Rigid cohomology of locally noetherian schemes Part 1 : Geometry

TL;DR

This paper develops the geometric framework to extend rigid cohomology from algebraic varieties to general locally noetherian formal schemes, including a generalization of Berthelot's fibration theorem to adic spaces.

Contribution

It introduces the geometric setup for rigid cohomology on formal schemes and generalizes key theorems to adic spaces, broadening the scope of the theory.

Findings

Extended rigid cohomology to locally noetherian formal schemes.

Generalized Berthelot's strong fibration theorem to adic spaces.

Established local fibrations on strict neighborhoods under new conditions.

Abstract

We set up the geometric background necessary to extend rigid cohomology from the case of algebraic varieties to the case of general locally noetherian formal schemes. In particular, we generalize Berthelot's strong fibration theorem to adic spaces: we show that if we are given a morphism of locally noetherian formal schemes which is partially proper and formally smooth around a formal subscheme, and we pull back along a morphism from an analytic space which is locally of noetherian type, then we obtain locally a fibration on strict neighborhoods.