A 6-functor formalism for solid quasi-coherent sheaves on the Fargues-Fontaine curve

TL;DR

This paper develops a 6-functor formalism for solid quasi-coherent sheaves on the Fargues-Fontaine curve, advancing the understanding of p-adic geometry and local Langlands correspondence.

Contribution

It introduces a new 6-functor formalism with $Z_p$-linear coefficients on small v-stacks, enabling deeper duality and finiteness results in p-adic cohomology.

Findings

Establishes a formalism for pro-étale cohomology of rigid-analytic varieties.

Provides initial examples related to p-adic geometrization of the local Langlands program.

Discusses implications for duality and finiteness in p-adic sheaf theory.

Abstract

We develop a 6-functor formalism $\mathcal{D}_{[0,\infty)}(-)$ with $\mathbb{Z}_p$-linear coefficients on small v-stacks, and discuss consequences for duality and finiteness for pro-\'etale cohomology of rigid-analytic varieties of general pro-\'etale $\mathbb{Q}_p$-local systems as well as first examples motivated by a potential $p$-adic analog of Fargues-Scholze's geometrization program of the local Langlands correspondence.