Connecting homomorphism (boundary map) lemma

Let

0 \longrightarrow A_\bullet \xrightarrow{i} B_\bullet \xrightarrow{p} C_\bullet \longrightarrow 0

be a short exact sequence of chain complexes in an abelian category (in particular, this holds for $R$ -modules). Then there is a natural long exact sequence in homology:

\cdots \to H_n(A) \to H_n(B) \to H_n(C) \xrightarrow{\ \delta\ } H_{n-1}(A) \to H_{n-1}(B)\to \cdots

where $\delta$ is the connecting homomorphism.

The map $\delta$ is natural with respect to morphisms of short exact sequences of complexes.

Construction of the connecting map

Given a class $[c]\in H_n(C)$ , choose a cycle representative $c\in Z_n(C)$ .

Pick $b\in B_n$ with $p(b)=c$ (possible since $p$ is degreewise surjective).
Since $c$ is a cycle, $0=d_C(c)=d_C(p(b))=p(d_B(b))$ , so $d_B(b)\in \ker(p)=\operatorname{im}(i)$ .
Choose $a\in A_{n-1}$ with $i(a)=d_B(b)$ .
One checks $a$ is a cycle and that its homology class $[a]\in H_{n-1}(A)$ is independent of all choices.

Define $\delta([c]) := [a]$ .

This is the mechanism behind the connecting maps in long exact sequences from derived functors, such as Tor and Ext long exact sequences.

Examples

Example 1 (Tor: identifying the connecting map with a kernel)

In $\mathbf{Ab}$ , start with

0\to \mathbb Z \xrightarrow{\cdot n} \mathbb Z \to \mathbb Z/n \to 0

and tensor with $\mathbb Z/m$ . Because tensor is right exact, you get an exact sequence

\operatorname{Tor}_1^\mathbb Z(\mathbb Z/n,\mathbb Z/m)\xrightarrow{\ \delta\ } \mathbb Z/m \xrightarrow{\cdot n} \mathbb Z/m \to (\mathbb Z/n)\otimes(\mathbb Z/m)\to 0.

Exactness forces

\operatorname{im}(\delta)=\ker(\cdot n : \mathbb Z/m\to \mathbb Z/m),

so $\delta$ identifies $\operatorname{Tor}_1^\mathbb Z(\mathbb Z/n,\mathbb Z/m)$ with the $n$ -torsion in $\mathbb Z/m$ , which is cyclic of order $\gcd(n,m)$ .

Example 2 (Ext: boundary map from a short exact sequence in the second variable)

Given a short exact sequence $0\to N'\to N\to N''\to 0$ , applying the left exact functor $\operatorname{Hom}_R(M,-)$ produces a connecting map

\delta:\operatorname{Hom}_R(M,N'') \to \operatorname{Ext}^1_R(M,N')

appearing in the long exact Ext sequence. Concretely, a map $f:M\to N''$ defines a pullback extension of $M$ by $N'$ , and $\delta(f)$ is precisely its class in $\operatorname{Ext}^1$ (compare Ext $^1$ classifies extensions).

Example 3 (homology of a mapping cone short exact sequence)

Given a chain map $u:A_\bullet\to B_\bullet$ , there is a short exact sequence of complexes

0\to B_\bullet \to \mathrm{Cone}(u)_\bullet \to A_\bullet[-1]\to 0.

The connecting homomorphism in the associated long exact sequence recovers the map on homology induced by $u$ . This is a standard way to package “ $u_*$ ” as a boundary map.