Valuations on Algebras with Involution

TL;DR

This paper characterizes the existence and uniqueness of sigma-special v-gauges on central simple algebras with involution, linking it to the anisotropic property of the involution and the Henselian property of the valuation.

Contribution

It provides necessary and sufficient conditions for the existence and uniqueness of sigma-special v-gauges based on involution anisotropy and valuation properties.

Findings

Existence of sigma-special v-gauges is equivalent to involution anisotropy under Henselian valuations.

Uniqueness of the gauge is established when conditions are met.

The gauge's existence depends on the involution's behavior after scalar extension.

Abstract

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.