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Content available remote Elementary Matrix-computational Proof of Quillen-Suslin Theorem for Ore Extensions
EN
In this short note we present an elementary matrix-constructive algorithmic proof of the Quillen-Suslin theorem for Ore extensions A := K[x; σ, δ], where K is a division ring, σ : K → K is a division ring automorphism and σ : K → K is a σ-derivation of K. It asserts that every finitely generated projective A-module is free. We construct a symbolic algorithm that computes the basis of a given finitely generated projective A-module. The algorithm is implemented in a computational package. Its efficiency is illustrated by four representative examples.
EN
In this two parts article with the same title we continue the Coxeter spectral study of the category UBigrm of loop-free edge-bipartite (signed) graphs Δ, with m ≥ 2 vertices, we started in [SIAM J. Discr. Math. 27(2013), 827-854] for corank r = 0 and r = 1. Here we study the class of all non-negative edge-bipartite graphs Δ ∈ UBigrn+r of corank r ≥ 0, up to a pair of the Gram Z-congruences ;~z and ≈z, by means of the non-symmetric Gram matrix ĞΔ∈Mn+r(Z) of Δ, the symmetric Gram matrix GΔ:=1/2[ĞΔ+ĞΔ-tr]∈Mn+r(Z), the Coxeter matrix CoxΔ:[formula...], its spectrum speccΔ⊂C, called the Coxeter spectrum of Δ, and the Dynkin type DynΔ∈{An,Dn,E6,E7,E8} associated in Part 1 of this paper. One of the aims in the study of the category UBigrn+r is to classify the equivalence classes of the non-negative edge-bipartite graphs in UBigrn+r with respect to each of the Gram congruences ~Z and ≈Z. In particular, the Coxeter spectral analysis question, when the congruence Δ≈ZΔ′ holds (hence also Δ~ZΔ′ holds), for a pair of connected non-negative graphs Δ,Δ′∈uBigrn+rsuch that speccΔ=speccΔ′ and DynΔ=DynΔ′, is studied in the paper. One of our main aims in this Part 2 of the paper is to get an algorithmic description of a matrix B defining the strong Gram Z-congruence Δ≈ZΔ′, that is, a Z-invertible matrix B∈Mn+r(Z) such that [formula...]. We obtain such a description for a class of non-negative connected edge-bipartite graphs Δ∈uBigrn+r of corank r = 0 and r = 1. In particular, we construct symbolic algorithms for the calculation of the isotropy mini-group ..., for a class of edge-bipartite graphs Δ∈uBigrn+r. Using the algorithms, we calculate the isotropy mini-groupGl(n,Z)D where D is any of the Dynkin bigraphs An, Bn, Cn, Dn, E6, E7, E8, F4, G2 and .D is any of the Euclidean graphs .[formula...].
EN
We continue the Coxeter spectral study of the category UBigrm of loop-free edge-bipartite (signed) graphs Δ, with m ≥ 2 vertices, we started in [SIAM J. Discr. Math. 27(2013), 827-854] for corank r = 0 and r = 1. Here we study the class of all non-negative edge-bipartite graphs Δ ∈ UBigrn+r of corank r ≥ 0, up to a pair of the Gram Z-congruences ~ z and ≈z, by means of the non-symmetric Gram matrix GΔ∈ Mn+r(Z), the symmetric Gram matrix GΔ:=[formula..]..., the Coxeter matrix CoxΔ:=[formula...]... and its spectrum speccΔ ⊂ C, called the Coxeter spectrum of Δ. One of the aims in the study of the category UBigrn+r is to classify the equivalence classes of the non-negative edge-bipartite graphs in UBigrn+r with respect to each of the Gram congruences ~Z and ≈Z. In particular, the Coxeter spectral analysis question, when the strong congruence Δ≈ZΔ′ holds (hence also ΔZΔ′ holds), for a pair of connected non-negative graphs Δ, Δ′ ∈ UBigrn+r such that speccΔ = speccΔ′, is studied in the paper. One of our main aims is an algorithmic description of a matrix B defining the Gram Z-congruences Δ≈ZΔ′ and ΔZΔ′, that is, a Z-invertible matrix B∈Mn+r(Z) such that ..., respectively. We show that, given a connected non-negative edge-bipartite graph Δ in UBigrn+r of corank r ≥ 0 there exists a simply laced Dynkin diagram D, with n vertices, and a connected canonical r-vertex extension ... of D of corank r (constructed in Section 2) such that Δ~ZD. We also show that every matrix B defining the strong Gram Z-congruence Δ≈ZΔ′ in UBigrn+r has the form [formula...], where CΔ,CΔ′∈Mn+r(Z) are fixed Z-invertible matrices defining the weak Gram congruences Δ~Z ... and Δ′~ZD with an r-vertex extended graph ..., respectively, and B ∈Mn+r(Z) is Z-invertible matrix lying in the isotropy group ... Moreover, each of the columns k∈Zn+r of B is a root of Z, i.e., ... Algorithms constructing the set of all such matrices B are presented in case when r = 0. We essentially use our construction of a morsification reduction map ... that reduces (up to ≈Z) the study of the set UBigr... of all connected non-negative edge-bipartite graphs Δ in UBigrD such that ... to the study of G1(n+r,Z)D-orbits in the set MorD⊆G1(n+r,Z) of all matrix morsifications of the graph D.
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