On $q$-de Rham cohomology via $\Lambda$-rings

TL;DR

This paper develops a functorial $q$-de Rham cohomology theory for $ abla$-rings and formal schemes, connecting it with $ heta$-deformations, Adams operations, and lifts of Frobenius, extending previous work by Bhatt, Morrow, and Scholze.

Contribution

It introduces a natural cohomology theory for $ abla$-rings that refines existing $q$-de Rham theories and is independent of certain lifts of Frobenius.

Findings

$q$-de Rham cohomology arises from $ abla$-rings.

The theory depends only on Adams operations at each residue characteristic.

It provides a fully functorial cohomology theory with a lift of the Cartier isomorphism.

Abstract

We show that Aomoto's $q$-deformation of de Rham cohomology arises as a natural cohomology theory for $\Lambda$-rings. Moreover, Scholze's $(q-1)$-adic completion of $q$-de Rham cohomology depends only on the Adams operations at each residue characteristic. This gives a fully functorial cohomology theory, including a lift of the Cartier isomorphism, for smooth formal schemes in mixed characteristic equipped with a suitable lift of Frobenius. If we attach $p$-power roots of $q$, the resulting theory is independent even of these lifts of Frobenius, refining a comparison by Bhatt, Morrow and Scholze.