Value functions and associated graded rings for semisimple algebras

Transactions of the American Mathematical Society (Impact Factor: 1.02). 01/2009; 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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    ABSTRACT: Suppose $F$ is a field with a nontrivial valuation $v$ and valuation ring $O_{v}$, $E$ is a finite field extension and $w$ is a quasi-valuation on $E$ extending $v$. We study the topology induced by $w$. We prove that the quasi-valuation ring determines the topology, independent of the choice of its quasi-valuation. Moreover, we prove the weak approximation theorem for quasi-valuations.
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    ABSTRACT: Suppose $F$ is a field with valuation $v$ and valuation ring $O_{v}$, $E$ is a finite field extension and $w$ is a quasi-valuation on $E$ extending $v$. We study quasi-valuations on $E$ that extend $v$; in particular, their corresponding rings and their prime spectrums. We prove that these ring extensions satisfy INC (incomparability), LO (lying over), and GD (going down) over $O_{v}$; in particular, they have the same Krull Dimension. We also prove that every such quasi-valuation is dominated by some valuation extending $v$. Under the assumption that the value monoid of the quasi-valuation is a group we prove that these ring extensions satisfy GU (going up) over $O_{v}$, and a bound on the size of the prime spectrum is given. In addition, a 1:1 correspondence is obtained between exponential quasi-valuations and integrally closed quasi-valuation rings. Given $R$, an algebra over $O_{v}$, we construct a quasi-valuation on $R$; we also construct a quasi-valuation on $R \otimes_{O_{v}} F$ which helps us prove our main Theorem. The main Theorem states that if $R \subseteq E$ satisfies $R \cap F=O_{v}$ and $E$ is the field of fractions of $R$, then $R$ and $v$ induce a quasi-valuation $w$ on $E$ such that $R=O_{w}$ and $w$ extends $v$; thus $R$ satisfies the properties of a quasi-valuation ring.
    Journal of Algebra. 09/2012; 372.
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    ABSTRACT: For a simple graded algebra A=M_n(E) over a graded division algebra E, a short exact sequence relating the reduced Whitehead group of the homogeneous part of A to that of E is established. In particular it is shown that the homogeneous SK1 is not in general Morita invariant.
    New York Journal of Mathematics 10/2011; · 0.44 Impact Factor


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