Classes of extensions and resolutions

Abstract

A class of resolutions of objects of an abelian category determines a theory of derived functors if each morphism between objects extends to a morphism, unique to within homotopies, between their resolutions. This paper is primarily concerned with resolutions canonically associated with certain natural classes of extensions (E-functors), and the known examples are constructed by using pairs of adjoint functors. An inclusion between two E-functors on the same category induces natural transformations between functors derived from their associated resolutions, and other relations exist in the form of invariant exact couples. The relations simplify for the special and frequently occurring class of ‘central’ inclusions of E-functors; in particular the operations of forming satellites of a functor on the two resolutions commute. Amongst various applications the general theory provides generalizations of: results on groups of extensions of modules over Dedekind domains; the Hochschild—Serre spectral sequences in the homology theory of groups; the spectral sequences for coherent algebraic sheaves that determine Ext by means of vector bundle resolutions and affine coverings.