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Extensions of vector bundles on the Fargues-Fontaine curve II

Serin Hong

Abstract

Given two arbitrary vector bundles on the Fargues-Fontaine curve, we completely classify all vector bundles which arise as their extensions.

Extensions of vector bundles on the Fargues-Fontaine curve II

Abstract

Given two arbitrary vector bundles on the Fargues-Fontaine curve, we completely classify all vector bundles which arise as their extensions.
Paper Structure (4 sections, 17 theorems, 47 equations, 7 figures)

This paper contains 4 sections, 17 theorems, 47 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Vector bundles on the Fargues-Fontaine curve
  3. Extensions and permutations of HN polygons
  4. Classification theorems for extensions

Key Result

Theorem 1.1

Let $\mathcal{D}$, $\mathcal{E}$ and $\mathcal{F}$ be vector bundles on $X_{E, F}$ such that $\mathcal{D}$ or $\mathcal{F}$ is semistable. There exists a short exact sequence if and only if the line segments of $\mathrm{HN}(\mathcal{D} \oplus \mathcal{F})$ can be rearranged so that the resulting (possibly non-concave) polygon $\mathscr{P}$ satisfies the following properties:

Figures (7)

  • Figure 1: Illustration of the conditions in Theorem \ref{['classification of extensions, semistable case intro']}
  • Figure 2: Illustration of the conditions in Definition \ref{['suitable permutation of HN polygon for extension']}
  • Figure 3: Construction of $\mathscr{P}$ in the case $\mathrm{rk}(\overline{\mathcal{F}}) = \mathrm{rk}(\mathcal{F}) -1$
  • Figure 4: Construction of $\mathscr{P}$ in the case $\mathrm{rk}(\overline{\mathcal{F}}) = \mathrm{rk}(\mathcal{F})$
  • Figure 5: Construction of $\overline{\mathcal{E}}$ and $\overline{\mathscr{P}}$ in the case $\mu(\mathcal{D}) > \mu(\mathcal{F}_r)$
  • ...and 2 more figures

Theorems & Definitions (49)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Remark
  • Theorem 2.2: KL_relpadic1
  • Definition 2.3
  • Proposition 2.4: FF_curve
  • Remark
  • Definition 2.5
  • Proposition 2.6: FF_curve,Kedlaya_slopefiltrations
  • ...and 39 more