## Division algebras and quadratic forms over fraction fields of two-dimensional henselian domains.(English)Zbl 1294.11035

Let $$R$$ be a two-dimensional, henselian, excellent local domain with finite residue field $$k$$, and $$K$$ be the fraction field of $$R$$. Let $$\Omega_R$$ be the set of discrete valuations of $$K$$ that correspond to codimension-1 points of regular proper models of Spec $$R$$. The following two theorems are the main results of the paper.
Theorem 1.1. If char $$k \neq 2$$, then a quadratic form of rank $$5$$ over $$K$$ has a nontrivial zero over $$K$$ if and only if it has a nontrivial zero over the completion $$K_v$$ for every $$v \in \Omega_R$$.
Theorem 1.2. If char $$k \neq 2$$, then every quadratic form over $$K$$ of rank $$\geq 9$$ has a nontrivial zero over $$K$$.
The proof is based on a careful analysis of ramification and cyclicity of division algebras over $$K$$ culminating in the following local-global principle.
Theorem 1.3. Let $$q$$ be a prime number, $$q \neq$$ char $$k$$. For a Brauer class $$\alpha \in Br(K)$$ of order $$q$$, if $$\alpha_v \in Br(K_v)$$ is represented by a cyclic algebra of degree $$q$$ over the completion $$K_v$$ for every $$v \in \Omega_R$$, then $$\alpha$$ is represented by a cyclic algebra of degree $$q$$ over $$K$$.
This is proved by using methods developed by D. J. Saltman [J. Ramanujan Math. Soc. 12, No. 1, 25–47 (1997; Zbl 0902.16021), J. Algebra 314, No. 2, 817–843 (2007; Zbl 1129.16014), J. Algebra 320, No. 4, 1543–1585 (2008; Zbl 1171.16011)].

### MSC:

 11E04 Quadratic forms over general fields 16K99 Division rings and semisimple Artin rings

### Citations:

Zbl 0902.16021; Zbl 1129.16014; Zbl 1171.16011
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