Noncommutative motives of separable algebras

TL;DR

This paper explores the structure of noncommutative motives of separable algebras, providing models, structural insights, and applications to invariants and motivic relations, with implications for algebraic K-theory and cyclic sieving.

Contribution

It introduces new models and structural descriptions of noncommutative motives of separable algebras, linking them to Brauer groups and invariants, and applies these to compute motives of various algebraic objects.

Findings

Established motivic relations between central simple algebras for all additive invariants.

Computed additive invariants of twisted flag varieties using Brauer classes.

Categorified the cyclic sieving phenomenon and computed motives of inseparable extensions.

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.