Noncommutative algebraic geometry. (English) Zbl 1056.14001

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 examples are given in chapters 7 and 8.
The paper is compactly written, and contains a theory and ideas that will prove essential in the years to come.


14A22 Noncommutative algebraic geometry
16S38 Rings arising from noncommutative algebraic geometry
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