Motivic rigidity of Severi-Brauer varieties

TL;DR

This paper investigates the invariance and rigidity of motivic decompositions of Severi-Brauer varieties associated with central division algebras, focusing on coefficient rings and field extensions, revealing their dependence on characteristic and stability under extensions.

Contribution

It demonstrates that motivic decompositions depend only on the characteristic of the coefficient ring and remain unchanged under certain field extensions.

Findings

Decompositions depend solely on the characteristic of the coefficient ring.

Motivic decompositions are invariant under field extensions where D remains division.

The results establish conditions for the rigidity of motivic structures of Severi-Brauer varieties.

Abstract

Let D be a central division algebra over a field F. We study in this note the rigidity of the motivic decompositions of the Severi-Brauer varieties of D, with respect to the ring of coefficients and to the base field. We first show that if the ring of coefficient is a field, these decompositions only depend on its characteristic. In a second part we show that if D remains division over a field extension E/F, the motivic decompositions of several Severi-Brauer varieties of D remain the same when extending the scalars to E.