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


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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    ABSTRACT: We study possible decompositions of totally decomposable algebras with involution, that is, tensor products of quaternion algebras with involution. In particular, we are interested in decompositions in which one or several factors are the split quaternion algebra $M_2(F)$, endowed with an orthogonal involution. Using the theory of gauges, developed by Tignol-Wadsworth, we construct examples of algebras isomorphic to a tensor product of quaternion algebras with $k$ split factors, endowed with an involution which is totally decomposable, but does not admit any decomposition with $k$ factors $M_2(F)$ with involution. This extends an earlier result of Sivatski where the algebra considered is of degree $8$ and index $4$, and endowed with some orthogonal involution.
    Full-text · Article · Dec 2015
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    • "Then, as B 0 is simple, by Cor. 2.3 of [8], θ E ′ = θ B where θ B is the map defined in (2.5) of [8]. So, the commutativity of the above diagram is followed. "
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    ABSTRACT: Let E be a graded central division algebra (GCDA) over a grade field R. Let S be an unramified graded field extension of R. We describe the grading on the underlying GCDA E' of which is analogous to the valuation on a tame division algebra over Henselian valued field.
    Preview · Article · Jan 2014 · Communications of the Korean Mathematical Society
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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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    ABSTRACT: A quadratic form over a Henselian-valued field of arbitrary residue characteristic is tame if it becomes hyperbolic over a tame extension. The Witt group of tame quadratic forms is shown to be canonically isomorphic to the Witt group of graded quadratic forms over the graded ring associated to the filtration defined by the valuation, hence also isomorphic to a direct sum of copies of the Witt group of the residue field indexed by the value group modulo 2. KeywordsWitt group–Henselian valuation
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