Quasicoherent Sheaves on Perfectoid Spaces

We present v-descent results for quasicoherent sheaves on perfectoid spaces and show some applications. For sheaves over $\mathcal O^{+a}/\pi$ we obtain a 6-functor formalism for p-torsion étale cohomology on diamonds and v-stacks. If one is willing to slightly modify the definition of solid modules over adic rings, then similar descent results can be established for $\mathcal O^{+a}$-modules and $\mathcal O$-modules, thus providing 6-functor formalisms for completed cohomology and pro-étale $\mathbb{Z}_p$- and $\mathbb{Q}_p$-cohomology. This is ongoing joint work with J. Anschütz and A.-C. Le Bras.