Ext and Tor - Examples and Constructions

Explicit computations of Ext and Tor reveal patterns and provide insight into module structure.

ExampleExt over $\mathbb{Z}$

For abelian groups (i.e., $\mathbb{Z}$ -modules):

$\text{Ext}^1_{\mathbb{Z}}(\mathbb{Z}/n\mathbb{Z}, M) \cong M/nM$ (quotient by $n$ -multiplication)
$\text{Ext}^n_{\mathbb{Z}}(M, N) = 0$ for $n \geq 2$ (since $\text{gl.dim}(\mathbb{Z}) = 1$ )
$\text{Ext}^1_{\mathbb{Z}}(\mathbb{Q}, M) = 0$ for any $M$ (since $\mathbb{Q}$ is flat)

ExampleTor over $\mathbb{Z}$

For abelian groups:

$\text{Tor}_1^{\mathbb{Z}}(\mathbb{Z}/m\mathbb{Z}, \mathbb{Z}/n\mathbb{Z}) \cong \mathbb{Z}/\gcd(m,n)\mathbb{Z}$
$\text{Tor}_n^{\mathbb{Z}}(M, N) = 0$ for $n \geq 2$
$\text{Tor}_1^{\mathbb{Z}}(\mathbb{Q}/\mathbb{Z}, M) \cong M_{\text{tors}}$ (torsion subgroup)

Definition5.11Universal Coefficient Theorem (Homology)

For a chain complex $C_\bullet$ of free abelian groups and any abelian group $G$ : $0 \to H_n(C_\bullet) \otimes G \to H_n(C_\bullet \otimes G) \to \text{Tor}_1(H_{n-1}(C_\bullet), G) \to 0$

This sequence splits (non-naturally).

ExampleExt over Polynomial Rings

Over $k[x]$ where $k$ is a field:

$\text{Ext}^n_{k[x]}(k, k) \cong k$ for all $n \geq 0$
$\text{Ext}^1_{k[x]}(k[x]/(x^2), k[x]/(x)) \cong k$

Theorem5.12Flatness and Tor

An $R$ -module $M$ is flat if and only if $\text{Tor}_1^R(M, R/I) = 0$ for every finitely generated ideal $I \subseteq R$ .

ExampleLocal Cohomology via Ext

For a Noetherian ring $R$ and ideal $I$ , the local cohomology modules can be expressed as: $H^n_I(M) \cong \varinjlim_k \text{Ext}^n_R(R/I^k, M)$

This connects local cohomology to Ext groups.

Definition5.13Ext and Deformation Theory

In deformation theory, $\text{Ext}^1$ classifies first-order deformations while $\text{Ext}^2$ measures obstructions. For an algebra $A$ over $k$ : $\text{Ext}^1_{A^e}(A, A)$ parametrizes infinitesimal deformations of $A$ .

Remark

Ext and Tor appear throughout mathematics: in algebraic geometry (sheaf Ext), algebraic topology (Tor in homology), representation theory (extension groups), and commutative algebra (depth and dimension theory).