Surjectivity of the total Clifford invariant and Brauer dimension

TL;DR

This paper extends Merkurjev's theorem to certain algebraic varieties over local and finite fields by using the total Witt group, showing that the 2-torsion of the Brauer group is represented by line bundle-valued quadratic forms.

Contribution

It proves that replacing the Witt group with the total Witt group recovers Merkurjev's theorem for smooth curves over local fields and surfaces over finite fields.

Findings

2-torsion of the Brauer group is represented by even Clifford algebras of line bundle-valued quadratic forms

Merkurjev's theorem holds for smooth curves over local fields and surfaces over finite fields with the total Witt group

The total Witt group generalizes the classical result to broader algebraic settings.

Abstract

Merkurjev's theorem--the statement that the 2-torsion of the Brauer group is represented by Clifford algebras of quadratic forms--is in general false when the base is no longer a field. The work of Parimala, Scharlau, and Sridharan proves the existence of smooth complete curves over local fields, over which Merkurjev's theorem is equivalent to the existence of a rational theta characteristic. Here, we prove that for smooth curves over a local field or surfaces over a finite field, replacing the Witt group by the total Witt group of all line bundle-valued quadratic forms recovers Merkurjev's theorem: the 2-torsion of the Brauer group is always represented by even Clifford algebras of line bundle-valued quadratic forms.