$\mathrm{Ext}^0(A, B) = \mathrm{Hom}(A, B)$. This follows from $R^0\mathrm{Hom}(A, -) = \mathrm{Hom}(A, -)$ since $\mathrm{Hom}(A, -)$ is left exact.

$\mathrm{Ext}^1_R(A, B)$ classifies short exact sequences $0 \to B \to E \to A \to 0$ up to equivalence. The zero element corresponds to the split extension $E = A \oplus B$. For abelian groups: $\mathrm{Ext}^1_{\mathbb{Z}}(\mathbb{Z}/n, \mathbb{Z}/m) \cong \mathbb{Z}/\gcd(n, m)$.

Over $\mathbb{Z}$: $\mathrm{Ext}^n_{\mathbb{Z}}(A, B) = 0$ for all $n \geq 2$ (since $\mathbb{Z}$ has global dimension 1). $\mathrm{Ext}^1_{\mathbb{Z}}(\mathbb{Z}/n, B) \cong B/nB$. In particular, $\mathrm{Ext}^1_{\mathbb{Z}}(\mathbb{Z}/2, \mathbb{Z}) \cong \mathbb{Z}/2$, reflecting the non-split extension $0 \to \mathbb{Z} \xrightarrow{2} \mathbb{Z} \to \mathbb{Z}/2 \to 0$.

Over a field $k$: $\mathrm{Ext}^n_k(V, W) = 0$ for all $n \geq 1$ and all vector spaces $V, W$. Every short exact sequence of vector spaces splits, so there are no non-trivial extensions. This reflects $\mathrm{gl.dim}(k) = 0$.

Over $R = k[x]/(x^2)$ (dual numbers), $\mathrm{Ext}^n_R(k, k) \cong k$ for all $n \geq 0$. The periodic projective resolution $\cdots \xrightarrow{x} R \xrightarrow{x} R \to k \to 0$ gives $\mathrm{Hom}_R(R, k) \cong k$ in each degree. This ring has infinite global dimension.

$H^n(G, M) = \mathrm{Ext}^n_{\mathbb{Z}[G]}(\mathbb{Z}, M)$. The bar resolution of $\mathbb{Z}$ as a $\mathbb{Z}[G]$-module provides the standard complex computing group cohomology. For $G = \mathbb{Z}/n$, $H^{2k}(G, \mathbb{Z}) = \mathbb{Z}/n$ and $H^{2k+1}(G, \mathbb{Z}) = 0$ for $k \geq 1$.

For sheaves on $X$: $\mathrm{Ext}^n_{\mathbf{Sh}(X)}(\mathbb{Z}_X, \mathcal{F}) \cong H^n(X, \mathcal{F})$. Sheaf cohomology is Ext from the constant sheaf. More generally, $\mathrm{Ext}^n(\mathcal{E}, \mathcal{F})$ for coherent sheaves on a scheme governs deformations of morphisms $\mathcal{E} \to \mathcal{F}$.

On a smooth projective variety $X$ of dimension $n$, Serre duality gives $\mathrm{Ext}^i(\mathcal{E}, \mathcal{F}) \cong \mathrm{Ext}^{n-i}(\mathcal{F}, \mathcal{E} \otimes \omega_X)^*$. For a curve ($n = 1$): $\mathrm{Ext}^1(\mathcal{O}_X, \mathcal{L}) \cong H^0(X, \mathcal{L}^{-1} \otimes \omega_X)^*$ by Serre duality.

$\mathrm{Ext}^1(\mathcal{E}, \mathcal{E})$ is the tangent space to the moduli of deformations of a coherent sheaf $\mathcal{E}$. $\mathrm{Ext}^2(\mathcal{E}, \mathcal{E})$ contains the obstructions to deformation. If $\mathrm{Ext}^2 = 0$, the moduli space is smooth at $\mathcal{E}$.

In the derived category, $\mathrm{Ext}^n(A, B) = \mathrm{Hom}_{D(\mathcal{A})}(A, B[n])$. This gives a single formula unifying all Ext groups. The composition $\mathrm{Ext}^m(B, C) \otimes \mathrm{Ext}^n(A, B) \to \mathrm{Ext}^{m+n}(A, C)$ is the Yoneda product, corresponding to composition of morphisms in the derived category.

The Yoneda product $\mathrm{Ext}^m(B, C) \times \mathrm{Ext}^n(A, B) \to \mathrm{Ext}^{m+n}(A, C)$ is defined by splicing extensions. For $m = n = 1$: splice $0 \to C \to E' \to B \to 0$ with $0 \to B \to E \to A \to 0$ to get $0 \to C \to E' \to E \to A \to 0 \in \mathrm{Ext}^2(A, C)$. The Ext algebra $\mathrm{Ext}^\bullet(A, A) = \bigoplus_n \mathrm{Ext}^n(A, A)$ is a graded ring.

The projective dimension of $A$ is $\mathrm{pd}(A) = \sup\{n : \mathrm{Ext}^n(A, -) \neq 0\}$. For $R$-modules, $\mathrm{pd}(A) \leq d$ iff $A$ has a projective resolution of length $\leq d$. The global dimension $\mathrm{gl.dim}(R) = \sup_A \mathrm{pd}(A)$.

For a ring homomorphism $\phi : R \to S$ and $S$-modules $A, B$, there is a change-of-rings spectral sequence $\mathrm{Ext}^p_S(A, \mathrm{Ext}^q_R(S, B)) \Rightarrow \mathrm{Ext}^{p+q}_R(A, B)$. When $S = R/I$ for a regular sequence, this collapses and gives explicit formulas.