The six functors for Zariski-constructible sheaves in rigid geometry

TL;DR

This paper develops a six functor formalism for Zariski-constructible sheaves in rigid geometry, enabling a robust theory of perverse sheaves and advancing the understanding of sheaf theory in nonarchimedean analytic spaces.

Contribution

It introduces a novel notion of generic points in rigid geometry and establishes a six functor formalism for Zariski-constructible sheaves in characteristic zero.

Findings

Proves a generic smoothness result in rigid analytic geometry.

Develops a six functor formalism for Zariski-constructible sheaves.

Supports a well-behaved theory of perverse sheaves in rigid spaces.

Abstract

We prove a generic smoothness result in rigid analytic geometry over a characteristic zero nonarchimedean field. The proof relies on a novel notion of generic points in rigid analytic geometry which are well-adapted to "spreading out" arguments, in analogy with the use of generic points in scheme theory. As an application, we develop a six functor formalism for Zariski-constructible \'etale sheaves on characteristic zero rigid spaces. Among other things, this implies that characteristic zero rigid spaces support a well-behaved theory of perverse sheaves.