Complexity and Multiple Complexes

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.