J.-P. Tignol and A. R. Wadsworth: Value Functions and Associated Graded Rings for Semisimple Algebras

jean-pierre.tignol@uclouvain.be, arwadsworth@ucsd.edu

Submission: 2007, Apr 13

We introduce a type of value function y called a gauge on a finite-dimensional semisimple algebra A over a field F with valuation v. The filtration on A induced by y yields an associated graded ring gr_y(A) which is a graded algebra over the graded field gr_v(F). Key requirements for y to be a gauge are that gr_y(A) be graded semisimple and that dim_{gr_v(F)}(gr_y(A)) = dim_F(A). It is shown that gauges behave well with respect to scalar extensions and tensor products. When v is Henselian and A is central simple over F, it is shown that gr_y(A) is simple and graded Brauer equivalent to gr_w(D) where D is the division algebra Brauer equivalent to A and w is the valuation on D extending v on F. The utility of having a good notion of value function for central simple algebras, not just division algebras, and with good functorial properties, is demonstrated by giving new and greatly simplified proofs of some difficult earlier results on valued division algebras.

2000 Mathematics Subject Classification: 16K20, 16W60, 16W50

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