Overview
The research explores the application of Bridgeland stability conditions to the derived category of coherent sheaves on Fargues-Fontaine curves. This investigation introduces the study of quasi-coherent sheaves, specifically $\mathcal{O}(\theta^{\pm})$, which are derived from the convergents of an irrational number $\theta$. A central concept introduced is the continuum envelope, denoted as $\mathrm{QCoh}_{\mathbb{R}}(X_{FF})$, defined as the smallest abelian subcategory within $\mathrm{QCoh}(X_{FF})$ that encompasses both $\mathrm{Coh}(X_{FF})$ and the specified $\mathcal{O}(\theta^{\pm})$ sheaves.
Research Context
The theoretical foundation for this work is the theory of Bridgeland stability conditions. This theory, initially motivated by insights from string theory, is employed to analyze the structure of the derived category of coherent sheaves present on Fargues-Fontaine curves.
Approach
The methodology involved applying Bridgeland stability conditions to the derived category of coherent sheaves on Fargues-Fontaine curves. This led to the consideration of quasi-coherent sheaves $\mathcal{O}(\theta^{\pm})$, which are characterized by the convergents of an irrational number $\theta$. The continuum envelope $\mathrm{QCoh}_{\mathbb{R}}(X_{FF})$ was defined as the minimal abelian subcategory of $\mathrm{QCoh}(X_{FF})$ containing both coherent sheaves $\mathrm{Coh}(X_{FF})$ and the identified $\mathcal{O}(\theta^{\pm})$ sheaves. The homological algebra of $\mathrm{QCoh}_{\mathbb{R}}(X_{FF})$ was investigated using Farey diagrams as an analytical tool.
Findings
- The application of Bridgeland stability conditions to the derived category of coherent sheaves on Fargues-Fontaine curves leads to the consideration of quasi-coherent sheaves $\mathcal{O}(\theta^{\pm})$.
- These quasi-coherent sheaves are related to the convergents of an irrational number $\theta$.
- The continuum envelope $\mathrm{QCoh}_{\mathbb{R}}(X_{FF})$ is defined as the smallest abelian subcategory of $\mathrm{QCoh}(X_{FF})$ that contains $\mathrm{Coh}(X_{FF})$ and $\mathcal{O}(\theta^{\pm})$.
- The study of the homological algebra of $\mathrm{QCoh}_{\mathbb{R}}(X_{FF})$ was conducted via Farey diagrams.
- The homological properties of $\mathcal{O}(\theta^{\pm})$ are heavily dependent on the Diophantine properties of $\theta$.
- Fargues-Fontaine curves exhibit strong similarities to complex elliptic curves from this viewpoint concerning Diophantine properties and homological characteristics.