Relative $A_{\inf }$-cohomology

Abstract

We construct a relative version of the $A_{\inf }$-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 étale comparison of the $q$-crystalline pushforward after inverting $\mu \in A_{\inf }$.