Fitting ideals and various notions of equivalence for modules

Recently there has been a lot of work on determining the Fitting ideals of arithmetic modules over groups rings, first and foremost of class groups and their Pontryagin duals. In particular, it has turned out that these Fitting ideals are usually non-principal and may be described, up to principal ideals, in terms of group-theoretical information only. The involved principal ideal factors are essentially given by values of equivariant L-functions. The present paper is not concerned with these L-functions but rather focuses on a systematic understanding of the Fitting ideals up to principal factors. To this end, we develop a certain notion of “equivalence of modules” over suitable commutative rings R. We establish that understanding the equivalence of R-modules is closely related to the classification of R-lattices. We also offer a construction of a category, inspired by derived categories, which embodies our new notion of “equivalence”.