Revista Matemática Iberoamericana

Noncommutative algebraic geometry

Olav A. Laudal

Full-text: Open access

Abstract

The need for a noncommutative algebraic geometry is apparent 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 the closure of another, the orbit space cannot, in a natural way, be given a scheme structure. In this paper we shall show that one may overcome these difficulties 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 proposed by the author in \cite{Laudal2002}. More generally, the geometry in the theory is represented by a {\it swarm}, i.e. a diagram (finite or infinite) of objects (and if one wants, arrows) in a given $k$-linear Abelian category ($k$ a field), satisfying some reasonable conditions. The noncommutative deformation theory refered to above, permits the construction of a presheaf of associative $k$-algebras, locally {\it parametrizing} the diagram. It is shown that this theory, in a natural way, generalizes the classical scheme theory. Moreover it provides a promising framework for treating problems of invariant theory and moduli problems. In particular it is shown that many moduli spaces in classical algebraic geometry are commutativizations of noncommutative schemes containing additional information.

Article information

Source
Rev. Mat. Iberoamericana Volume 19, Number 2 (2003), 509-580.

Dates
First available in Project Euclid: 8 September 2003

Permanent link to this document
http://projecteuclid.org/euclid.rmi/1063050166

Mathematical Reviews number (MathSciNet)
MR2023198

Zentralblatt MATH identifier
02109608

Subjects
Primary: 14A22: Noncommutative algebraic geometry [See also 16S38] 16E 16D90: Module categories [See also 16Gxx, 16S90]; module theory in a category-theoretic context; Morita equivalence and duality 16G 13D

Keywords
modules noncommutative deformations of modules Massey products swarm of modules algebra of observables noncommutative schemes invariant theory moduli spaces

Citation

Laudal, Olav A. Noncommutative algebraic geometry. Rev. Mat. Iberoamericana 19 (2003), no. 2, 509--580. http://projecteuclid.org/euclid.rmi/1063050166.


Export citation

References

  • Artin, M.: On Azumaya Algebras and Finite Dimensional Representations of Rings. J. Algebra 11 (1969), 532-563.
  • Artin, M., Tate, J. and Van den Bergh, M.: Some algebras associated to automorphisms of elliptic curves. In The Grothendieck Festschrift, Vol 1, 33-85. Progr. Math. 86, Birkhäuser, Boston, 1990.
  • Connes, A.: Géometrie Noncommutative. InterEditions, 1990.
  • Formanek, E.: The polynomial identities and invariants of $n\times n$ matrices. CBMS Regional Conference Series in Mathematics 78. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1991.
  • Ile, R.: Noncommutative deformations. $ab\neq ba$. Master Thesis, University of Oslo, 1990.
  • Laudal, O. A.: Sur les limites projectives et inductives. Ann. Sci. École Norm. Sup. (3) 82 (1965), 241-296.
  • Laudal, O. A.: Formal moduli of algebraic structures. Lecture Notes in Math. 754, Springer Verlag, 1979.
  • Laudal, O. A.: Matric Massey products and formal moduli. In Algebra, Algebraic Topology and their interactions (Roos, J. E., ed.), 218-240. Lecture Notes in Math. 1183, Springer Verlag, 1986.
  • Laudal, O. A. and Pfister, G.: Local moduli and singularities. Lecture Notes in Math. 1310, Springer Verlag, 1988.
  • Laudal, O. A.: Noncommutative deformations of modules. The Roos Festschrift volume (2). Homology Homotopy Appl. 4 (2002), no. 2, part 2, 357-396.
  • Laudal, O. A.: A generalized Burnside theorem. Preprint Series vol. 42, Institute of Mathematics, University of Oslo, 1995.
  • Laudal, O. A.: Noncommutative Algebraic Geometry. Max-Planck-Institut für Mathematik, Preprint Series 2000 (115).
  • Manin, Y. I.: Topics in Noncommutative Geometry. Princeton University Press, 1991.
  • Procesi, C.: Non-commutative affine rings. Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. I (8) 8 (1967), 237-255.
  • Romstad, T.: Non-commutative Algebraic Geometry Applied to Invariant Theory of Finite Group Actions. Cand. Scient. Thesis. Institute of Mathematics, University of Oslo, 2000.
  • Rosenberg, A. L.: Noncommutative algebraic geometry and representations of quantized algebra. Mathematics and its applications 330, Klüwer, 1995.
  • Schlessinger, M.: Functors of Artin rings. Trans. Amer. Math. Soc. 130 (1968), 208-222.
  • Siqveland, A.: Matric Massey products and formal moduli. Thesis, Institute of Mathematics, University of Oslo, 1995.
  • Siqveland, A.: The Noncommutative Moduli of Rank 3 Endomorphisms. Report Series No. 26, Buskerud College, Norway, 2001.
  • Van Oystaeyen, F. and Verschoren A.: Non-commutative Algebraic Geometry. An Introduction. Lecture Notes in Math. 887, Springer-Verlag, 1981.