Value functions and associated graded rings for semisimple algebras

Abstract:

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 $\operatorname {dim}_{gr_v(F)}(gr_y(A)) = \operatorname {dim}_F(A)$. 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.