Overconvergent prismatic cohomology

TL;DR

This paper introduces an overconvergent variant of prismatic cohomology, connecting it to various p-adic cohomologies and extending the framework of Bhatt and Scholze to overconvergent settings.

Contribution

It defines overconvergent prismatic cohomology and demonstrates its specialization to key p-adic cohomologies, extending the existing theory to overconvergent contexts.

Findings

Overconvergent prismatic cohomology specializes to Monsky-Washnitzer and rigid cohomology.

It relates overconvergent prismatic cohomology to de Rham and étale cohomologies.

An overconvergent version of the AΩ complex is constructed and connected to the new cohomology.

Abstract

In this note I define an overconvergent version of prisms and prismatic cohomology as introduced by Bhatt and Scholze and show that overconvergent prismatic cohomology specialises to $p$-adic cohomologies, like Monsky-Washnitzer resp. rigid cohomology for smooth varieties over a perfect field, the de Rham cohomology of smooth weak formal schemes over a perfectoid ring and the \'{e}tale cohomology of its generic fibre. Besides, I give an overconvergent version of the complex $A\Omega$ of Bhatt-Morrow-Scholze and relate it to overconvergent prismatic cohomology.