Cyclic factorizability theories

Abstract

Let r denote a finite group and R a commutative ring. Factorizability theories seek to describe similarities between the local structure of R1-modules M and N, where M and N are related by, for example, being isomorphic when tensored up with Q. In the first three chapters of this thesis, we define two families of factorizability theories, the invariance and coinvariance factorizability theories. We will consider three members of these families. We demonstrate that monomial invariance factorizability is equivalent to monomial factorizability as defined in [19]. We go on to consider the two cyclic cases. We demonstrate that the weak cyclic invariance factorizability theory is strict and is identical to the weak cyclic coinvariance factorizability theory. We also demonstrate that the strong cyclic invariance factorizability theory and the strong cyclic coinvariance factorizability theory are not identical but are equivalent. In chapters 4 and 5, we discuss C.M.M. F-functors over R. Thus we find relations which can simplify the calculation of the invariance and coinvariance factorizability theories. An index of the less well known definitions used in this thesis is included as an appendix.