Noncommutative motives of separable algebras

Abstract

Abstract. In this article we study in detail the category of noncommutative motives of separable algebras Sep(k) over a base field k. We start by constructing four different models of the full subcategory of commutative separable algebras CSep(k). Making use of these models, we then explain how the category Sep(k) can be described as a "fibered Z-order" over CSep(k). This viewpoint leads to several computations and structural properties of the category Sep(k). For example, we obtain a complete dictionary between directs sums of noncommutative motives of central simple algebras (=CSA) and sequences of elements in the Brauer group of k. As a first application, we establish two families of motivic relations between CSA which hold for every additive invariant (e.g. algebraic K-theory, cyclic homology, and topological Hochschild homology). As a second application, we compute the additive invariants of twisted flag varieties using solely the Brauer classes of the corresponding CSA. Along the way, we categorify the cyclic sieving phenomenon and compute the (rational) noncommutative motives of purely inseparable field extensions and of dg Azumaya algebras.

Noncommutative rigidity

“…Let AM(k) Q be the idempotent completion of the full subcategory of NNum(k) Q consisting of the objects U (A) Q with A a commutative separable kalgebra. As proved in [17,Prop. 2.3], AM(k) Q is equivalent to the classical category of Artin motives.…”

Untitled

“…U(A) ‫ކ‬ p U(B) ‫ކ‬ p in NNum(k) ‫ކ‬ p ). On one hand, Corollary B.14 of [Tabuada and Van den Bergh 2014] implies that U(B) ‫ޑ‬ U(k) ‫ޑ‬ in NNum(k) ‫ޑ‬ . This contradicts the left-hand side of (6.9).…”

Amitsur subgroup and noncommutative motives

“…From a motivic point of view, it is quite natural to ask how birational geometry of a given variety X is detected by its noncommutative motive. And indeed, there are results in this direction for (generalized) Brauer-Severi varieties [30] and [31]. In the present paper we want to consider twisted flags and shed some light to the case of arbitrary proper k-schemes admitting a certain type of semiorthogonal decomposition.…”

“…A source of examples for dg categories is provided by schemes since the derived category of perfect complexes perf(X) of any quasi-projective scheme X admits a canonical (unique) dg enhancement perf dg (X). In [30] it is proved that if two Brauer-Severi varieties X and Y (see Section 2 for a definition) are birational, then U (perf dg (X)) = U (perf dg (Y )). In view of the Amitsur conjecture for central simple algebras (two Brauer-Severi varieties X and Y are birational if and only if the corresponding central simple algebras A and B generate the same subgroup in Br(k)), it is conjectured in loc.cite that U is actually a complete birational invariant for Brauer-Severi varieties.…”

“…Note that these sums are finite according to Theorem 1.1. Furthermore, let Sep(k) be the full subcategory of the category of noncommutative Chow motives (see [30] for a definition) consisting of objects U (F ) with F a separable k-algebra. Now let CSA(k) be the full subcategory of Sep(k) consisting of objects U (A) with A a central simple k-algebra (see Section 2 for a definition of central simple algebra) and denote by CSA(k) ⊕ its closure under finite sums.…”