Amitsur subgroup and noncommutative motives

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.