Cofinality Properties of Categories of Chain Complexes

Abstract: This thesis treats a family of categories, the chain categories of an A-module M, and functors indexed by them. Among the chain categories are two classical constructions; the category of finitely generated projective Amodules, and the category of finitely generated free A-modules, here denoted by P0(0) and Sing(0) respectively. The focus of this thesis is on how to construct homotopy colimits of functors indexed by chain categories, and taking values in non-negative chain complexes of A-modules.One consequence of Lazard’s theorem is that if M is flat, then all functors over Sing(M) are flat; that is, the homotopy colimits of these functors are weakly equivalent to the ordinairy colimits. A motivating question has been to understand when functors over Sing(M) are flat for non-flat M. In particular, when the forgetful functor UM is flat. One of the results obtain is that if A is Noetherian, then UM is flat over many chain categories, and this property is independent of M. In contrast, if A is commutative, then the pointwise tensor product UM UM is defined, and this is not a flat functor in general, even if UM is flat.The key notion used to study these questions is that of a cofinal functor. Among the main results are the cofinality of various inclusion functors among the chain categories themselves, and the existence, construction and classification of cofinal simplicial objects in P0(M) and Sing(M). Also, a method to construct flat resolutions of functors indexed by P0 and taking values in A-modules is developed (but applicability of this construction depends on severe restrictions on M). These methods are used to compute the homotopy colimits of several functors defined over various chain categories.