The sequence $0 \to \mathbb{Z} \xrightarrow{\times n} \mathbb{Z} \xrightarrow{\pi} \mathbb{Z}/n\mathbb{Z} \to 0$ is a short exact sequence. For $n \geq 2$, it does not split: $\mathbb{Z} \not\cong \mathbb{Z} \oplus \mathbb{Z}/n\mathbb{Z}$.

The sequence $0 \to A \xrightarrow{\iota_A} A \oplus C \xrightarrow{\pi_C} C \to 0$ always splits: the inclusion $\iota_C : C \to A \oplus C$ is a section of $\pi_C$.

$0 \to \mathbb{Z}/2\mathbb{Z} \to \mathbb{Z}/4\mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} \to 0$ does not split, since $\mathbb{Z}/4\mathbb{Z} \not\cong \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$.

In $\mathbf{Vect}_k$, every short exact sequence $0 \to V \to W \to W/V \to 0$ splits. This is because every subspace has a complement. Equivalently, every module over a field is projective and injective.

$1 \to A_n \to S_n \to \mathbb{Z}/2\mathbb{Z} \to 1$ is a short exact sequence of groups (here $A_n$ is the alternating group). For $n \geq 3$, this does not split when viewed in $\mathbf{Grp}$ (though $S_n$ contains elements of order 2, the extension is not a direct product for $n \geq 3$). For $n = 2$, $S_2 \cong \mathbb{Z}/2\mathbb{Z}$ and the sequence is trivial.

On a complex manifold $X$, the exponential sequence is:

$0 \to \underline{\mathbb{Z}} \xrightarrow{2\pi i} \mathcal{O}_X \xrightarrow{\exp} \mathcal{O}_X^* \to 0$

The long exact sequence in cohomology gives $\cdots \to H^1(X, \mathcal{O}_X) \to H^1(X, \mathcal{O}_X^*) \xrightarrow{c_1} H^2(X, \mathbb{Z}) \to \cdots$, connecting line bundles to topology via the first Chern class.

On $\mathbb{P}^n$, the Euler sequence is $0 \to \Omega^1_{\mathbb{P}^n} \to \mathcal{O}(-1)^{n+1} \to \mathcal{O} \to 0$, relating the cotangent sheaf to twists of the structure sheaf. This is fundamental for computing cohomology on projective space.

A short exact sequence of chain complexes $0 \to A_\bullet \to B_\bullet \to C_\bullet \to 0$ (exact in each degree) induces the long exact sequence in cohomology via the Snake Lemma.

The extensions of $\mathbb{Z}/2\mathbb{Z}$ by $\mathbb{Z}/2\mathbb{Z}$ are classified by $\mathrm{Ext}^1_\mathbb{Z}(\mathbb{Z}/2\mathbb{Z}, \mathbb{Z}/2\mathbb{Z}) \cong \mathbb{Z}/2\mathbb{Z}$. The two classes are the split extension $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$ and the non-split extension $\mathbb{Z}/4\mathbb{Z}$.

Given a SES $0 \to A \to B \to C \to 0$ and projective resolutions $P_\bullet \to A$ and $Q_\bullet \to C$, the horseshoe lemma constructs a projective resolution $P_\bullet \oplus Q_\bullet \to B$. This is essential for computing derived functors.

A module $P$ is projective if and only if every SES $0 \to A \to B \to P \to 0$ splits. Equivalently, $\mathrm{Ext}^1(P, A) = 0$ for all $A$. Free modules are projective, and over a PID, projective modules are free.

A module $I$ is injective if and only if every SES $0 \to I \to B \to C \to 0$ splits. Equivalently, $\mathrm{Ext}^1(A, I) = 0$ for all $A$. By Baer's criterion, $I$ is injective iff every map from an ideal of $R$ to $I$ extends to $R$.