ is just a graph of some morphism $f\in\mathrm{Hom}(\mathrm{B},\mathrm{G})$. (3) implies that this morphism must preserve induced filtration. (2) implies that this morphism must have target $\mathfrak{m}\mathrm{G}$. So we see that $f\in\mathrm{Hom}_{\mathrm{F}^{\bullet}}(\mathrm{B},\mathfrak{m}\mathrm{G})=\mathfrak{m}\mathrm{Hom}_{\mathrm{F}^{\bullet}}(\mathrm{B},\mathrm{G})$. By the same reasoning any such $f$ would give rise to a lifting satisfying (1), (2) and (3). ∎

The second half of the lemma above can be generalized to multi-filtered situation in the following way:

Lemma 2.4**.**

In the situation of 2.2, suppose $0\to\overline{\mathrm{B}}\to\overline{\mathrm{M}}\to\overline{\mathrm{G}}\to 0$ is liftable modulo $\pi\in\mathcal{O}$, and fix a splitting $\widetilde{\mathrm{M}}=\widetilde{\mathrm{B}}\oplus\widetilde{\mathrm{G}}$. Then all different liftings correspond to graphs of morphisms in $\mathrm{Hom}_{\operatorname{Fil}^{\bullet}}(\widetilde{\mathrm{B}},\widetilde{\mathrm{G}})\cap\mathfrak{m}\mathrm{Hom}(\widetilde{\mathrm{B}},\widetilde{\mathrm{G}})$.

It’s nice to prove the following lemma directly:

Lemma 2.5**.**

In the situation of the lemma above, $\mathrm{Hom}_{\mathrm{Fil^{\bullet}}}(\widetilde{\mathrm{B}},\widetilde{\mathrm{G}})$ is always a finitely generated $(\mathcal{O}/\pi)$-module.

Proof.

We will prove by induction on the number of filtrations. The homomorphisms preserve all the $n$ filtrations are exactly those preserve the first $n-1$ filtrations and preserve the last filtration. Hence as a submodule of $\mathrm{Hom}(\widetilde{\mathrm{B}},\widetilde{\mathrm{G}})$ (which is abstractly isomorphic to ${(\mathcal{O}/\pi)}^{n}$), we see that

[TABLE]

By induction hypothesis we see that $\mathrm{Hom}_{\mathrm{Fil^{\bullet}_{1},\dots,Fil^{\bullet}_{n-1}}}(\widetilde{\mathrm{B}},\widetilde{\mathrm{G}})$ is a finitely generated submodule of $\mathrm{Hom}(\widetilde{\mathrm{B}},\widetilde{\mathrm{G}})$ and by Lemma 2.3 $\mathrm{Hom}_{\mathrm{Fil^{\bullet}_{n}}}(\widetilde{\mathrm{B}},\widetilde{\mathrm{G}})$ is a saturated submodule of $\mathrm{Hom}(\widetilde{\mathrm{B}},\widetilde{\mathrm{G}})$. Therefore it suffices to prove that the intersection of a finitely generated submodule $H_{1}$ and a saturated submodule $H_{2}$ of $H={(\mathcal{O}/\pi)}^{n}$ is again finitely generated. Fix a splitting $H=H_{2}\oplus H_{3}$, we see that $H_{1}\cap H_{2}$ is the kernel of projection map $H_{1}\to H_{3}$. This is finitely generated because $\mathcal{O}/\pi$ is a coherent ring. ∎

Lemma 2.6**.**

Let $H={(\mathcal{O}/\pi)}^{N}$. Let $H_{1}$ be a finitely generated submodule in $H$ and let $H_{2}$ be a saturated finitely generated submodule in $H$. Then

[TABLE]

where the direct sum is finite and $\pi_{i}^{\prime},\pi_{j}\in\mathfrak{m}-\{0\}$.

Proof.

Because $H_{2}$ is saturated, we may assume that $H=H_{2}\oplus H_{3}$. Then we see $H^{\prime}:=H/\mathfrak{m}H_{2}=H_{2}/\mathfrak{m}H_{2}\oplus H_{3}$, and let us fix a basis of $H_{3}$ as $\{f_{i}\}$. Notice that $\overline{H_{1}}\cap\mathfrak{m}\overline{H}=\overline{H_{1}\cap\mathfrak{m}H}$, where “bar” of a module denote its image in $H/\mathfrak{m}H_{2}$. Let us consider the image of $H_{1}$ (suppose there are $L$ generators) inside $H^{\prime}$. Project further to $H_{2}^{\prime}:=H_{2}/\mathfrak{m}H_{2}$, we may assume the image of the first $l$ generators form a basis of the image in $H_{2}^{\prime}$. Without loss of generality we can now assume the image of the generators are of the form $e_{i}+\sum a_{ij}f_{j}$ and $\sum a_{ij}f_{j}$, where $e_{1},\ldots,e_{l}$ is the image of the first $l$ generators.

Now let us perform Gauss elimination carefully as following: choose $a_{ij}$ with smallest valuation (if it is possible then we would prefer to a choice with $i>l$), then do Gauss elimination to cancel all the other $f_{j}$ coordinates appearing in other generators. Keep doing this procedure. After rearranging basis of both $H_{2}^{\prime}$ and $H_{3}$, we can assume the image of $L$ generators of $H_{1}$ in $H^{\prime}$ are of the form $e_{i}+\pi_{i}^{\prime}f_{i}$ and $\pi_{j}f_{j}$. Now we can finally conclude that the quotient is of the form as stated in this lemma. ∎

Lemma 2.7**.**

In the situation of 2.2, there exists a $\pi\in\mathfrak{m}$ with biggest valuation such that the sequence can be lifted modulo $\pi$.

Proof.

Let us do induction on the number of filtrations. Suppose there exists a $\pi^{\prime}$ with biggest valuation such that the first $n-1$ filtrations are lifted, say, $0\to\widetilde{\mathrm{B}}\to\widetilde{\mathrm{M}}=\mathrm{M}/(\pi\mathrm{M})\to\widetilde{\mathrm{G}}\to 0$. Let us fix a splitting $\widetilde{\mathrm{M}}=\widetilde{\mathrm{B}}\oplus\widetilde{\mathrm{G}}$. By Lemma 2.3, the n-th filtration can be lifted modulo $\pi$ also. By the same reasoning as in Lemma 2.3, these two liftings differ by an $A\in\mathfrak{m}\mathrm{Hom}(\widetilde{\mathrm{B}},\widetilde{\mathrm{G}}):=\mathfrak{m}H$. Both of these two liftings are not unique. They differ by elements in $H_{1}\cap\mathfrak{m}H$ and $H_{2}\cap\mathfrak{m}H=\mathfrak{m}H_{2}$, where $H_{1}=\mathrm{Hom}_{\mathrm{Fil_{1},\ldots,Fil_{n-1}}}(\widetilde{\mathrm{B}},\widetilde{\mathrm{G}})$ which is finitely generated by Lemma 2.5 and $H_{2}=\mathrm{Hom}_{\mathrm{Fil_{n}}}(\widetilde{\mathrm{B}},\widetilde{\mathrm{G}})$ which is saturated. We see that by Lemma 2.6, $H/(H_{1}\cap\mathfrak{m}H+\mathfrak{m}H_{2})=\oplus\mathcal{O}/\pi_{k}\bigoplus\oplus\mathcal{O}/\pi_{k}^{\prime}\mathfrak{m}$ in which residue of $A$ lives. The coordinates of residue of $A$ generate a principal ideal $(\pi)$ in $(\mathcal{O}/\pi^{\prime})$. Because $A\in\mathfrak{m}H$ we see that $\pi\in\mathfrak{m}$. It is easy to see that $\pi$ satisfies the lemma. ∎

The following lemma will give another characterization of liftable modulo $\pi$.

Lemma 2.8**.**

Given $0\to\overline{\mathrm{B}}\to\overline{\mathrm{M}}\to\overline{\mathrm{G}}\to 0$ in the situation of 2.2. Then the following are equivalent:

1. (1)

the sequence above is liftable modulo $\pi$ 2. (2)

there exists a splitting $\mathrm{M}=\mathrm{B}\oplus\mathrm{G}$ that reduces to the above sequence and for all $\overline{v}\in\overline{\mathrm{B}}\cap\overline{\mathrm{Fil_{i}}}$ there exists a lifting $v\in\mathrm{Fil_{i}}$ such that $p_{2}(v)\in\pi G$ where $p_{2}$ is the obvious projection onto $G$. 3. (3)

the same as above except $\overline{v}$ only have to run over a set of chosen basis of the induced filtration.

Proof.

(2) and (3) are obviously equivalent by finiteness of dimension and linearity.

(1) implies (2): liftable means one can find splitting $\widetilde{\mathrm{M}}=\widetilde{\mathrm{B}}\oplus\widetilde{\mathrm{G}}$, lift this splitting over $\mathcal{O}$ we see that (2) is fulfilled.

(2) implies (1): one just modulo this lifting by $\pi$. ∎

Now we are ready to prove the analogue of Langton’s theorem for multi-filtered vector spaces.

Proof of the Main Theorem.

Choose an arbitrary integral model of $(V,\mathrm{Fil}^{\bullet}_{i})$, let us denote it as $(\mathrm{M},\mathrm{Fil}^{\bullet}_{i})$. Then consider its reduction $\overline{\mathrm{M}}$ with induced multi-filtrations. If it is semistable, then we are done. Otherwise, there is a maximal destabilizing subspace $\overline{\mathrm{B}}$ inside $\overline{\mathrm{M}}$ (cf.[Eve, Lemma 2.5]): $0\to\overline{\mathrm{B}}\to\overline{\mathrm{M}}\to\overline{\mathrm{G}}\to 0$. By Lemma 2.7 there exists a $\pi$ with biggest valuation, such that the sequence above is liftable modulo $\pi$. Then $\pi$ is not [math] because $(V,\mathrm{Fil}^{\bullet}_{i})$ is semistable. Let us denote a lifting as $0\to\widetilde{\mathrm{B}}\to\widetilde{\mathrm{M}}=\mathrm{M}/(\pi\mathrm{M})\to\widetilde{\mathrm{G}}\to 0$. This lifting gives us a splitting $\widetilde{\mathrm{M}}=\widetilde{\mathrm{B}}\oplus\widetilde{\mathrm{G}}$, we can and do lift this splitting further to get $\mathrm{M}=\mathrm{B}\oplus\mathrm{G}$. Let $\mathrm{M}^{\prime}=\ker(\mathrm{M}\to\widetilde{\mathrm{G}})$ be another integral model.

Now we have the following sequence

[TABLE]

tensoring with $\kappa$ gives

[TABLE]

After computing in the single filtered case we see that it is a sequence of multi-filtered vector spaces.

We claim that there is no splitting $\overline{\mathrm{B}}\to\overline{\mathrm{M^{\prime}}}$ as multi-filtered vector spaces. Otherwise there exists such a splitting $\overline{h}:\overline{\mathrm{B}}\to\overline{\mathrm{M^{\prime}}}$. Then lift this splitting to a morphism $h:\mathrm{B}\to\pi\mathrm{G}$. Now let us consider the splitting $\mathrm{M}=\Gamma_{h}\oplus\mathrm{G}$. Let $\overline{v}\in\overline{\mathrm{B}}\cap\overline{\mathrm{F_{i}}}$, then by assumption we can lift $\overline{h}(\overline{v})$ to $v\in\mathrm{F_{i}}$ and $h(p_{1}(v))-p_{2}(v)\in\pi\mathfrak{m}G$. Hence under the new splitting $\mathrm{M}=\Gamma_{h}\oplus\mathrm{G}$, $p^{\prime}_{2}(v)\in\pi^{\prime}G$ for some $\pi^{\prime}\in\pi\mathfrak{m}$ where $p^{\prime}_{2}=h\circ p_{1}-p_{2}$ is the new obvious projection onto $G$. Now by Lemma 2.8 we can lift the original sequence modulo $\pi^{\prime}$ for some $\pi^{\prime}\in\pi\mathfrak{m}$ contradicting our assumption that $\pi$ has the biggest valuation modulo which one can lift the sequence.

Our last claim is that the maximal destabilizing subspace $\overline{B^{\prime}}$ of $\overline{\mathrm{M}^{\prime}}$ has either slope or dimension less than $\overline{B}$. Denote the image of $\overline{B^{\prime}}$ in $\overline{B}$ by $\operatorname{Im}(\overline{B^{\prime}})$. We have

[TABLE]

where the first equality can be obtained only if $\overline{B^{\prime}}\cap\overline{G}=0$. In that case, we see immediately that $w(\overline{B^{\prime}})\leq w(\operatorname{Im}(\overline{B^{\prime}}))$ where equality is obtained only if $\overline{B^{\prime}}$ is a section of its image inside $\overline{B}$ as a multi-filtered vector space. Therefore if $\mu(\overline{B^{\prime}})=\mu(\overline{B})$ then $\overline{B^{\prime}}$ is isomorphic to $\operatorname{Im}(\overline{B^{\prime}})$ (as two multi-filtered vector spaces) which is a proper subspace of $\overline{B}$ (by last paragraph). Hence in the situation of $\mu(\overline{B^{\prime}})=\mu(\overline{B})$ the dimension of $\overline{B^{\prime}}$ must be strictly less than that of $\overline{B}$.

Because the ranges of slope and dimension are finite, after doing the procedure above finitely many times, we see that the reduction would become semistable. ∎

3. Stacks of semistable multi-filtered vector bundles

In this section, let us make the following definition.

Definition 3.1**.**

Fix two natural numbers $s$ and $n$ and $s$ non-decreasing functions $l_{i}:\mathbb{N}\to\{0,\dots,n\}$ such that $l_{i}(0)=0$ and $l_{i}(\infty)=n$, we call such a datum type and denote it as $(s,n;l_{i})$.

Given a type $(s,n;l_{i})$, we can consider the stack $\mathcal{M}_{(s,n;l_{i})}$ which associates any scheme $X$ the groupoid of multi-filtered rank $n$ vector bundles of type $(s,n;l_{i})$ on $X$, i.e., there are $s$ decreasing exhaustive separated filtrations $\operatorname{Fil}_{j}^{\bullet}$ satisfying the following two properties:

1. (1)

the graded pieces $\operatorname{Gr}_{i}^{j}=\operatorname{Fil}_{i}^{j}/\operatorname{Fil}_{i}^{j+1}$ are flat over $X$; 2. (2)

the ranks of $\operatorname{Fil}_{i}^{j}$ are $n-l_{i}(j)$.

It is not hard to see that $\mathcal{M}_{(s,n;l_{i})}=[\prod_{i}\mathrm{Fl}(n;l_{i})/\mathrm{GL}_{n}]$ where $\mathrm{Fl}(n;l_{i})$ is the flag variety with numerical conditions as above. Therefore the stack $\mathcal{M}_{(s,n;l_{i})}$ is an algebraic stack and it is quasi-separated and of finite type over $\operatorname{Spec}(\mathbb{Z})$.

There is an open subset ${(\prod_{i}\mathrm{Fl}(n;l_{i}))}^{s.s.}$ in $\prod_{i}\mathrm{Fl}(n;l_{i})$ whose points correspond to semistable multi-filtered vector spaces. As being semistable is independent of choice of basis we see that this open subset defines an open substack ${\mathcal{M}_{(s,n;l_{i})}}^{s.s.}=[{(\prod_{i}\mathrm{Fl}(n;l_{i}))}^{s.s.}/\mathrm{GL}_{n}]$ in $\mathcal{M}_{(s,n;l_{i})}$ which is just the stack of rank $n$ multi-filtered vector bundles of type $(s,n;l_{i})$ whose fibers are semistable.

Theorem 3.2**.**

${\mathcal{M}_{(s,n;l_{i})}}^{s.s.}$* is a quasi-separated algebraic stack which satisfies the existence part of the valuative criterion (c.f. [Sta17, Tag 0CL9]) over $\operatorname{Spec}(\mathbb{Z})$.*

Proof.

From the discussion before theorem we know that ${\mathcal{M}_{(s,n;l_{i})}}^{s.s.}$ is an algebraic stack quasi-separated and of finite type over $\operatorname{Spec}(\mathbb{Z})$. By our Theorem 1.3, we see the structure morphism ${\mathcal{M}_{(s,n;l_{i})}}^{s.s.}\to\operatorname{Spec}(\mathbb{Z})$ satisfies the valuative criterion in [Sta17, Tag 0CL9]. ∎

Acknowledgements

The author wants to thank his advisor Johan de Jong for telling him this elementary yet interesting problem.