The loop-space functor in homological algebra

Abstract

This note constitutes a sequel to [AC], the terminology and notation of which are used throughout. Its purpose is to contribute some technical devices, viz. the notion of an ideal, and that of the loop-space functor, to the study of homological algebra. The concept of an ideal of an is introduced (?1) in analogy with the familiar notion of the theory of rings. In particular the quotient of a by a two-sided ideal is defined. Of especial interest are the ideals generated by identity maps, which are studied in ?2. For an with enough projectives a functor Q, defined on the quotient of the by its projectives (?3), bears a close analogy to the loop-space functor of topology. This functor, as an operation on modules, has also been considered by P. J. Hilton and B. Eckmann [H-E]. If 3e is an with enough projectives then for 'C8, the of proper s.e.s. in 3X, a functor r is introduced (?4). Pursuing the topological analogy, r plays the role of the operation which turns inclusion maps into fibre maps. This is related to the loop-space functor by the operation (?5) that its threefold iterate is just Q for the 3C' (a fact which is known for the topological analogue). Using this result it is possible to associate to a proper s.e.s. sequences of homomorphism groups in the original modulo its projectives. Finally the Ext groups of the original are related to the homomorphism groups in the quotient by projectives (?6) by a functor which exhibits the latter as quotient groups of the former. The kernel measures the extent to which projectives fail to be injective; if they are all injective the study of Ext is reduced to that of the quotient and the functor U. As in [AC] the theory is subject to straightforward dualization which however is barely indicated. The general reference is [AC]. To this should be added [G] which had not yet appeared when [AC] was written. It should be noted that additive category in [G] is used in a sense different from the present one, and that abelian category in [G] is equivalent to the exact category of Buchsbaum and [AC]. 1. Ideals in an category. A left ideal i of an 3C is a subset of the maps of 3C with the properties: