Chapter IV. Homological Algebra

Abstract

This chapter develops the rudiments of the subject of homological algebra, which is an abstraction of various ideas concerning manipulations with homology and cohomology. Sections 1–7 work in the context of good categories of modules for a ring, and Section 8 extends the discussion to abelian categories. Section 1 gives a historical overview, defines the good categories and additive functors used in most of the chapter, and gives a more detailed outline than appears in this abstract. Section 2 introduces some notions that recur throughout the chapter—complexes, chain maps, homotopies, induced maps on homology and cohomology, exact sequences, and additive functors. Additive functors that are exact or left exact or right exact play a special role in the theory. Section 3 contains the first main theorem, saying that a short exact sequence of chain or cochain complexes leads to a long exact sequence in homology or cohomology. This theorem sees repeated use throughout the chapter. Its proof is based on the Snake Lemma, which associates a connecting homomorphism to a certain kind of diagram of modules and maps and which establishes the exactness of a certain 6-term sequence of modules and maps. The section concludes with proofs of the crucial fact that the Snake Lemma and the first main theorem are functorial. Section 4 introduces projectives and injectives and proves the second main theorem, which concerns extensions of partial chain and cochain maps and also construction of homotopies for them when the complexes in question satisfy appropriate hypotheses concerning exactness and the presence of projectives or injectives. The notion of a resolution is defined in this section, and the section concludes with a discussion of split exact sequences. Section 5 introduces derived functors, which are the basic mathematical tool that takes advantage of the theory of homological algebra. Derived functors of all integer orders $\geq0$ are defined for any left exact or right exact additive functor when enough projectives or injectives are present, and they generalize homology and cohomology functors in topology, group theory, and Lie algebra theory. Section 6 implements the two theorems of Section 3 in the situation in which a left exact or right exact additive functor is applied to an exact sequence. The result is a long exact sequence of derived functor modules. It is proved that the passage from short exact sequences to long exact sequences of derived functor modules is functorial. Section 7 studies the derived functors of $\mathrm{Hom}$ and tensor product in each variable. These are called $\mathrm{Ext}$ and $\mathrm{Tor}$, and the theorem is that one obtains the same result by using the derived functor mechanism in the first variable as by using the derived functor mechanism in the second variable. Section 8 discusses the generalization of the preceding sections to abelian categories, which are abstract categories satisfying some strong axioms about the structure of morphisms and the presence of kernels and cokernels. Some generalization is needed because the theory for good categories is insufficient for the theory for sheaves, which is an essential tool in the theory of several complex variables and in algebraic geometry. Two-thirds of the section concerns the foundations, which involve unfamiliar manipulations that need to be internalized. The remaining one-third introduces an artificial definition of “member” for each object and shows that familiar manipulations with members can be used to verify equality of morphisms, commutativity of square diagrams, and exactness of sequences of objects and morphisms. The consequence is that general results for categories of modules in homological algebra requiring such verifications can readily be translated into results for general abelian categories. The method with members, however, does not provide for constructions of morphisms member by member. Thus the construction of the connecting homomorphism in the Snake Lemma needs a new proof, and that is given in a concluding example.