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.

In this article we prove that Kontsevich’s category NCnum(k)F of noncommutative numerical motives is equivalent to the one constructed by the authors in [Marcolli and Tabuada, Noncommutative motives, numerical equivalence, and semisimplicity, Amer. J. Math., to appear, available at arXiv:1105.2950]. As a consequence, we conclude that NCnum(k)F is abelian semi-simple as conjectured by Kontsevich.