Article

# 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

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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 view11/2014; - [Show abstract] [Hide abstract]

**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 - [Show abstract] [Hide abstract]

**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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