Complexity and Multiple Complexes

  • Jon F. Carlson
  • Lisa Townsley
  • Luis Valeri-Elizondo
  • Mucheng Zhang
Part of the Algebras and Applications book series (AA, volume 3)

Abstract

In this chapter we introduce the notion of the complexity of a module and explore some related ideas. Complexity was first defined by Jon Alperin in the late 1970’s and it helped to motivate much of the development of the homological properties of modules. In fact, the complexity of a finitely generated kG-module M is a rather crude invariant, in that it only measures the polynomial rate of growth of a minimal projective resolution of the module. The original theorem of Alperin and Evens which extended Quillen’s Dimension Theorem to modules, was proved only in terms of complexity. Later it was extended to support varieties as well. The theorem computes the complexity of a kG-module M as the maximum of the complexities of the restrictions of M to the elementary abelian subgroups of the group G.

Keywords

Exact Sequence Prime Ideal Maximal Ideal Homogeneous Element Cohomology Ring 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer Science+Business Media Dordrecht 2003

Authors and Affiliations

  • Jon F. Carlson
    • 1
  • Lisa Townsley
    • 2
  • Luis Valeri-Elizondo
    • 3
  • Mucheng Zhang
    • 1
  1. 1.University of GeorgiaAthensUSA
  2. 2.Benedictine UniversityLisleUSA
  3. 3.Instituto de MatematicasUNAMMoreliaUSA

Personalised recommendations