Over $R = \mathbb{Z}$, the module $\mathbb{Z}/n\mathbb{Z}$ has the projective (free) resolution

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

This is the simplest non-trivial projective resolution: two terms, with the map being multiplication by $n$.

Over $\mathbb{Z}$, the module $\mathbb{Z}$ has the injective resolution

$0 \to \mathbb{Z} \hookrightarrow \mathbb{Q} \to \mathbb{Q}/\mathbb{Z} \to 0$

Here $\mathbb{Q}$ is injective (divisible abelian group), and $\mathbb{Q}/\mathbb{Z}$ is also injective (divisible). This terminates in two steps because $\mathbb{Z}$ has global dimension 1.

Over $R = k[x]$, the module $k = k[x]/(x)$ has the projective resolution

$0 \to k[x] \xrightarrow{\cdot x} k[x] \to k \to 0$

Since $k[x]$ is a PID, every module has a projective resolution of length at most 1.

Over $R = k[x_1, \ldots, x_n]$, the residue field $k = R/(x_1, \ldots, x_n)$ has the Koszul resolution

$0 \to R \to R^n \to R^{\binom{n}{2}} \to \cdots \to R^n \to R \to k \to 0$

The $i$-th term is $\bigwedge^{n-i} R^n$, a free module of rank $\binom{n}{n-i}$. This resolution has length $n$, reflecting the Krull dimension.

For a group $G$ with group ring $\mathbb{Z}[G]$, the trivial module $\mathbb{Z}$ has the bar resolution

$\cdots \to \mathbb{Z}[G^3] \to \mathbb{Z}[G^2] \to \mathbb{Z}[G] \to \mathbb{Z} \to 0$

with differentials $d_n(g_0, \ldots, g_n) = \sum_{i=0}^{n} (-1)^i (g_0, \ldots, \hat{g}_i, \ldots, g_n)$. This is a free $\mathbb{Z}[G]$-resolution, and its cohomology computes group cohomology $H^n(G, M)$.

An abelian group $I$ is injective if and only if it is divisible: for every $n \geq 1$ and every $x \in I$, there exists $y \in I$ with $ny = x$. Examples include $\mathbb{Q}$, $\mathbb{Q}/\mathbb{Z}$, and $\mathbb{Z}[p^{-1}]/\mathbb{Z}$ for any prime $p$.

In the category of abelian groups, the projective objects are exactly the free abelian groups. Every abelian group $A$ admits a surjection $\mathbb{Z}^{(S)} \twoheadrightarrow A$ from a free group, so $\mathbf{Ab}$ has enough projectives.

Every module $M$ over a Noetherian ring has an injective hull (or injective envelope) $E(M)$, which is the smallest injective module containing $M$ as an essential submodule. For example, $E(\mathbb{Z}/p\mathbb{Z}) = \mathbb{Z}[p^{-1}]/\mathbb{Z}$ as a $\mathbb{Z}$-module, the Prüfer $p$-group.

Over a local ring $(R, \mathfrak{m}, k)$, a minimal free resolution of a finitely generated module $M$ is one where each differential satisfies $d(P_n) \subseteq \mathfrak{m} P_{n-1}$. The ranks of the free modules are the Betti numbers $\beta_n(M) = \dim_k \mathrm{Tor}_n^R(M, k)$. Minimal resolutions are unique up to isomorphism.

The category $\mathbf{Sh}(X)$ of sheaves on a general topological space $X$ typically has no nonzero projective objects. However, it always has enough injectives (via Godement resolution). This is why sheaf cohomology uses injective resolutions rather than projective ones.

Given a short exact sequence $0 \to A \to B \to C \to 0$ with projective resolutions $P_\bullet \to A$ and $Q_\bullet \to C$, the horseshoe lemma constructs a projective resolution of $B$ of the form $P_\bullet \oplus Q_\bullet$ (as graded modules, with a modified differential). This is essential for the long exact sequence of derived functors.

A resolution $0 \to A \to F^0 \to F^1 \to \cdots$ where each $F^n$ is $F$-acyclic (meaning $R^iF(F^n) = 0$ for $i > 0$) suffices to compute $R^nF(A)$. For example, flasque sheaves are $\Gamma$-acyclic, so the Godement resolution (which consists of flasque sheaves) computes sheaf cohomology without being an injective resolution.