Value functions and associated graded rings for semisimple algebras

Transactions of the American Mathematical Society (Impact Factor: 1.1). 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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    ABSTRACT: Given a place between two fields, the isotropy behaviour of Azumaya algebras with involu-tion over the valuation ring corresponding to the place is studied. In particular, it is shown that isotropic right ideals specialise in an appropriate way. The treatment is characteristic free, and it provides a natural analogue to the existing specialisation theory for non-singular symmetric bilinear forms. The rest of the paper then deals with the case where the residue characteristic is different from 2. In that case we show that if the valuation ring is Henselian then isotropy can be lifted from the residue field to the fraction field of the valuation ring, and this can then be used to show that rationally isomorphic Azumaya algebras with invo-lution over this Henselian valuation ring are already isomorphic. This then implies there is a notion of good reduction with respect to places for algebras with involution, just as for non-singular symmetric bilinear forms.
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    ABSTRACT: Let $T$ be a totally ordered set and let $D(T)$ denote the set of all cuts of $T$. We prove the existence of a discrete valuation domain $O_{v}$ such that $T$ is order isomorphic to two special subsets of Spec$(O_{v})$. We prove that if $A$ is a ring (not necessarily commutative) whose prime spectrum is totally ordered and satisfies (K2), then there exists a totally ordered set $U \subseteq \text{Spec}(A)$ such that the prime spectrum of $A$ is order isomorphic to $D(U)$. We also present equivalent conditions for a totally ordered set to be a Dedekind totally ordered set. At the end, we present an algebraic geometry point of view
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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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