Canonical q-deformations in arithmetic geometry

TL;DR

This paper discusses a new canonical q-deformation of the de Rham complex in p-adic cohomology, bridging etale and de Rham theories, with insights into its properties and conjectures.

Contribution

It introduces a canonical q-deformation of the de Rham complex within a new p-adic cohomology framework, connecting different cohomological theories.

Findings

The q-deformation can be computed in coordinates.

The cohomology theory interpolates between etale and de Rham cohomology.

The deformation is canonical in the derived category.

Abstract

In recent work with Bhatt and Morrow, we defined a new integral p-adic cohomology theory interpolating between etale and de Rham cohomology. An unexpected feature of this cohomology is that in coordinates, it can be computed by a q-deformation of the de Rham complex, which is thus canonical, at least in the derived category. In this short survey, we try to explain what we know about this phenomenon, and what can be conjectured to hold.