The six-functor formalism for rigid analytic motives

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.

The de Rham–Fargues–Fontaine cohomology

“…Definitions and formal properties. Our conventions and notation are mostly taken from [Ayoub 2015;Ayoub et al 2022] even if we typically omit any visual reference to the étale topology and the ring of coefficients in what follows.…”

“…As a matter of fact, in all that follows one can replace the category Adic with any subcategory of adic spaces over ‫ޚ‬ p which are locally of finite Krull dimension that is stable under open immersions, finite étale extensions as well as relative discs, and that contains reduced rigid analytic varieties and relative Fargues-Fontaine curves. Alternatively, one may consider the (larger) category of rigid spaces as defined by [Fujiwara and Kato 2018] and considered in [Ayoub et al 2022]. In this article, we stick to an adic perspective and we leave it to the reader to extend the statements and definitions of the present article to any more general setting.…”

“…Remark 2.5. In [Ayoub et al 2022], the category RigDA (eff) (S) is denoted by RigSH (eff) (S, ‫. )ޑ‬ We use the notation DA which is more customary in the case of sheaves of -modules for a ring .…”

“…In particular, we deduce that it is motivic, i.e., it can be defined as a contravariant realization functor dR S : RigDA(S) → QCoh(S) op on the (unbounded, derived, stable, étale) category RigDA(S) of rigid analytic motives over S with values in the infinity-category of solid quasicoherent O S -modules. As a matter of fact, in order to prove the properties above we make extensive use of the theory of motives, and more specifically of their six-functor formalism [Ayoub et al 2022] and of a homotopy-based relative version of Artin's approximation lemma (Theorem 3.9) inspired by the absolute motivic proofs given in [Vezzani 2018]. If X is a proper smooth rigid variety over S, dR S (X ) is a perfect complex, whose cohomology groups are vector bundles.…”

“…If X is a proper smooth rigid variety over S, dR S (X ) is a perfect complex, whose cohomology groups are vector bundles. To prove this finiteness result, we combine the characterization of dualizable objects in QCoh(S) due to Andreychev [2021] (see also [Scholze 2020]), the motivic proper base change and the "continuity" property for rigid analytic motives (see [Ayoub et al 2022]). The latter result, which is based on the use of explicit rigid homotopies, states that whenever one has a weak limit of adic spaces (in the sense of Huber) X ∼ lim ← − − X i , any compact motive over X has a model over some X i .…”