Continuum envelopes on Fargues-Fontaine curves and elliptic curves

Abstract

In this paper, we apply the theory of Bridgeland stability conditions, which was motivated by ideas from string theory, to study the derived category of coherent sheaves on Fargues--Fontaine curves. This leads us to consider the quasi-coherent sheaves $\mathcal{O}({\theta }^{\pm })$ via the convergents of an irrational number $\theta$. We define the continuum envelope ${\mathrm{QCoh}}_{\mathbb{R}}(XFF)$ to be the smallest abelian subcategory of $\mathrm{QCoh}(X_{FF})$ containing $\mathrm{Coh}(X_{FF})$ and $\mathcal{O}({\theta }^{\pm })$. We study the homological algebra of ${\mathrm{QCoh}}_{\mathbb{R}}(XFF)$ via Farey diagrams. We show that the homological properties of $\mathcal{O}({\theta }^{\pm })$ depend heavily on the Diophantine properties of $\theta$. From this point of view, Fargues--Fontaine curves exhibit strong similarities to complex elliptic curves.

arXiv commentThe original version of this work has been split into two papers. The first is available as arXiv:2606.12220, and the present paper is the second. All comments are welcome!