On the Derived Category of the Fargues-Fontaine Curve

Let $X$ be the schematic Fargues-Fontaine curve. Following arguments of Bondal and van den Bergh we show that $\mathcal{O}_X \oplus \mathcal{O}_X(1)$ generates the derived category of quasi-coherent $\mathcal{O}_X$-modules. By a theorem of Keller the latter is equivalent to the derived category of an associated differential graded algebra. We give an explicit description of this algebra in terms of rings of adèles on $X$ and determine the dg modules corresponding to all coherent $\mathcal{O}_X$-modules. We also take first steps in determining the multiplicative structure of a huge skew field first discovered by Colmez. This involves explicit computations in the heart of a $t$-structure constructed by Le Bras.


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