The algebra of Green and Mackey functors

Abstract

We investigate various Green and Mackey functor analogs of concepts from the theory of rings and modules. In particular, we consider ideals, chain conditions, Krull dimension, decomposition theorems and completion for these functors. We characterize the Jacobson radical and the prime and maximal ideals of an arbitrary Green functor A. We prove various properties of these ideals. We also investigate the Krull dimension of a commutative Green functor. We analyze the Green and Mackey functors satisfying various chain conditions. For left-modules over Green functors A satisfying a certain noetherian-like condition we study the analog of the tertiary decomposition theorem. For the case when A is commutative we study the analog of the primary decomposition theorem. We also give induction theorems for various special types of Green and Mackey functors such as, prime and simple Green functors A, simple left-A-modules, cotertiary and coprimary left-A-modules. We end with an induction theory for the completion of a Green functor in a left ideal. This work generalizes most of the major topics from classical algebra to the category of Green and Mackey functors.