Relative p-adic Hodge theory, II: Imperfect period rings

TL;DR

This paper extends p-adic Hodge theory to imperfect period rings on adic spaces, establishing a pseudocoherent (phi, Gamma)-modules framework and connecting it to existing theories with applications to etale cohomology.

Contribution

It develops a theory of pseudocoherent (phi, Gamma)-modules on adic spaces, generalizing previous constructions and relating them to known frameworks in p-adic Hodge theory.

Findings

Constructed a theory of pseudocoherent sheaves on adic spaces.

Established an equivalence with existing relative (phi, Gamma)-modules.

Proved the category of pseudocoherent (phi, Gamma)-modules is abelian and stable under derived functors.

Abstract

In a previous paper, we constructed a category of (phi, Gamma)-modules associated to any adic space over Q_p with the property that the etale (phi, Gamma)-modules correspond to etale Q_p-local systems; these involve sheaves of period rings for Scholze's pro-etale topology. In this paper, we first extend Kiehl's theory of coherent sheaves on rigid analytic spaces to a theory of pseudocoherent sheaves on adic spaces, then construct a corresponding theory of pseudocoherent (phi, Gamma)-modules. We then relate these objects to a more explicit construction in case the space comes equipped with a suitable infinite etale cover; in this case, one can decomplete the period sheaves and establish an analogue of the theorem of Cherbonnier-Colmez on the overconvergence of p-adic Galois representations. As an application, we show that relative (phi, Gamma)-modules in our sense coincide with the relative (phi, Gamma)-modules constructed by Andreatta and Brinon in the geometric setting where the latter can be constructed. As another application, we establish that the category of pseudocoherent (phi, Gamma)-modules on an arbitrary rigid analytic space over a p-adic field is abelian, satisfies the ascending chain condition, and is stable under various natural derived functors (including Hom, tensor product, and pullback). Applications to the etale cohomology of pro-etale local systems will be given in a subsequent paper.