Finitely presented modules over Leavitt algebras.
type: Item
instance of: scholarly article
title: Finitely presented modules over Leavitt algebras.
zbMATH Open document ID: 1072.16012
author: Pere Ara
published in: Journal of Pure and Applied Algebra
publication date: 2004-08-19T00:00:00Z
review text: In this paper the author studies the structure of the Abelian category \(\text{fp}(L)\) of finitely presented left modules over a Leavitt algebra \(L\) of type \((1,n)\) over a field \(k\) by obtaining it as the quotient category of the category of finitely presented modules over the free algebra \(k\langle Y\rangle:=k\langle y_0,y_1,\dots,y_n\rangle\). A special subcategory \(\text{fp}(L)_{fl}\) of \(\text{fp}(L)\) of all finitely presented left \(L\)-modules of finite length turns out to be the category of torsion modules with respect to the localization \(L\to Q\). The author also obtains a category equivalence between \(\text{fp}(L)_{fl}\) and the quotient of an Abelian category of finitely presented left \(k\langle Y\rangle\)-modules of negative Euler characteristic. Finally, as an application, the author computes \(K_1(Q)\) by getting a formula which is analogous to the one for \(K_1\) of the universal field of fractions of \(k\langle X\rangle\).
Mathematics Subject Classification ID: 16E20
Mathematics Subject Classification ID: 16E50
Mathematics Subject Classification ID: 19A49
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 18E10
zbMATH DE Number: 2093436
zbMATH Keywords: Leavitt algebras
zbMATH Keywords: free algebras
zbMATH Keywords: finitely presented modules
zbMATH Keywords: Grothendieck groups
zbMATH Keywords: power series fields
zbMATH Keywords: Laurent series fields
zbMATH Keywords: categories of finitely presented modules
zbMATH Keywords: von Neumann regular algebras
zbMATH Keywords: universal localizations
reviewed by: Veereshwar A. Hiremath
MaRDI profile type: Publication
full work available at URL: j.jpaa.2003.12.001
OpenAlex ID: W2059379309
cites work: The exchange property for purely infinite simple rings
cites work: K1of Corner Skew Laurent Polynomial Rings and Applications
cites work: \(K_0\) of purely infinite simple regular rings
cites work: Coproducts and Some Universal Ring Constructions
cites work: Universal derivations and universal ring constructions
cites work: Q3994777
cites work: Q3708140
cites work: Q3808246
cites work: On \(n\)-simple rings.
cites work: Localization in Semifirs II
cites work: Free algebras over Bezout domains are Sylvester domains
cites work: Sylvester domains
cites work: Hermitian forms on link modules
cites work: Noncommutative rational functions and boundary links
cites work: The Cohn localization of the free group ring
cites work: Algebras over zero-dimensional rings
cites work: Modules without Invariant Basis Number
cites work: The Module Type of a Ring
cites work: Free Modules over Free Algebras and Free Group Algebras: The Schreier Technique
cites work: Lifting Idempotents and Exchange Rings
cites work: On the Abelianized multiplicative group of a universal field of fractions
cites work: Classification of nuclear \(C^*\)-algebras. Entropy in operator algebras
cites work: Q4300594
cites work: Ideals of finite codimension in free algebras and the \(fc\)-localization
cites work: Free Coverings and Modules of Boundary Links
cites work: Q3688509
cites work: NON-COMMUTATIVE CHARACTERISTIC POLYNOMIALS AND COHN LOCALIZATION
cites work: Whitehead groups of localizations and the endomorphism class group.
cites work: Invariants of boundary link cobordism
cites work: Q5294643
cites work: Q4027069
cites work: Exchange rings and decompositions of modules
DOI: 10.1016/J.JPAA.2003.12.001
description: scientific article; zbMATH DE number 2093436