Chain Complexes and Exact Sequences

Understanding how to measure the "holes" in mathematical structures—from topological spaces to algebraic systems—requires precise tools. Chain complexes and exact sequences provide the fundamental language for this task, forming the backbone of homological algebra with profound applications in algebraic topology and commutative algebra. By studying sequences of homomorphisms and their failures of exactness, you gain a powerful framework for computing invariants that classify and relate complex objects.

From Exact Sequences to Homology

An exact sequence is a sequence of module homomorphisms (or, more generally, homomorphisms in an abelian category) where the image of one map equals the kernel of the next. A short exact sequence, denoted $$0\to A\stackrel{f}{\to }B\stackrel{g}{\to }C\to 0,$$ encapsulates the idea that $B$ is built from $A$ and $C$ in a structured way: $f$ is injective, $g$ is surjective, and $\mathrm{im}(f)=\ker (g)$. Exactness means there is no "loss" or "gain" of information between consecutive steps.

A chain complex is a sequence where this condition is relaxed. It is a collection of modules $\{C_n\}$ connected by boundary homomorphisms ${\partial }_n:C_n\to C_{n-1}$ such that composing two consecutive maps yields zero: ${\partial }_{n-1}\circ {\partial }_n=0$. We denote a complex as $C_{\bullet }=\cdots \to C_{n+1}\stackrel{{\partial }_{n+1}}{\to }{C}_n\stackrel{{\partial }_n}{\to }{C}_{n-1}\to \cdots$. The condition ${\partial }^2=0$ ensures $\mathrm{im}({\partial }_{n+1})\subseteq \ker ({\partial }_n)$.

The failure of a chain complex to be exact is precisely measured by its homology. For each index $n$, the $n$-th homology group is defined as the quotient: $$H_n(C_{\bullet })=\frac{\ker ({\partial }_n)}{\mathrm{im}({\partial }_{n+1})}.$$ Elements of the kernel, called cycles, are those mapped to zero by ${\partial }_n$. Elements of the image, called boundaries, are those that come from a higher dimension. Homology $H_n$ is the group of cycles modulo boundaries; it is trivial (zero) if and only if the sequence is exact at $C_n$. Thus, nontrivial homology classes represent "holes" or structural features that are not filled in.

The Long Exact Sequence in Homology

A cornerstone result arises when you have a short exact sequence of chain complexes. A morphism of chain complexes induces maps between their homology groups. If you have a short exact sequence of complexes: $$0\to A_{\bullet }\stackrel{i}{\to }{B}_{\bullet }\stackrel{p}{\to }{C}_{\bullet }\to 0,$$ where $i$ and $p$ are chain maps and the sequence is exact at every module level, then this situation gives rise to a long exact sequence in homology: $$\cdots \to H_n(A_{\bullet })\stackrel{i_{\ast }}{\to }{H}_n(B_{\bullet })\stackrel{p_{\ast }}{\to }{H}_n(C_{\bullet })\stackrel{\delta }{\to }{H}_{n-1}(A_{\bullet })\to \cdots .$$

The maps $i_{\ast }$ and $p_{\ast }$ are the induced homomorphisms on homology. The crucial new map, $\delta$, is the connecting homomorphism. Its construction is both subtle and natural. Given a cycle $c\in C_n$, the surjectivity of $p$ lets you lift it to an element $b\in B_n$. Applying the boundary map $\partial$ to $b$ gives an element in $B_{n-1}$ that, because $p(\partial b)=\partial (p(b))=\partial c=0$, must actually lie in the kernel of $p$. By exactness, it comes from a unique element $a\in A_{n-1}$. This element $a$ is a cycle in $A_{\bullet }$, and its homology class defines $\delta ([c])=[a]$.

The long exact sequence is a powerful computational tool. It allows you to compute the homology of a complex $B_{\bullet }$ if you know the homology of a subcomplex $A_{\bullet }$ and the quotient complex $C_{\bullet }$, by analyzing the relationships woven by the connecting maps.

Construction and Naturality of the Connecting Homomorphism

Understanding the connecting homomorphism $\delta$ is key to wielding long exact sequences effectively. Its definition can be summarized in a commutative diagram chase, a standard proof technique in homological algebra. The process is algorithmic:

1. Start: Take a homology class $[c]\in H_n(C_{\bullet })$, represented by a cycle $c\in C_n$ with ${\partial }_n^C(c)=0$.
2. Lift: Since $p_n:B_n\to C_n$ is surjective, choose a lift $b\in B_n$ such that $p_n(b)=c$.
3. Apply Boundary: Compute ${\partial }_n^B(b)\in B_{n-1}$.
4. Push Down: Observe $p_{n-1}({\partial }_n^B(b))={\partial }_n^C(p_n(b))={\partial }_n^C(c)=0$. Thus, ${\partial }_n^B(b)\in \ker (p_{n-1})$.
5. Pull Back: By exactness at $B_{n-1}$, $\ker (p_{n-1})=\mathrm{im}(i_{n-1})$. Therefore, there exists a unique $a\in A_{n-1}$ such that $i_{n-1}(a)={\partial }_n^B(b)$.
6. Verify Cycle: Check that $a$ is a cycle in $A_{\bullet }$: $i_{n-2}({\partial }_{n-1}^A(a))={\partial }_{n-1}^B(i_{n-1}(a))={\partial }_{n-1}^B({\partial }_n^B(b))=0$, and since $i_{n-2}$ is injective, ${\partial }_{n-1}^A(a)=0$.
7. Define: Set $\delta ([c])=[a]\in H_{n-1}(A_{\bullet })$.

This construction is natural: if you have a morphism of short exact sequences of complexes, the induced map on homology groups commutes with the connecting homomorphisms. This property makes $\delta$ a well-defined, functorial tool, essential for more advanced arguments.

Applications in Topology and Algebra

The utility of these concepts is vast. In algebraic topology, chain complexes are built from geometric data. For a topological space $X$, the singular chain complex $C_{\bullet }(X)$ has $C_n(X)$ as the free abelian group on continuous maps from the standard $n$-simplex into $X$. Its homology $H_n(X)$ is the singular homology, a topological invariant. The long exact sequence appears, for example, in the relative homology of a pair $(X,A)$ (a space $X$ and a subspace $A$), yielding the sequence: $$\cdots \to H_n(A)\to H_n(X)\to H_n(X,A)\stackrel{\delta }{\to }{H}_{n-1}(A)\to \cdots .$$ This sequence is instrumental in proving excision theorems and calculating homology groups via Mayer-Vietoris sequences.

In commutative algebra, chain complexes model resolutions of modules. A free resolution of a module $M$ is an exact sequence $$\cdots \to F_2\to F_1\to F_0\to M\to 0,$$ where the $F_i$ are free modules. Truncating the $M$ gives a chain complex whose homology at $F_0$ is $M$, and elsewhere is zero. When the resolution is not free but flat or projective, its failure to be exact in other positions defines derived functors like $\mathrm{Tor}$ and $\mathrm{Ext}$. For an $R$-module $M$ and a short exact sequence of $R$-modules $0\to N^{\prime }\to N\to N^{\prime \prime }\to 0$, you obtain long exact sequences for these functors, such as: $$\cdots \to {\mathrm{Tor}}_n^R(M,N^{\prime })\to {\mathrm{Tor}}_n^R(M,N)\to {\mathrm{Tor}}_n^R(M,N^{\prime \prime })\to {\mathrm{Tor}}_{n-1}^R(M,N^{\prime })\to \cdots .$$ These sequences are vital for comparing modules and understanding their structure over a ring.

Common Pitfalls

1. Confusing the Level of Exactness: A common error is to apply the definition of a short exact sequence of modules to a diagram of chain complexes without checking exactness at each module degree. Remember, a short exact sequence of complexes $0\to A_{\bullet }\to B_{\bullet }\to C_{\bullet }\to 0$ means that for every integer $n$, the sequence $0\to A_n\to B_n\to C_n\to 0$ is exact. The induced long exact sequence is in homology, not at the chain level.

1. Misunderstanding the Connecting Homomorphism's Domain: The connecting map $\delta$ goes from the homology of the quotient complex to the homology of the subcomplex, but it decreases the degree: $\delta :H_n(C_{\bullet })\to H_{n-1}(A_{\bullet })$. Forgetting this index shift leads to incorrect diagrams and computations. A useful mnemonic is that $\delta$ "closes the loop" back to the subcomplex, stepping down one dimension.

1. Overlooking Naturality in Applications: When using the long exact sequence for computations, it's tempting to treat $\delta$ as an abstract arrow. However, its naturality property is what allows you to compare sequences from different spaces or modules via induced maps. Ignoring this can make proofs non-functorial and messy. Always consider if a morphism between two setups induces a commutative ladder of long exact sequences.

1. Assuming All Lifts are Cycles: In the construction of $\delta$, you lift a cycle $c\in C_n$ to some $b\in B_n$. This element $b$ is almost never a cycle itself. Its boundary is precisely what gets mapped via $i$ to define the class in $H_{n-1}(A_{\bullet })$. The fact that $\partial b\ne 0$ is the whole point; if $b$ were a cycle, then $\delta ([c])$ would be zero.

Summary

• A chain complex is a sequence where composing consecutive boundary maps gives zero (${\partial }^2=0$). Its homology groups $H_n=\ker {\partial }_n/\mathrm{im}{\partial }_{n+1}$ measure the failure of exactness at each stage.
• A short exact sequence of chain complexes induces a long exact sequence in homology, featuring the crucial connecting homomorphism $\delta$. This $\delta$ is constructed via a diagram chase: lift a cycle, apply the boundary, and pull back through the injection.
• The connecting homomorphism is natural, meaning it commutes with morphisms between different short exact sequences, making it a powerful functorial tool.
• These structures are foundational in algebraic topology, where they define homology theories and yield computational tools like the relative homology long exact sequence.
• In commutative algebra, they underpin the theory of derived functors (Tor, Ext), providing long exact sequences that analyze module properties and relationships over a ring.