Noncommutative algebraic geometry
type: Item
instance of: scholarly article
title: Noncommutative algebraic geometry
zbMATH Open document ID: 1056.14001
DOI: 10.4171/RMI/360
published in: Revista Matemática Iberoamericana
publication date: 2004-10-25T00:00:00Z
full work available at URL: https://eudml.org/doc/39675
review text: In classical invariant and moduli theory, it is in general impossible to find commuting parameters parametrizing all orbits of a Lie group acting on a scheme. When one orbit is contained in another, the orbit space cannot be given a scheme structure in a natural way. In this paper these difficulties are overcome by introducing a noncommutative algebraic geometry, where affine schemes are modeled on associative algebras. The points of such an affine scheme are the simple modules of the algebra, and the local structure of the scheme at a finite family of points, is expressed in terms of a noncommutative deformation theory. The geometry in the theory is represented by a swarm, i.e. a diagram of objects in e.g. the category of \(A\)-modules, satisfying reasonable conditions. The noncommutative deformation theory permits the construction of a presheaf of associative \(k\)-algebras, locally parametrizising the diagram. Chapter 1 of the paper contains the homological preparations needed for this theory, and is also of interest in its self. The theory of \(\text{Ext}\) and Hochschild cohomology modules are considered, and also their interaction with the resolving functors of \(\varinjlim\). Chapter 2 contains the deformation theory, which in some sense is the affine part in noncommutative geometry. This comes to its full right when the formal rings are replaced by their algebraizations. Chapter 3 contains the definition of the presheaves corresponding to the regular functions in the commutative situation. It contains the definition of noncommutative schemes and the most important results of the theory. Also some nice examples are presented. Chapter 4 proves that the noncommutative scheme-theory is a good extension of classical commutative scheme-theory. In chapter 5, the theory is modeled on the category of \(A-G\)-modules, where \(G\) is a Lie-group. This is used to solve classical problems in geometric invariant theory by use of noncommutative theory in chapter 6. Several…
reviewed by: Arvid Siqveland
Mathematics Subject Classification ID: 14A22
Mathematics Subject Classification ID: 16S38
zbMATH DE Number: 2109608
zbMATH Keywords: deformation theory
zbMATH Keywords: formal methods
zbMATH Keywords: representation theory
author: Olav Arnfinn Laudal
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2109608