Amitsur subgroup and noncommutative motives

TL;DR

This paper computes the Amitsur subgroup for proper varieties, relates it to the Brauer group, classifies absolutely split vector bundles, and explores implications for noncommutative motives and twisted flags.

Contribution

It provides new calculations of the Amitsur subgroup, classifies AS-bundles on twisted flags, and links these to noncommutative motives and birational invariants.

Findings

Amitsur subgroup computed for certain proper varieties.

Brauer group is isomorphic to the quotient of the Brauer group of the base field.

Noncommutative motives of twisted flags are birational invariants.

Abstract

This paper addresses the problem of calculating the Amitsur subgroup of a proper $k$-scheme. Under mild hypothesis, we calculate this subgroup for proper $k$-varieties $X$ with $\mathrm{Pic}(X)\simeq \mathbb{Z}^{\oplus m}$, using a classification of so called absolutely split vector bundles ($AS$-bundles for short). We also show that the Brauer group of $X$ is isomorphic to $\mathrm{Br}(k)$ modulo the Amitsur subgroup, provided $X$ is geometrically rational. Our results also enable us to classify $AS$-bundles on twisted flags. Moreover, we find an alternative proof for a result due to Merkurjev and Tignol, stating that the Amitsur subgroup of twisted flags is generated by a certain subset of the set of classes of Tits algebras of the corresponding algebraic group. This result of Merkurjev and Tignol is actually a corollary of a more general theorem that we prove. The obtained results have also consequences for the noncommutative motives of the twisted flags under consideration. In particular, we show that a certain noncommutative motive of a twisted flag is a birational invariant, generalizing in this way a result of Tabuada. We generalize this result for $X$ having a certain type of semiorthogonal decomposition.