Let be a complete algebraically closed extension, and let be the Fargues-Fontaine curve associated with . If is any vector bundle on , the cohomology groups vanish for all and are naturally Banach-Colmez Spaces for . Recall that the latter things are roughly “finite-dimensional -vector spaces up to finite-dimensional -vector spaces”. By a hard and wonderful theorem of Colmez, these Spaces form an abelian category, and they have a well-defined Dimension valued in which is (componentwise-) additive in short exact sequences. The Dimension roughly records the -dimension and the -dimension, respectively. Typical examples are , which has Dimension , and , which has Dimension .
Here I want to record the following beautiful Riemann-Roch formula.
Theorem. If is any vector bundle on , then .
One can prove this by induction on the rank of , reducing to line bundles; the latter were classified by Fargues-Fontaine, and one concludes by an explicit calculation in that case. In particular, the proof doesn’t require the full classification of bundles.