Algebr

Chapter 10

Introducing Homological Algebra
Roughly speaking, homological algebra consists of (A) that part of algebra that is fundamental in building the foundations of algebraic topology, and (B) areas that arise naturally in studying (A).

10.1

Categories

We have now encountered many algebraic structures and maps between these structures. There are ideas that seem to occurregardless of the particular structure under consideration. Category theory focuses on principles that are common to all algebraic systems.

10.1.1

Definitions and Comments

A category C consists of objects A, B, C, . . . and morphisms f : A → B (where A and B are objects). If f : A → B and g : B → C are morphisms, we have a notion of composition, in other words, there is a morphism gf = g ◦ f : A →C, such that the following axioms are satisfied. (i) Associativity: If f : A → B, g : B → C, h : C → D, then (hg)f = h(gf ); (ii) Identity: For each object A there is a morphism 1A : A → A such that for each morphism f : A → B, we have f 1A = 1B f = f . A remark for those familiar with set theory: For each pair (A, B) of objects, the collection of morphisms f : A → B is required to be a set ratherthan a proper class. We have seen many examples: 1. Sets: The objects are sets and the morphisms are functions. 2. Groups: The objects are groups and the morphisms are group homomorphisms. 3. Rings: The objects are rings and the morphisms are ring homomorphisms. 1

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CHAPTER 10. INTRODUCING HOMOLOGICAL ALGEBRA

4. Fields: The objects are fields and the morphisms are field homomorphisms [=field monomorphisms; see (3.1.2)]. 5. R-mod: The objects are left R-modules and the morphisms are R-module homomorphisms. If we use right R-modules, the corresponding category is called mod-R. 6. Top: The objects are topological spaces and the morphisms are continuous maps. 7. Ab: The objects are abelian groups and the the morphisms are homomorphisms from one abelian group to another. A morphism f : A→ B is said to be an isomorphism if there is an inverse morphism g : B → A, that is, gf = 1A and f g = 1B . In Sets, isomorphisms are bijections, and in Top, isomorphisms are homeomorphisms. For the other examples, an isomorphism is a bijective homomorphism, as usual. In the category of sets, a function f is injective iff f (x1 ) = f (x2 ) implies x1 = x2 . But in an abstract category, we don’t haveany elements to work with; a morphism f : A → B can be regarded as simply an arrow from A to B. How do we generalize injectivity to an arbitrary category? We must give a definition that does not depend on elements of a set. Now in Sets, f is injective iff it has a left inverse; equivalently, f is left cancellable, i.e. if f h1 = f h2 , then h1 = h2 . This is exactly what we need, and a similar ideaworks for surjectivity, since f is surjective iff f is right cancellable, i.e., h1 f = h2 f implies h1 = h2 .

10.1.2

Definitions and Comments

A morphism f is said to be monic if it is left cancellable, epic if it is right cancellable. In all the categories listed in (10.1.1), a morphism f is monic iff f is injective as a mapping of sets. If f is surjective, then it is epic, but the conversecan fail. See Problems 2 and 7–10 for some of the details. In the category R-mod, the zero module {0} has the property that for any Rmodule M , there is a unique module homomorphism from M to {0} and a unique module homomorphism from {0} to M . Here is a generalization of this idea.

10.1.3

Definitions and Comments

Let A be an object in a category. If for every object B, there is a uniquemorphism from A to B, then A is said to be an initial object. If for every object B there is a unique morphism from B to A, then A is said to be a terminal object. A zero object is both initial and terminal. In the category of sets, there is only one initial object, the empty set. The terminal objects are singletons {x}, and consequently there are no zero objects. In the category of groups,...