Noncommutative motives of Azumaya algebras
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Tabuada_Noncommutative motives.pdf
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Author(s)
Van den Bergh, Michel
Trigo Neri Tabuada, Goncalo Jorge
Date Issued
March 2014
Journal
Journal of the Institute of Mathematics of Jussieu
Publisher
Cambridge University Press
Citation
Tabuada, Gonçalo, and Michel Van den Bergh. “Noncommutative Motives of Azumaya Algebras.” Journal of the Institute of Mathematics of Jussieu 14.2 (2015): 379–403.
Version
Original manuscript
Abstract
Let k be a base commutative ring, R a commutative ring of coefficients, X a quasi-compact quasi-separated k-scheme with m connected components, A a sheaf of Azumaya algebras over X of rank (r[subscript 1], . . . , r[subscript m]), and Hmo0(k)[subscript R] the category of noncommutative motives with R-coefficients. Assume that 1/r ∈ R with r := r[subscript 1] ×· · ·×r[subscript m]. Under these assumptions, we prove that the noncommutative motives with R-coefficients of X and A are isomorphic. As an application, we show that all the R-linear additive invariants of X and A are exactly the same. Examples include (nonconnective) algebraic K-theory, cyclic homology (and all its variants), topological Hochschild homology, etc. Making use of these isomorphisms, we then compute the R-linear additive invariants of differential operators in positive characteristic, of cubic fourfolds containing a plane, of Severi-Brauer varieties, of Clifford algebras, of quadrics, and of finite dimensional k-algebras of finite global dimension. Along the way we establish two results of independent interest. The first one asserts that every element α ∈ K[subscript 0](X) of rank (r[subscript 1], . . . , r[subscript m]) becomes invertible in the R-linearized Grothendieck group K[subscript 0](X)[subscript R], and the second one that every additive invariant of finite dimensional algebras of finite global dimension is unaffected under nilpotent extensions.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1017/S147474801400005X
