Article

Value functions and associated graded rings for semisimple algebras

Transactions of the American Mathematical Society (Impact Factor: 1.12). 02/2009; 362(02):687-726. DOI: 10.1090/S0002-9947-09-04681-9

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 gry(A) which is a graded algebra over the graded field grv(F). Key requirements for y to be a gauge are that gry(A) be graded semisimple and that dimgrv(F)(gry(A)) = dimF(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 gry(A) is simple and graded Brauer equivalent to grw(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.

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    • "Gauges on algebras with involution play a major role in the next sections. In this section we recall the notions of value functions and gauges introduced in [11], [16], [17] and [18] and gather some results for the sequel. We fix a divisible totally ordered abelian group Γ, which will contain the value of all the valuations and the degree of all the gradings we consider. "
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    • "Let (F, v) be an arbitrary valued field and V a finite-dimensional F -vector space. We recall from [7] and [11] that a v-value function on V is a map α : V → Γ ∪ {∞} satisfying the following properties, for x, y ∈ V and λ ∈ F : (i) α(x) = ∞ if and only if x = 0; (ii) α(xλ) = α(x) + v(λ); (iii) α(x + y) ≥ min α(x), α(y) . The v-value function α is called a v-norm if there is a base (e i ) n i=1 of V that splits α in the following sense: "
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