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Unitary SK_1 of semiramified graded and valued division algebras
We prove formulas for the unitary SK_1 of a semiramified graded division
algebra (or valued division algebra over a Henselian field) with a unitary
involution. These formulas generalize earlier formulas of Yanchevskii, (and
Platonov and Ershov for the nonunitary SK_1).Comment: 34 pages. Correction in References in v.
Valuations on Algebras with Involution
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.Comment: 27 page
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