Surjectivity of the total Clifford invariant and Brauer dimension

A celebrated theorem of Merkurjev—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 first counterexamples, when the base is among certain arithmetically subtle hyperelliptic curves over local fields, were constructed by Parimala, Scharlau, and Sridharan. We prove that considering Clifford algebras of all line bundle-valued quadratic forms, such counterexamples disappear and we recover Merkurjev's theorem in these cases: for any smooth curve over a local field or any smooth surface over a finite field, the 2-torsion of the Brauer group is always represented by Clifford algebras of line bundle-valued quadratic forms.