Division Algebras and Quadratic Forms over Fraction Fields of Two-dimensional Henselian Domains

Abstract

Let $R$ be a 2-dimensional, henselian, excellent local domain with finite residue field $k$. Let $K$ be the fraction field of $R$. Building on Saltman's work on division algebras over function fields of surfaces, the following variants of earlier results are proved: (1) any Brauer class over $K$ of prime index $q$ which is invertible in $k$ is represented by a cyclic algebra of the same degree; (2) if $n>0$ is invertible in $k$, then any Brauer class over $K$ of order $n$ has index dividing $n^2$. The method also yields a local-global principle for cyclicity: (3) for any Brauer class $\alpha\in \Br(K)$ of prime order $q$ (invertible in $k$), if $\alpha$ is cyclic of degree $q$ over the completed field $K_v$ for every discrete valuation $v$ of $K$, then $\alpha$ is cyclic of degree $q$ over $K$. As applications, when the characteristic of $k$ is not 2, we obtain the local-global principle for isotropy of quadratic forms of rank 5 with respect to discrete valuations of $K$ and we show that every quadratic form of rank $\ge 9$ is isotropic over $K$. These results concerning quadratic forms are proved using methods of Parimala and Suresh.