Valuations on algebras with involution
(2011) Mathematische Annalen — Vol. 351, n° 1, p. 109-148 (2011)
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Details
- Authors
- Abstract
- Let A be a central simple algebra with involution σ of the first or second kind. Let v be a valuation on the σ-fixed part F of Z(A). A σ-special v-gauge g on A is a kind of value function on A extending v on F, such that g(σ (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 σ-special v-gauge g if and only if σ is anisotropic, and g is unique. If v is not Henselian, it is shown that there is a σ-special v-gauge g if and only if σ remains anisotropic after scalar extension from F to the Henselization of F with respect to v; when this occurs, g is the unique σ-invariant v-gauge on A.
- Affiliations
University of California San Diego Department of Mathematics
Citations
Tignol, J.-P., & Wadsworth, A. R. (2011). Valuations on algebras with involution. Mathematische Annalen, 351(1), 109-148. https://doi.org/10.1007/s00208-010-0580-9 (Original work published 2011)