Classification of quotient bundles on the Fargues–Fontaine curve

Abstract

We completely classify all quotient bundles of a given vector bundle on the Fargues–Fontaine curve. As consequences, we have two additional classification results: a complete classification of all vector bundles that are generated by a fixed number of global sections, and a nearly complete classification of subsheaves of a given vector bundle. For the proof, we combine the dimension counting argument for moduli of bundle maps developed in Birkbeck et al. (J Inst Math Jussieu 21:487–532, 2022) with a series of reduction arguments based on some reinterpretation of the classifying conditions.

Appendix A. Classification of quotient bundles on \({\mathbb {P}}^1\) by Serin Hong and Hannah Larson

1.1 A.1. Main statement

In this appendix, we establish an analogue of Theorem 4.1.1 for vector bundles on the projective line ${\mathbb {P}}^1$ over an arbitrary field k. For each integer d, we denote by ${\mathcal {O}}(d)$ the d-th Serre twist of the trivial line bundle on ${\mathbb {P}}^1$. It is a classical theorem (often attributed to Grothendieck) that every vector bundle ${\mathcal {V}}$ on ${\mathbb {P}}^1$ admits a direct sum decomposition

$$\begin{aligned} {\mathcal {V}}\simeq \bigoplus _{i = 1}^r {\mathcal {O}}(d_i) \quad \quad \text { with } d_i \in {\mathbb {Z}}. \end{aligned}$$

(A.1)

Now we can state the main statement of this appendix as follows:

Theorem A.1.1

Let ${\mathcal {E}}$ and ${\mathcal {F}}$ be vector bundles on ${\mathbb {P}}^1$ with direct sum decompositions

$$\begin{aligned} {\mathcal {E}}\simeq \bigoplus _{i = 1}^r {\mathcal {O}}(a_i) \quad \quad \text { and } \quad \quad {\mathcal {F}}\simeq \bigoplus _{j=1}^s {\mathcal {O}}(b_j) \end{aligned}$$

(A.2)

for some integers $a_1 \le \cdots \le a_r$ and $b_1 \le \cdots \le b_s$. Then ${\mathcal {F}}$ arises as a quotient of ${\mathcal {E}}$ if and only if for each $j = 1, \ldots , s$, we have either $b_j \ge a_{j+1}$ or $b_i = a_i$ for all $i = 1, \ldots , j$.

This theorem is indeed an analogue of Theorem 4.1.1 for vector bundles on ${\mathbb {P}}^1$. For every vector bundle ${\mathcal {V}}$ on ${\mathbb {P}}^1$, we can use a direct sum decomposition as in (A.1) to define its Harder-Narasimhan polygon $\textrm{HN}({\mathcal {V}})$ and vector bundles ${\mathcal {V}}^{\le \mu }$ for every $\mu \in {\mathbb {Q}}$. Then we can state Theorem A.1.1 in exact accordance with Theorem 4.1.1.

In the subsequent sections, we present two proofs of Theorem A.1.1. The first proof is based on some elementary linear algebra, and is largely inspired by the argument of Eisenbud-Harris [6, Proposition 6.30]. The second proof is based on dimension analysis on moduli spaces of bundle maps, and is essentially identical to our proof of Theorem 4.1.1.

1.2 A.2 First proof: elementary linear algebra

For the rest of this appendix, we take ${\mathcal {E}}$ and ${\mathcal {F}}$ to be vector bundles on ${\mathbb {P}}^1$ with direct sum decompositions as in (A.2).

Lemma A.2.1

The existence of a surjective bundle map ${\mathcal {E}}\twoheadrightarrow {\mathcal {F}}$ amounts to the existence of an $s \times r$ matrix M over the polynomial ring k[x, y] with the following properties:

• (i) The (p, q)-th entry of M is either zero or homogeneous of degree $b_p - a_q$.

• (ii) The $s \times s$ minors of M have no common zeros.

Proof

For each integer $d \ge 0$, we can canonically identify $H^0({\mathbb {P}}^1, {\mathcal {O}}(d))$ as the space of degree d homogeneous polynomials in k[x, y]. In addition, we have a natural identification $\text {Hom}_{{\mathbb {P}}^1}({\mathcal {O}}(a_q), {\mathcal {O}}(b_p)) \cong H^0({\mathbb {P}}^1, {\mathcal {O}}(b_p - a_q))$ for each $p = 1, \ldots , s$ and $q = 1, \ldots , r$. Therefore every bundle map ${\mathcal {E}}\rightarrow {\mathcal {F}}$ can be represented by an $s \times r$ matrix M over k[x, y] with the property (i). Moreover, the map ${\mathcal {E}}\rightarrow {\mathcal {F}}$ is surjective if and only if M has rank s at all points on ${\mathbb {P}}^1$, which amounts to having the property (ii). $\square$

Remark

In the proof, when we identify $H^0({\mathbb {P}}^1, {\mathcal {O}}(d))$ as the space of degree d homogeneous polynomials over k, we take the convention that the zero polynomial is homogeneous of all nonnegative degrees.

Proposition A.2.2

If ${\mathcal {F}}$ arises as a quotient of ${\mathcal {E}}$, then for each $j = 1, \ldots , s$, we have either $b_j \ge a_{j+1}$ or $b_i = a_i$ for all $i = 1, \ldots , j$.

Proof

Suppose that we have $b_j < a_{j+1}$ for some $j = 1, \ldots , s$. We wish to show $b_i = a_i$ for $i = 1, \ldots , j$. Since ${\mathcal {F}}$ arises as a quotient of ${\mathcal {E}}$, we can take an $s \times r$ matrix M over k[x, y] as in Lemma A.2.1. Then for each $p = 1, \ldots , j$ and $q = j+1, \ldots , r$, we find $b_p \le b_j < a_{j+1} \le a_q$ and consequently deduce that the (p, q)-th entry of M is zero by the property (i) in Lemma A.2.1. In other words, M has a block decomposition

$$\begin{aligned} M = \begin{pmatrix} A &{} 0 \\ B &{} C \end{pmatrix}\end{aligned}$$

where A is a $j \times j$ matrix over k[x, y]. Hence any nonzero $s \times s$ minor of M should be divisible by the determinant of A. Now the property (ii) in Lemma A.2.1 implies that the determinant of A should be a constant nonzero polynomial; otherwise it has a nontrivial zero at which all $s \times s$ minors of M vanish.

We then observe that for each $i = 1, \ldots , j$ we must have $b_i \ge a_i$; otherwise, we find $b_p \le b_i < a_i \le a_q$ for each $p = 1, \ldots , i$ and $q = i+1, \ldots , r$, and consequently deduce by the property (i) in Lemma A.2.1 that A has a block decomposition

$$\begin{aligned} A = \begin{pmatrix} A_{11} &{} 0 \\ A_{21} &{} A_{22} \end{pmatrix}\end{aligned}$$

with $A_{11}$ being an $i \times (i-1)$ matrix and thus has a zero determinant. Similarly, for each $i = 1, \ldots , j$ we must have $b_i \le a_i$; otherwise, we find $b_p \ge b_i > a_i \ge a_q$ for each $p = i, \ldots , s$ and $q = 1, \ldots , i$, and consequently deduce by the property (i) in Lemma A.2.1 that A has a block decomposition

$$\begin{aligned} A = \begin{pmatrix} A_{11} &{} A_{12} \\ A_{21} &{} A_{22} \end{pmatrix}\end{aligned}$$

where $A_{21}$ is an $(s-i+1) \times i$ matrix with entries of positive degree, and thus has a determinant which is either zero or of positive degree. Therefore we conclude that $a_i$ and $b_i$ are equal for each $i = 1, \ldots , j$ as desired. $\square$

Proposition A.2.3

Assume that for each $j = 1, \ldots , s$, we have either $b_j \ge a_{j+1}$ or $b_i = a_i$ for all $i = 1, \ldots , j$. Then ${\mathcal {F}}$ arises as a quotient of ${\mathcal {E}}$.

Proof

Let l be the largest integer with the property $a_l = b_l$. Take M to be the $s \times r$ matrix whose nonzero entries are given as follows:

$$\begin{aligned} M = \left( \begin{array}{@{}c|c@{}} \begin{matrix} 1 &{} &{} \\ {} &{} \ddots &{} \\ {} &{} &{} 1 \end{matrix} &{} \\ \hline &{} \begin{matrix} x^{b_{l+1} - a_{l+1}} &{} y^{b_{l+1} - a_{l+2}} &{} &{} &{} \\ &{} \ddots &{} \ddots &{} &{} \\ &{} &{} x^{b_s - a_s} &{} y^{b_s - a_{s+1}} &{} \quad \quad \end{matrix} \end{array}\right) \end{aligned}$$

It suffices to show that M satisfies the properties (i) and (ii) in Lemma A.2.1. The property (i) is evident by construction. The property (ii) follows from the fact that the $s \times s$ minor with columns $1, \ldots , s$ is a power of x while the $s \times s$ minor with columns $1, \ldots , l, l+2, \ldots , s+1$ is a power of y. $\square$

We now deduce Theorem A.1.1 from Proposition A.2.2 and Proposition A.2.3.

1.3 A.3. Second proof: dimension analysis on moduli spaces of bundle maps

Let us denote by $\textrm{Sch}_{/ k}$ the category of k-schemes. For every vector bundle ${\mathcal {V}}$ on ${\mathbb {P}}^1$, we will write $\chi ({\mathcal {V}}):= \textrm{rk}({\mathcal {V}}) + \deg ({\mathcal {V}})$. In addition, for arbitrary vector bundles ${\mathcal {V}}$ and ${\mathcal {W}}$ on ${\mathbb {P}}^1$, we can define the moduli functors ${\mathcal {H}}\textrm{om}_{{\mathbb {P}}^1}({\mathcal {V}}, {\mathcal {W}})$, ${\mathcal {S}}\textrm{urj}_{{\mathbb {P}}^1}({\mathcal {V}}, {\mathcal {W}})$, and ${\mathcal {I}}\textrm{nj}_{{\mathbb {P}}^1}({\mathcal {V}}, {\mathcal {W}})$ on $\textrm{Sch}_{/ k}$ as in Definition 3.1.1. Then we have the following analogue of Proposition 3.1.2:

Proposition A.3.1

Let ${\mathcal {V}}$ and ${\mathcal {W}}$ be vector bundles on ${\mathbb {P}}^1$.

• (1) ${\mathcal {H}}\textrm{om}_{{\mathbb {P}}^1}({\mathcal {V}}, {\mathcal {W}})$ is represented by the affine scheme ${\mathbb {A}}_k^{\chi ({\mathcal {V}}^\vee \otimes {\mathcal {W}})^{\ge 0}}$.

• (2) ${\mathcal {S}}\textrm{urj}_{{\mathbb {P}}^1}({\mathcal {V}}, {\mathcal {W}})$ and ${\mathcal {I}}\textrm{nj}_{{\mathbb {P}}^1}({\mathcal {V}}, {\mathcal {W}})$ are represented by an open subscheme of ${\mathcal {H}}\textrm{om}_{{\mathbb {P}}^1}({\mathcal {V}}, {\mathcal {W}})$, and thus are either empty or of dimension $\chi ({\mathcal {V}}^\vee \otimes {\mathcal {W}})^{\ge 0}$.

Proof

The functor ${\mathcal {H}}\textrm{om}_{{\mathbb {P}}^1}({\mathcal {V}}, {\mathcal {W}})$ is indeed represented by ${{\,\mathrm{Spec\,}\,}}( {{\,\textrm{Sym}\,}}_k H^0({\mathbb {P}}^1, {\mathcal {V}}^\vee \otimes {\mathcal {W}})^\vee )$, whose Krull dimension is given by $\dim _k H^0({\mathbb {P}}^1, {\mathcal {V}}^\vee \otimes {\mathcal {W}}) = \chi ({\mathcal {V}}^\vee \otimes {\mathcal {W}})^{\ge 0}$. Now we can argue exactly as in [1, §3.3] to establish the second statement. $\square$

Moreover, we have an analogue of Proposition 3.1.3 as follows:

Proposition A.3.2

${\mathcal {F}}$ arises as a quotient of ${\mathcal {E}}$ if the following conditions are satisfied:

• (i) There exists a nonzero bundle map ${\mathcal {E}}\rightarrow {\mathcal {F}}$.

• (ii) For any ${\mathcal {Q}}\subsetneq {\mathcal {F}}$ which also occurs as a quotient of ${\mathcal {E}}$ we have an inequality

  $$\begin{aligned} \chi ({\mathcal {E}}^\vee \otimes {\mathcal {Q}})^{\ge 0}+ \chi ({\mathcal {Q}}^\vee \otimes {\mathcal {F}})^{\ge 0}< \chi ({\mathcal {E}}^\vee \otimes {\mathcal {F}})^{\ge 0}+ \chi ({\mathcal {Q}}^\vee \otimes {\mathcal {Q}})^{\ge 0}.\end{aligned}$$

Proof

Let S be the set of isomorphism classes of subsheaves ${\mathcal {Q}}\subseteq {\mathcal {F}}$ which also occur as a quotient of ${\mathcal {E}}$. We assert that S is finite. The Harder-Narasimhan theory for vector bundles on ${\mathbb {P}}^1$ is almost identical to the Harder-Narasimhan theory for vector bundles on ${\mathcal {X}}$, as we only need an additional requirement that all Harder-Narasimhan slopes are integers. In particular, for vector bundles on ${\mathbb {P}}^1$ we can define the notion of slopewise dominance and obtain analogues of Propositions 4.3.2 and 4.3.3. It follows that the slopes of every ${\mathcal {Q}}\in S$ are bounded by $\mu _\text {min}({\mathcal {E}})$ and $\mu _\text {max}({\mathcal {F}})$. Since vector bundles on ${\mathbb {P}}^1$ have integer slopes in their HN polygons, we deduce that S is finite.

Let us now assume for contradiction that ${\mathcal {F}}$ does not arise as a quotient of ${\mathcal {E}}$. Then S does not contain the isomorphism class of ${\mathcal {F}}$. For each ${\mathcal {Q}}\in S$, we define ${\mathcal {H}}\textrm{om}_{{\mathbb {P}}^1}({\mathcal {E}}, {\mathcal {F}})_{{\mathcal {Q}}}$ to be the image of the natural map

$$\begin{aligned}{\mathcal {S}}\textrm{urj}_{{\mathbb {P}}^1}({\mathcal {E}},{\mathcal {Q}}) \times _{{{\,\mathrm{Spec\,}\,}}(k)} {\mathcal {I}}\textrm{nj}_{{\mathbb {P}}^1}({\mathcal {Q}},{\mathcal {F}}) \rightarrow {\mathcal {H}}\textrm{om}_{{\mathbb {P}}^1}({\mathcal {E}},{\mathcal {F}})\end{aligned}$$

induced by composition of bundle maps, and write $\overline{{\mathcal {H}}\textrm{om}_{{\mathbb {P}}^1}({\mathcal {E}}, {\mathcal {F}})_{{\mathcal {Q}}}}$ for its closure in ${\mathcal {H}}\textrm{om}_{{\mathbb {P}}^1}({\mathcal {E}},{\mathcal {F}})$. Since ${\mathcal {H}}\textrm{om}_{{\mathbb {P}}^1}({\mathcal {E}},{\mathcal {F}})$ is represented by ${\mathbb {A}}_k^{\chi ({\mathcal {V}}^\vee \otimes {\mathcal {W}})^{\ge 0}}$ as noted in Proposition A.3.1, for each locally closed subscheme $\dim {\mathcal {H}}\textrm{om}_{{\mathbb {P}}^1}({\mathcal {E}}, {\mathcal {F}})_{{\mathcal {Q}}}$ with ${\mathcal {Q}}\in S$ we find

$$\begin{aligned}\dim {\mathcal {H}}\textrm{om}_{{\mathbb {P}}^1}({\mathcal {E}}, {\mathcal {F}})_{{\mathcal {Q}}} = \dim \overline{{\mathcal {H}}\textrm{om}_{{\mathbb {P}}^1}({\mathcal {E}}, {\mathcal {F}})_{{\mathcal {Q}}}}.\end{aligned}$$

By construction, ${\mathcal {H}}\textrm{om}_{{\mathbb {P}}^1}({\mathcal {E}},{\mathcal {F}})$ is covered by the subschemes $\overline{{\mathcal {H}}\textrm{om}_{{\mathbb {P}}^1}({\mathcal {E}}, {\mathcal {F}})_{{\mathcal {Q}}}}$ with ${\mathcal {Q}}\in S$. As the set S is finite, we find

$$\begin{aligned} \dim {\mathcal {H}}\textrm{om}_{{\mathbb {P}}^1}({\mathcal {E}}, {\mathcal {F}}) = \sup _{{\mathcal {Q}}\in S} \dim \overline{{\mathcal {H}}\textrm{om}_{{\mathbb {P}}^1}({\mathcal {E}}, {\mathcal {F}})_{\mathcal {Q}}} = \sup _{{\mathcal {Q}}\in S} \dim {\mathcal {H}}\textrm{om}_{{\mathbb {P}}^1}({\mathcal {E}}, {\mathcal {F}})_{\mathcal {Q}}. \end{aligned}$$

(A.3)

In addition, we can argue exactly as in [1, Lemma 3.3.10] to show that ${\mathcal {H}}\textrm{om}_{{\mathbb {P}}^1}({\mathcal {E}}, {\mathcal {F}})_{{\mathcal {Q}}}$ for each ${\mathcal {Q}}\in S$ is either empty or satisfies

$$\begin{aligned}\dim {\mathcal {H}}\textrm{om}_{{\mathbb {P}}^1}({\mathcal {E}}, {\mathcal {F}})_{{\mathcal {Q}}} = \chi ({\mathcal {E}}^\vee \otimes {\mathcal {Q}})^{\ge 0}+ \chi ({\mathcal {Q}}^\vee \otimes {\mathcal {F}})^{\ge 0}- \chi ({\mathcal {Q}}^\vee \otimes {\mathcal {Q}})^{\ge 0}.\end{aligned}$$

Now the assumption (ii) and Proposition A.3.1 together imply

$$\begin{aligned}\dim {\mathcal {H}}\textrm{om}_{{\mathbb {P}}^1}({\mathcal {E}}, {\mathcal {F}})_{{\mathcal {Q}}} < \dim {\mathcal {H}}\textrm{om}_{{\mathbb {P}}^1}({\mathcal {E}},{\mathcal {F}}) \quad \text { for every } {\mathcal {Q}}\in S,\end{aligned}$$

thereby yielding a contradiction by (A.3) as desired. $\square$

Remark

The proof of Proposition A.3.2 is slightly different from the original proof of [1, Theorem 3.3.11]. Here we established (A.3) by using the fact that ${\mathcal {H}}\textrm{om}_{{\mathbb {P}}^1}({\mathcal {E}},{\mathcal {F}})$ is represented by an affine algebraic space. For [1, Theorem 3.3.11], we get an analogous identity from the fact that the topological space $|{\mathcal {H}}\textrm{om}({\mathcal {E}}, {\mathcal {F}})_{\mathcal {Q}}|$ is stable under generalization and specialization inside $|{\mathcal {H}}\textrm{om}({\mathcal {E}}, {\mathcal {F}})|$.

Let us also record some basic properties of the function $\chi$.

Lemma A.3.3

Let ${\mathcal {V}}$ and ${\mathcal {W}}$ be vector bundles on ${\mathbb {P}}^1$.

• (1) We have $\chi ({\mathcal {V}}\oplus {\mathcal {W}}) = \chi ({\mathcal {V}}) + \chi ({\mathcal {W}})$.

• (2) We have $\chi ({\mathcal {V}}^\vee \otimes {\mathcal {W}})^{\ge 0}= 0$ if and only if $\mu _\text {min}({\mathcal {V}})$ is greater than $\mu _\text {max}({\mathcal {W}})$.

• (3) For any $d \in {\mathbb {Z}}$, we have $\chi ({\mathcal {V}}(d)^\vee \otimes {\mathcal {W}}(d))^{\ge 0}= \chi ({\mathcal {V}}^\vee \otimes {\mathcal {W}})^{\ge 0}$.

• (4) If ${\mathcal {V}}$ slopewise dominates ${\mathcal {W}}$, then we have $\chi ({\mathcal {V}})^{\ge 0}\ge \chi ({\mathcal {W}})^{\ge 0}$.

• (5) For any $d \in {\mathbb {Z}}$, we have $\chi ({\mathcal {V}}^\vee \otimes {\mathcal {W}}^{>d})^{\ge 0}\ge \chi ({\mathcal {V}}^\vee \otimes {\mathcal {O}}(d)^{\textrm{rk}({\mathcal {W}}^{>d})})^{\ge 0}$.

Proof

The statement (1) is evident by the additivity of rank and degree for vector bundles. The statements (2), (3) and (4) are straightforward to check by arguing as in their analogues, namely Corollary 3.2.5, Lemma 3.2.7 and Lemma 4.2.3. The statement (5) is an analogue of the inequality (4.15), and can be verified by arguing as in the proof of Proposition 4.4.10; in fact, we immediately find

$$\begin{aligned}\deg ({\mathcal {V}}^\vee \otimes {\mathcal {W}}^{>d})^{\ge 0}\ge \deg ({\mathcal {V}}^\vee \otimes {\mathcal {O}}(d)^{\textrm{rk}({\mathcal {W}}^{>d})})^{\ge 0}\end{aligned}$$

as the inequality (4.15) holds verbatim for vector bundles on ${\mathbb {P}}^1$, and also find

$$\begin{aligned}\textrm{rk}({\mathcal {V}}^\vee \otimes {\mathcal {W}}^{>d})^{\ge 0}\ge \textrm{rk}({\mathcal {V}}^\vee \otimes {\mathcal {O}}(d)^{\textrm{rk}({\mathcal {W}}^{>d})})^{\ge 0}\end{aligned}$$

by a similar argument that considers the x-coordinates of HN vectors. $\square$

Remark

The function $\chi$ does not admit an analogue of Lemma 3.2.8. However, this won’t be a problem for us. Indeed, we used Lemma 3.2.8 only once in the proof of Theorem 4.1.1 for reduction to the case of integer slopes. For Theorem A.1.1, we don’t need such a reduction step as vector bundles on ${\mathbb {P}}^1$ only have integer slopes.

Now we present another proof of Theorem A.1.1.

Proposition A.3.4

Theorem A.1.1 follows from Proposition A.3.2 and Lemma A.3.3.

Proof

We can reformulate Theorem A.1.1 as an analogue of Theorem 4.1.1 using vector bundles ${\mathcal {E}}^{\le \mu }$ and ${\mathcal {F}}^{\le \mu }$ for $\mu \in {\mathbb {Q}}$. Then the necessity part of Theorem A.1.1 becomes an analogue of Proposition 4.3.2, which is a formal consequence of the Harder-Narasimhan theory for vector bundles on ${\mathbb {P}}^1$ as noted in the proof of Proposition A.3.2. It remains to show that the sufficiency part of Theorem A.1.1 follows from Proposition A.3.2 and Lemma A.3.3. Assume that for each $j = 1, \ldots , s$, we have either $b_j \ge a_{j+1}$ or $b_i = a_i$ for all $i = 1, \ldots , j$. As noted already, our assumption precisely means that we have $\textrm{rk}({\mathcal {E}}^{\le \mu }) \ge \textrm{rk}({\mathcal {F}}^{\le \mu })$ for each $\mu \in {\mathbb {Q}}$ with equality if and only if ${\mathcal {E}}^{\le \mu }$ and ${\mathcal {F}}^{\le \mu }$ are isomorphic. We wish to show that ${\mathcal {E}}$ and ${\mathcal {F}}$ satisfy the conditions of Proposition A.3.2. This is essentially an analogue of Proposition 4.3.5, after arguing as in Lemma 4.3.4 to add an assumption that ${\mathcal {E}}$ and ${\mathcal {F}}$ have no common slopes. Our proof of Proposition 4.3.5 relies only on the Harder-Narasimhan theory for vector bundles on ${\mathcal {X}}$ and some basic properties of the degree function such as Corollary 3.2.5, Lemma 3.2.7, Lemma 4.2.3 and some inequalities in the proof of Proposition 4.4.10. In the context of Proposition A.3.2, the function $\chi$ takes the role of the degree function in Proposition 4.3.5 and has analogous properties as summarized in Lemma A.3.3. Therefore we can verify the conditions of Proposition A.3.2 for ${\mathcal {E}}$ and ${\mathcal {F}}$ by only using the Harder-Narasimhan theory for vector bundles on ${\mathbb {P}}^1$ and Lemma A.3.3. $\square$

Remark

Our argument in this subsection suggests that Theorem 4.1.1 (and Theorem A.1.1) should extend to any Harder-Narasimhan categories where certain moduli spaces of morphisms exist with locally spectral topological spaces that admit a nice dimension formula as in Proposition 3.1.2 or Proposition A.3.1 with some nice algebraic properties as in Lemma A.3.3.