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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    • "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.
    Communications of the Korean Mathematical Society 01/2014; 29(1). DOI:10.4134/CKMS.2014.29.1.023
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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
    Mathematische Zeitschrift 10/2011; 269(1):309-323. DOI:10.1007/s00209-010-0729-y · 0.69 Impact Factor
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    • "As it is mentioned in [15], even though computations in the graded setting are easier (and discrete), it seems not so much is lost in passage from D to its corresponding graded division algebra grD. This was a motivation to the systematic study of graded central simple algebras and their correspondences, notably by Boulagouaz [4], Hwang, Tignol and Wadsworth [13] [14] [15] [19] and to the comparison of certain functors defined on these objects, notably the Brauer group and the reduced Whitehead group. Recall that a unital ring R = γ∈Γ R γ is called a graded ring if Γ is a group, each R γ is a subgroup of (R, +) and R γ · R δ ⊆ R γ+δ for all γ, δ ∈ Γ. "
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    ABSTRACT: For an Azumaya algebra A which is free over its centre R, we prove that K-theory of A is isomorphic to K-theory of R up to its rank torsions. We conclude that Ki(A,Z/m) = Ki(R,Z/m) for any m relatively prime to the rank and i ≥ 0. This covers, for example, K-theory of division algebras, K-theory of Azumaya algebras over semi-local rings and K-theory of graded central simple algebras indexed by a totally ordered abelian group. 1. introduction Let R be a ring and Ki(R), i ≥ 0, be Quillen’s K-groups. The construction of K-groups is functorial. Furthermore, Ki functors induce identity maps on inner-automorphisms of a ring and Ki(R) → Ki(MtR) → Ki(R) is multiplication by t, where R → MtR is the diagonal homomorphism, r ↦ → rIt. Fortheclassofdivision ringsfinitedimensional over their centres (which arefields), Green et. al. [8] proved that K-theory of a division algebra is essentially the same as K-theory of its centre, i.e., for a division algebra D over its centre F of index n, (1.1) Ki(D)⊗Z[1/n] ∼ = Ki(F)⊗Z[1/n]. Their proof combines the above observations with the Skolem-Noether theorem which guarantees that algebra homomorphisms in the setting of central simple algebras are inner, and then uses the main result of [7] which states that lim i→ ∞ Mn2iF ∼ = lim Mn2(i+1)D. i→∞ This note is a continuation of [10] where it was observed that one can naturally deduce (1.1) by using the fact that an F-central simple algebra A is a twisted form of a matrix algebra, i.e., there is a finite field extension L/F such that A⊗F L = MkL. However, using
    Communications in Algebra 03/2010; 38(3):919-926. DOI:10.1080/00927870902828710 · 0.39 Impact Factor
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