The six-functor formalism for rigid analytic motives

TL;DR

This paper develops a six-functor formalism for rigid analytic motives, extending the theory to general spaces without noetherianity assumptions, and introduces a reduction technique to algebraic motives.

Contribution

It introduces a systematic framework for rigid analytic motives with a new reduction technique to algebraic motives, broadening applicability without noetherianity constraints.

Findings

Extended proper base change theorem for rigid analytic motives

Reduction technique from rigid analytic to algebraic motives

Framework applicable to non-noetherian rigid analytic spaces

Abstract

We offer a systematic study of rigid analytic motives over general rigid analytic spaces, and we develop their six-functor formalism. A key ingredient is an extended proper base change theorem that we are able to justify by reducing to the case of algebraic motives. In fact, more generally, we develop a powerful technique for reducing questions about rigid analytic motives to questions about algebraic motives, which is likely to be useful in other contexts as well. We pay special attention to establishing our results without noetherianity assumptions on rigid analytic spaces. This is indeed possible using Raynaud's approach to rigid analytic geometry.