Noncommutative motives of Azumaya algebras

Abstract

Abstract Let $k$ be a base commutative ring, $R$ a commutative ring of coefficients, $X$ a quasi-compact quasi-separated $k$ -scheme, and $A$ a sheaf of Azumaya algebras over $X$ of rank $r$ . Under the assumption that $1/r\in R$ , we prove that the noncommutative motives with $R$ -coefficients of $X$ and $A$ are isomorphic. As an application, we conclude that a similar isomorphism holds for every $R$ -linear additive invariant. This leads to several computations. Along the way we show that, in the case of finite-dimensional algebras of finite global dimension, all additive invariants are nilinvariant.