We completely classify the possible extensions between semistable vector bundles on the Fargues–Fontaine curve (over an algebraically closed perfectoid field), in terms of a simple condition on Harder–Narasimhan (HN) polygons. Our arguments rely on a careful study of various moduli spaces of bundle maps, which we define and analyze using Scholze’s language of diamonds. This analysis reduces our main results to a somewhat involved combinatorial problem, which we then solve via a reinterpretation in terms of the Euclidean geometry of HN polygons.

中文翻译：

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向量束在 FARGUES-FONTAINE 曲线上的扩展

根据 Harder-Narasimhan (HN) 多边形的简单条件，我们完全分类了 Fargues-Fontaine 曲线（在代数封闭的完美域上）半稳态向量丛之间的可能扩展。我们的论点依赖于对束映射的各种模空间的仔细研究，我们使用 Scholze 的钻石语言对其进行定义和分析。这种分析将我们的主要结果简化为一个有点复杂的组合问题，然后我们通过重新解释 HN 多边形的欧几里得几何来解决这个问题。