Extended Modules and Ore Extensions
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Extended Modules and Ore Extensions Viacheslav Artamonov1 William Fajardo2
· Oswaldo Lezama2 ·
Received: 21 August 2015 / Revised: 15 December 2015 / Accepted: 23 December 2015 © School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag Berlin Heidelberg 2016
Abstract In this paper, we study extended modules for a special class of Ore extensions. We will assume that R is a ring and A will denote the Ore extension A := R[x1 , . . . , xn ; σ ] for which σ is an automorphism of R, xi x j = x j xi and xi r = σ (r )xi , for every 1 ≤ i, j ≤ n. With some extra conditions over the ring R, we will prove Vaserstein’s, Quillen’s patching, Horrocks’, and Quillen–Suslin’s theorems for this type of non-commutative rings. Keywords Extended modules and rings · Quillen–Suslin’s methods · Ore extensions Mathematics Subject Classification 16U20 · 16S80 · 16N60 · 16S36
1 Introduction The study of finitely generated projective modules over commutative polynomial rings occupied the attention of many important researchers in the last decades. The class of extended modules and rings was very useful for this problem. This special class arises when we try to generalize the famous Quillen–Suslin theorem about projective modules over polynomial rings with coefficients in principal ideal domains (PIDs) to a
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Viacheslav Artamonov [email protected] Oswaldo Lezama [email protected] William Fajardo [email protected]
1
Moscow State University, Moscow, Russia
2
Universidad Nacional de Colombia, Bogotá, Colombia
123
V. Artamonov et al.
wider classes of coefficients, or yet, to the non-commutative rings of polynomial type (see [1–3,8]). In this work, we are interested in investigating extended modules and rings for some special classes of Ore extensions. If nothing contrary is assumed, we will suppose that R is a ring and A will denote the Ore extension A := R[x1 , . . . , xn ; σ ] for which σ is an automorphism of R, xi x j = x j xi and xi r = σ (r )xi , for every 1 ≤ i, j ≤ n. In some places, we will assume some extra conditions on R. For this particular type of Ore extensions, we will extend some key theorems of commutative algebra, namely, Vaserstein’s (Theorem 3.3), Quillen’s patching (Theorem 4.4), Horrocks’ (Theorem 5.1), and Quillen–Suslin’s (Theorem 5.3) theorems, to this type of non-commutative polynomial rings. In addition, we present in Theorem 2.2 a matrix description of extended modules and rings for this class of Ore extensions. Some notation is needed as well as to recall some definitions and basic facts about finitely generated projective modules. Let S be a ring. S is PSF if any finitely generated projective left S-module is stably free; we say that S is PF if any finitely generated projective left S-module is free; S is Hermite (H) if any stably free left S-module is free. The ring S is d-Hermite if any stably free left S-module of rank ≥ d is free. Note that H ∩ PSF = PF. For the Ore extensions, we are interested in this paper, the PSF, H, and d-Hermite
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