Homological Algebra Introduction

Homological algebra provides a powerful toolkit for studying algebraic structures by analyzing their "shadows" in sequences of maps. While rooted in algebraic topology, its techniques—centered on chain complexes, exact sequences, and derived functors—have become indispensable in commutative algebra, representation theory, and geometry. This field transforms problems about objects into often simpler problems about the relationships between them, allowing you to classify structures, measure obstructions, and transfer information across mathematical domains.

Chain Complexes and Homology: The Basic Machinery

At the heart of homological algebra lies the chain complex. Formally, a chain complex is a sequence of abelian groups (or modules, or objects in any abelian category) connected by homomorphisms, called boundary maps or differentials:

$$\cdots \to C_{n+1}\stackrel{d_{n+1}}{\to }{C}_n\stackrel{d_n}{\to }{C}_{n-1}\to \cdots$$

such that the composition of any two consecutive maps is zero: $d_n\circ d_{n+1}=0$ for all $n$. This condition, $d^2=0$, means the image of one map is contained in the kernel of the next: $\text{im }{d}_{n+1}\subseteq \ker d_n$.

The failure of this containment to be an equality is precisely what we measure. The homology of the complex at the $n$-th spot is defined as the quotient group:

$$H_n(C_{\bullet })=\ker d_n/\text{im }{d}_{n+1}.$$

Elements of the kernel are called cycles (they get mapped to zero), while elements of the image are called boundaries (they come from the previous level). Homology, therefore, consists of cycles modulo boundaries—the "interesting" cycles that are not filling in a higher-dimensional shape. In topology, this counts holes; in algebra, it measures the failure of exactness.

Exact Sequences and Their Implications

A special and crucial type of chain complex is an exact sequence. A sequence is exact at an object if the image of the incoming map equals the kernel of the outgoing map. If it is exact at every object, it is an exact sequence. The condition $d^2=0$ holds, but the stronger condition $\text{im}=\ker$ means the homology is zero everywhere. Exact sequences package relationships like isomorphism, injection, and surjection into a single, flowing structure.

A short exact sequence is of the form:

$$0\to A\stackrel{i}{\to }B\stackrel{p}{\to }C\to 0.$$

Exactness at $A$ means $i$ is injective (its kernel is the image of $0$, which is $0$). Exactness at $C$ means $p$ is surjective (its image is all of $C$, which is the kernel of the map to $0$). Most importantly, exactness at $B$ tells us that $\text{im }i=\ker p$. This implies that $C\cong B/i(A)$. You can think of $B$ as extending $A$ by $C$; $B$ is built from $A$ and $C$ in a potentially twisted way. Short exact sequences are the fundamental units for building and decomposing objects in homological algebra.

The Snake Lemma and Five Lemma: Diagram Chasing Tools

Working with complexes and sequences often involves commutative diagrams. Two fundamental lemmas allow you to extract new exact sequences from diagrams of existing ones.

The Snake Lemma takes a commutative diagram of two short exact sequences connected by vertical maps:

0 --> A --> B --> C --> 0
      f|    g|    h|
      v     v     v
0 --> A'--> B'--> C'--> 0

It states that if the rows are exact, then there is an induced connecting homomorphism $\delta :\ker h\to \text{coker }f$ that yields a long exact sequence in homology:

$$0\to \ker f\to \ker g\to \ker h\stackrel{\delta }{\to }\text{coker }f\to \text{coker }g\to \text{coker }h\to 0.$$

The proof involves meticulous diagram chasing, following elements around the diagram. The connecting map $\delta$ is constructed by lifting an element backwards through the surjection and then pushing it forward using the maps in the bottom row. The Snake Lemma is the primary engine for constructing long exact sequences, which are ubiquitous.

The Five Lemma is a powerful tool for showing maps are isomorphisms. Consider a commutative diagram with exact rows:

A --> B --> C --> D --> E
f|    g|    h|    i|    j|
v     v     v     v     v
A'--> B'--> C'--> D'--> E'

The lemma states that if $f,g,i,$ and $j$ are isomorphisms, then $h$ is also an isomorphism. A weaker but more commonly used variant is that if $f$ and $j$ are surjective and $g$ and $i$ are injective, then $h$ is injective; conversely, if $f$ and $j$ are injective and $g$ and $i$ are surjective, then $h$ is surjective. This lemma allows you to prove an isomorphism in the middle of a complex structure by checking conditions on the "ends."

Derived Functors: Ext and Tor

A core objective of homological algebra is to repair the failure of functors to preserve exactness. A functor is left exact if it turns a short exact sequence $0\to A\to B\to C\to 0$ into an exact sequence $0\to F(A)\to F(B)\to F(C)$. It need not preserve the surjection on the right. Right exact functors preserve the left part instead: $F(A)\to F(B)\to F(C)\to 0$ is exact.

Derived functors systematically measure this failure and extend the exact sequence. For a left exact functor $F$, we define its right derived functors $R^{n}F$. The construction uses injective resolutions—replacing an object with a long exact sequence of injective objects. The functor is applied to the resolution, the term at the start is removed, and the homology of the resulting complex defines $R^{n}F$.

Two archetypal examples are Ext and Tor. For modules over a ring, consider the functor $\text{Hom}(M,-)$. It is left exact. Its right derived functors are denoted ${\text{Ext}}^n(M,-)$. The group ${\text{Ext}}^1(M,N)$ classifies all short exact sequences $0\to N\to X\to M\to 0$ up to equivalence. It measures the ways $M$ can be extended by $N$.

Dually, the tensor product functor $M\otimes -$ is right exact. Its left derived functors are denoted ${\text{Tor}}_n(M,-)$. The group ${\text{Tor}}_1(M,N)$ detects torsion and measures the failure of tensor product to be exact. For example, if $R=\mathbb{Z}$, ${\text{Tor}}_1(\mathbb{Z}/2,\mathbb{Z}/2)\cong \mathbb{Z}/2$, reflecting the fact that tensoring the exact sequence $0\to \mathbb{Z}\stackrel{\times 2}{\to }\mathbb{Z}\to \mathbb{Z}/2\to 0$ with $\mathbb{Z}/2$ does not yield an exact sequence on the left.

Preview of Applications: Group Cohomology and Topology

Derived functors provide a unified lens for important applications. In group cohomology, let $G$ be a group. The functor of $G$-invariants, which takes a $G$-module $M$ to $M^G=\{m\in M:g\cdot m=m\forall g\in G\}$, is left exact. Its right derived functors are the group cohomology groups: $H^n(G;M)={\text{Ext}}_{\mathbb{Z}[G]}^n(\mathbb{Z},M)$. Here, $\mathbb{Z}$ is the trivial $G$-module. $H^1(G;M)$ classifies crossed homomorphisms modulo principal ones, while $H^2(G;M)$ classifies group extensions of $G$ by $M$.

In topology, homology and cohomology theories themselves are derived functors in a suitable sense. Singular cohomology $H^n(X;G)$ can be viewed as ${\text{Ext}}^n(\mathbb{Z},C_{\bullet }(X;G))$ in the category of chain complexes. More profoundly, the universal coefficient theorems for homology and cohomology are proved using the properties of $\text{Tor}$ and $\text{Ext}$, linking the homology with different coefficients. This showcases homological algebra's power to dissect and relate topological invariants.

Common Pitfalls

1. Confusing Homology with Cohomology: While formally dual, they answer different questions. Homology typically arises from chain complexes with descending indices ($C_n$), while cohomology comes from cochain complexes with ascending indices ($C^n$) where the differential increases degree. A cochain complex is still a sequence with $d^2=0$: $\cdots \to C^{n-1}\stackrel{d}{\to }{C}^n\stackrel{d}{\to }{C}^{n+1}\to \cdots$. Its cohomology is $H^n=\ker d_n/\text{im }{d}_{n-1}$. Remember, homology often relates to cycles and boundaries in a space, while cohomology often carries multiplicative structure and classifies higher-order obstructions.

1. Misapplying the Five Lemma: The most frequent error is assuming that if the outer four maps $f,g,i,j$ are isomorphisms, then the rows must be isomorphic. This is false. The lemma only concludes that the middle map $h$ is an isomorphism. The rows could be entirely different objects, but the diagram forces their middle terms to be isomorphic via $h$.

1. Ignoring the Category: Homological algebra works in any abelian category (a category with kernels, cokernels, and properties mimicking abelian groups). However, specific constructions like derived functors require enough injective or projective objects. Assuming that you can always take an injective resolution in any category is a mistake. For instance, the category of sheaves has enough injectives, which is crucial for sheaf cohomology, but constructing them is non-trivial.

1. Overlooking Naturality: The connecting homomorphism $\delta$ in the Snake Lemma and the isomorphisms provided by many theorems are natural. This means they commute with morphisms of the input data. This naturality is what allows these tools to be glued together consistently across larger mathematical structures and is key for proving functoriality of derived functors.

Summary

• Chain complexes $(C_{\bullet },d)$ with $d^2=0$ are the foundational objects, and their homology $H_n=\ker d_n/\text{im }{d}_{n+1}$ measures the failure of exactness, capturing essential structural information.
• Short exact sequences $0\to A\to B\to C\to 0$ represent $B$ as an extension of $A$ by $C$, and are the building blocks connected by powerful diagram-chasing tools like the Snake Lemma and Five Lemma.
• Derived functors, such as Ext and Tor, systematically correct the failure of functors like $\text{Hom}$ and $\otimes$ to be exact, providing long exact sequences that contain deep algebraic information.
• These techniques find major application in defining group cohomology $H^n(G;M)$ for classifying extensions and analyzing group actions, and they provide the formal underpinning for many constructions in algebraic topology.
• Successful application requires careful attention to the underlying category, the direction (homology vs. cohomology), and the naturality of the constructed maps.