Relative $A_{\rm inf}$-cohomology

TL;DR

This paper develops a relative $A_{inf}$-cohomology theory connecting it to prismatic theory, using fiber products of topoi, and demonstrates an étale comparison of $q$-crystalline pushforward after inverting $$ in $A_{inf}$.

Contribution

It introduces a relative version of $A_{inf}$-cohomology and links it to prismatic theory through fiber products of topoi, expanding the theoretical framework.

Findings

Established a relative $A_{inf}$-cohomology construction.

Connected the theory to prismatic cohomology.

Proved an étale comparison for $q$-crystalline pushforward.

Abstract

We construct a relative version of the $A_{\rm inf}$-cohomology theory developed by Bhatt-Morrow-Scholze and relate it to the prismatic theory of Bhatt-Scholze. The construction relies on the fiber product of topoi. As an application we show that there is an \'etale comparison of the $q$-crystalline pushforward after inverting $\mu \in A_{\rm inf}$.