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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
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