Noncommutative motives of Azumaya algebras

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 is an element of 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.