TY - JOUR

T1 - Noncommutative motives of Azumaya algebras

AU - Tabuada, Gonçalo Jorge Trigo Neri

AU - van den Bergh, Michel

N1 - Sem PDF.
G. Tabuada was partially supported by the NEC Award-2742738 and by the Portuguese Foundation for Science and Technology through the project PEst-OE/MAT/UI0297/2014 (CMA).
This material is based upon work supported by the National Science Foundation (NSF) under Grant No. 0932078 000, undertaken while the authors were in residence at the Mathematical Science Research Institute (MSRI) in Berkeley, California, during the spring semester of 2013.

PY - 2015/4

Y1 - 2015/4

N2 - 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.

AB - 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.

KW - algebraic K-theory

KW - Azumaya algebras

KW - cyclic homology

KW - nilinvariance

KW - noncommutative algebraic geometry

KW - noncommutative motives

U2 - 10.1017/S147474801400005X

DO - 10.1017/S147474801400005X

M3 - Article

VL - 14

SP - 379

EP - 403

JO - Journal of the Institute of Mathematics of Jussieu

JF - Journal of the Institute of Mathematics of Jussieu

IS - 2

ER -