Almost coherent modules and almost coherent sheaves

TL;DR

This paper develops a theory of almost coherent sheaves on schemes and formal schemes, extending the concept of almost coherent modules, and applies it to $p$-adic nearby cycles, providing new insights into rigid space cohomology.

Contribution

It introduces a global theory of almost coherent sheaves on schemes and formal schemes, including properties, functors, and applications to $p$-adic cohomology, extending previous almost ring theory.

Findings

Almost coherent sheaves satisfy properties similar to coherent sheaves.

Construction of an almost version of the Grothendieck $f^!$ functor.

Application to $p$-adic nearby cycles and a new proof of étale cohomology finiteness.

Abstract

We review the theory of almost coherent modules that was introduced in "Almost Ring Theory" by Gabber and Ramero. Then we globalize it by developing a new theory of almost coherent sheaves on schemes and on a class of "nice" formal schemes. We show that these sheaves satisfy many properties similar to usual coherent sheaves, i.e. the Almost Proper Mapping Theorem, the Formal GAGA, etc. We also construct an almost version of the Grothendieck twisted image functor $f^!$ and verify its properties. Lastly, we study sheaves of $p$-adic nearby cycles on admissible formal models of rigid spaces and show that these sheaves provide examples of almost coherent sheaves. This gives a new proof of the finiteness result for \'etale cohomology of proper rigid spaces obtained before in the work of Peter Scholze "$p$-adic Hodge Theory For Rigid-Analytic Varieties".