Continuum envelops on Fargues-Fontaine curves and elliptic curves

In this paper, we apply the theory of Bridgeland stability conditions, which was originated 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 envelop ${\mathrm{Q}\mathrm{C}\mathrm{o}\mathrm{h}}_{\mathbb{R}}(X_{FF})$ to be the smallest abelian subcategory in $\mathrm{Q}\mathrm{C}\mathrm{o}\mathrm{h}(X_{FF})$ containing $\mathrm{C}\mathrm{o}\mathrm{h}(X_{FF})$ and $\mathcal{O}({\theta }^{\pm })$. We study the homological algebra of ${\mathrm{Q}\mathrm{C}\mathrm{o}\mathrm{h}}_{\mathbb{R}}(X_{FF})$ via Farey diagrams. We show that the homological property of $\mathcal{O}({\theta }^{\pm })$ depends heavily on the arithmetic property of $\theta$. The Fargues–Fontaine curves present strong similarity with complex elliptic curves in this point of view.