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: 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: In this article, we define and study the discriminant of symplectic graded involutions on non-inertially split graded simple algebras with simple 0-component. In particular, we show that if F is a graded field of characteristic different from 2, D is a graded central division algebra over F with exp(D)=2exp(D)=2 and |ker(θD)|>4|ker(θD)|>4 (see the preliminaries below), A=Mn(D)A=Mn(D), and σ is a graded involution of symplectic type on A, then there is only a finite number of values for the discriminants Δσ(τ)Δσ(τ), where τ describes all graded involutions of symplectic type on A (see Proposition 2.11). Consequently, for any graded central simple algebra C over F with C0C0 simple non-split, exp(C)=2exp(C)=2, |ker(θC)|>4|ker(θC)|>4 and deg(C)ind(C) even, we have Δσ(τ)=0Δσ(τ)=0 for any graded involutions of symplectic type σ and τ on C (see Corollary 2.12). We prove also that if E is a Henselian valued field with residue characteristic different from 2, D is a central division algebra of exponent 2 over E with |ker(θD)|>4|ker(θD)|>4, and B=Mn(D)B=Mn(D) with n even, then for any symplectic involutions σ, τ on B, preserving a tame gauge defined on B, we have Δσ(τ)=0Δσ(τ)=0 (see Corollary 3.5).
    Journal of Algebra 02/2014; 400:17–32. · 0.60 Impact Factor
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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).


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