Azumaya descent and étale-local cyclotomic comparison: a noncommutative filtration

Abstract

Bhatt-Morrow-Scholze motivic filtration on topological Hochschild homology ($THH$) is extended to Azumaya algebras over quasisyntomic rings. We prove the Azumaya descent theorem: $THH$ of such an algebra is equivalent, as an underlying spectrum, to $THH$ of its center, normalized on ${\pi }_0$ by the reduced trace instead of the unit map. This comparison ${\Phi }_A$ is not shown to be étale-locally cyclotomic. But what is cyclotomic étale-locally is a separate Morita trace, agreeing with ${\Phi }_A$ only on ${\pi }_0$. The resulting transported filtration is exactly the center BMS filtration, well defined up to the automorphism ambiguity of ${\Phi }_A$. It is insensitive to the Brauer class, reflected instead by $K$-theoretic and module-categorical pairings $THH$ does not see. We establish the analogous statement for derived Azumaya algebras over connective ${\mathbb{E}}_{\infty }$-rings, under a connectivity hypothesis.