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 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.