Homological Algebra

Introduction

These notes provide an introduction to homological algebra and the category theory that underpins its modern structure. Central to homological algebra is the notion of a chain complex, which is to say a collection of abelian groups $C_{i}$ , with $C_{i}$ said to be “in degree $i$ ”, and differentials $d_{i} : C_{i} \to C_{i - 1}$ for $i \in ℤ$ , with the property that the composition of two consecutive differentials is trivial, i.e., $d_{i - 1} \circ d_{i} = 0$ . We may visualize this as

\dots \to C_{2} \to_{}^{d_{2}} C_{1} \to_{}^{d_{1}} C_{0} \to_{}^{d_{0}} C_{- 1} \to_{}^{d_{- 1}} C_{2} \to \dots .

For $j \in ℤ$ , the $j$ th homology group of the complex $C_{\cdot} = (C_{i}, d_{i})$ is the abelian group

H_{j} (C) = \frac{\ker d_{j}}{im d_{j + 1}} .

We do not feel it hyperbole to say that the introduction of these simple definitions led to a revolution in how much of mathematics is done.

Remarks 0.0.1. §

a.: Often, it is more convenient to have the differentials increase, rather than decrease, the degree by $1$ . In this case, we use subscripts instead of superscripts and obtain what are known as cochain complexes and cohomology groups.
b.: In many cases of interest, chain and cochain complexes are trivial in negative degree or sufficiently large positive degree, and are taken to be $0$ without explanation.
c.: More generally, chain complexes of abelian groups can be replaced by chain complexes of objects in an abelian category, which is a sort of category modeled on that of abelian groups. As well shall later discuss in depth, a category is a collection of objects and morphisms between them, such as abelian groups and homomorphisms.

The notion of homology first arose in topology. In a sense, the homology of a manifold had long been studied through invariants, prior to the introduction of homology groups in the mid-1920s. For instance, take the Betti number, introduced in 1871, but now defined as the rank of the first homology group of a manifold. Or take the work of Poincaré in 1899, in which he introduced a duality theorem that we now view as an isomorphism between homology and cohomology groups of closed oriented manifolds. After a suggestion of Emmy Noether in 1925, mathematicians began to study the homology groups themselves, and the names of the mathematicians who embarked upon this study in the ensuing decade now decorate some of the most fundamental theorems in algebraic topology: Hopf, Mayer, Vietoris, Alexander, Alexandroff, Čech, Lefschetz, and so on. Many different chain complexes were developed with homology groups that are isomorphic for sufficiently nice spaces to what we now consider the homology groups.

Example 0.0.2. §

We imagine the sphere as divided into two closed disks $b_{1}$ and $b_{2}$ , connected along the equator. Each of these disks is given an orientation such that the orientations of their boundary circles are opposite to each other. We next cover the circle with two $1$ -disks, or edges $e_{1}$ and $e_{2}$ , and again we give each an orientation agreeing with the orientation arising from $b_{1}$ . Finally, we have two vertices $v_{1}$ and $v_{2}$ that are the boundary of the edges. We think of each disk, edge, and vertex as contributing one generator to a free abelian group in degree the dimension that yields a complex $C$ :

0 \to ℤ b_{1} \oplus ℤ b_{2} \to_{}^{d_{1}} ℤ e_{1} \oplus ℤ e_{2} \to_{}^{d_{0}} ℤ v_{1} \oplus ℤ v_{2} \to 0 .

The boundary maps are given by the orientations of the boundaries of the corresponding generators. That is, we have

\begin{array}{rcl} d_{1} (b_{1}) = e_{1} - e_{2} = - d_{1} (b_{2}) & and & d_{0} (e_{1}) = v_{1} - v_{2} = - d_{0} (e_{2}) . \end{array}

One computes easily that $H_{0} (C) = ℤ$ , $H_{1} (C) = 0$ , and $H_{2} (C) = ℤ$ , and these are the homology groups of the sphere.

Homological algebra quickly passed from algebraic topology to the realm of differential topology through the work of de Rham in 1931, who introduced a cohomology theory computed by a complex of differential forms. Group cohomology, again long-studied in low degrees, was introduced in its full generality by Eilenberg and MacLane in the early 1940s. We relate its combinatorial definition.

Example 0.0.3. §

For $i \geq 0$ , the $i$ th (homogeneous) cochain group $C^{i} (G, A)$ of a group $G$ with coefficients in a module $A$ over its group ring $ℤ [G]$ is the set of maps $F : G^{i + 1} \to A$ such that

h \cdot F (g_{0}, \dots, g_{i}) = F (h g_{0}, \dots, h g_{i})

for $h \in G$ , and the $i$ th differential applied to $F$ satisfies

d^{i} (F) (g_{0}, \dots, g_{i + 1}) = \sum_{j = 0}^{i + 1} {(- 1)}^{j} F (g_{0}, \dots, g_{j - 1}, g_{j + 1}, \dots, g_{i + 1}) .

The $0$ th cohomology group of this complex is isomorphic to the set of elements of $A$ fixed by $G$ , and if the action of $G$ on $A$ is trivial, then $H^{1} (G, A)$ is isomorphic to the group of homomorphisms from $G$ to $A$ .

Group cohomology quickly found its application in field theory in the form of Galois cohomology, and through that in algebraic number theory. The main theorems of class field theory were reworked in terms of Galois cohomology by Artin and Tate in the early 1950s. Cohomology also became a crucial tool in algebraic geometry and commutative algebra through the work of many preeminent mathematicians, such as Zariski, Serre, and Grothendieck.

In the mid-1950s, a book of Cartan and Eilenberg advanced the field of homological algebra, and set it in line with category theory, with their introduction of derived functors through the use of projective and injective resolutions. The work of Leray had already introduced spectral sequences, which could be used to relate compositions of two derived functors to a third derived functor. In the late 1950s and 1960s, Grothendieck introduced abelian categories, and his more general viewpoint cemented category theory as the the foundation of homological algebra. Grothendieck also introduced the notion of a derived category, in which the chain complexes themselves, rather than the cohomology groups, play the central role.

Today, techniques in homological algebra continue to be developed and refined. Homological algebra stands an essential tool for mathematicians working in most any area of algebra, geometry, or topology. It would be a far-too-arduous and not necessarily enlightening task for this author to enumerate all of the various homology and cohomology theories in use at this time.

These notes will focus on the abstract foundation of homological algebra. We begin with the basics of category theory and abelian categories. We then develop the basic notions of chain complexes, injective and projective resolutions, and derived functors, following that with a treatment of spectral sequences. Finally, we will turn to derived categories. We hope to provide plenty of examples throughout, though we do not expect to develop specific cohomology theories in great depth.

Remark 0.0.4. §

The current version is missing important topics in several places. It is also unproofread and so likely far from error-free.

Contents

Introduction

Remarks 0.0.1. §

Example 0.0.2. §

Example 0.0.3. §

Remark 0.0.4. §

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