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Noncommutative motives of Azumaya algebras
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 14 / Issue 2 / April 2015
- Published online by Cambridge University Press:
- 10 March 2014, pp. 379-403
- Print publication:
- April 2015
-
- Article
- Export citation
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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.
