Quillen–Suslin theory for the special linear group

2017/07/05 by Jinwang Liu, Dongmei Li, Shexi Chen
Mathematics · #Advanced Topics in Algebra #Finite Group Theory Research #Geometric and Algebraic Topology

paper · doi:10.1080/00927872.2017.1347658

Abstract

We prove that for any valuation ring R of Krull dimension ≤1 or any commutative local ring R with Krull dimension 0 and n≥3, the special linear group SLn(R[x]) coincides with the group of elementary matrices En(R[x]), and for an arbitrary arithmetical ring R of Krull dimension ≤1 and n≥3, the group SLn(R[x])=SLn(R)⋅En(R[x]). These give partial solutions to two conjectures of Yengui.

Related

• The Hermite ring conjecture and special linear groups for valuation rings
• Quillen--Suslin Theory; Revisited over Graded Rings
• Quillen Suslin theory for algebraic fundamental group
• Structure theorem for the elementary 2 × n unimodular vector group
• K-theory and the Enriched Tits Building