Amitsur subgroup and noncommutative motives

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 Pic(X) ≃ Z ⊕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 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.