Special Year Workshop on p-adic Arithmetic Geometry

In joint work in progress with Anschütz and Le Bras we aim to construct a 6-functor formalism for quasicoherent sheaves on the relative Fargues-Fontaine curve over rigid-analytic varieties (and even general v-stacks), providing new insights into the p-adic cohomology theories appearing in p-adic Hodge theory. In particular, the 6-functor formalism should prove the conjectures by Colmez-Gilles-Niziol on dualities of pro-étale Q_p-cohomology on Stein spaces. While the details of the 6-functor formalism are still work in progress, we are able to prove powerful v-descent results for quasicoherent sheaves on perfectoid spaces, which provide the technical foundation to construct the six functors.