Valuations on Algebras with Involution

Details Valuations on Algebras with Involution Valuations on Algebras with Involution

2009-02-05

arxiv.org/abs/0902.0849v1

Mathematics

Rings and Algebras

27 pages

Scientific paper

Let A be a central simple algebra with involution sigma of first or second kind. Let v be a valuation on the sigma-fixed part F of Z(A). A sigma-special v-gauge g on A is a kind of value function on A extending v on F, such that g(sigma(x) x) = 2g(x) for all x in A. It is shown (under certain restrictions if the residue characteristic is 2) that if v is Henselian, then there is a sigma-special v-gauge g if and only if sigma is anisotropic, and g is unique. If v is not Henselian, it is shown that there is a sigma-special v-gauge g if and only if sigma remains anisotropic after scalar extension from F to the Henselization of F re v; when this occurs, g is the unique sigma-invariant v-gauge on A.

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