Ext and Tor - Core Definitions

Ext and Tor are the fundamental derived functors in homological algebra, measuring extensions and torsion phenomena.

Definition5.1Ext Groups

For $R$ -modules $M$ and $N$ , the Ext groups are defined as the right derived functors of Hom: $\text{Ext}^n_R(M, N) = R^n \text{Hom}_R(M, -)(N)$

Concretely, choose a projective resolution $P_\bullet \to M$ : $\cdots \to P_2 \to P_1 \to P_0 \to M \to 0$

Then: $\text{Ext}^n_R(M, N) = H^n(\text{Hom}_R(P_\bullet, N))$

where $\text{Hom}_R(P_\bullet, N)$ is the cochain complex with $d^n(f) = f \circ \partial_{n+1}$ .

ExampleExt and Extensions

There is a natural bijection: $\text{Ext}^1_R(M, N) \cong \{\text{extensions of } M \text{ by } N\} / \equiv$

where an extension is a short exact sequence $0 \to N \to E \to M \to 0$ , and $\equiv$ denotes equivalence by isomorphism over $M$ and $N$ . The zero element corresponds to split extensions.

Definition5.2Tor Groups

For $R$ -modules $M$ and $N$ , the Tor groups are defined as the left derived functors of tensor product: $\text{Tor}_n^R(M, N) = L_n(M \otimes_R -)(N)$

Concretely, choose a projective resolution $P_\bullet \to M$ and compute: $\text{Tor}_n^R(M, N) = H_n(P_\bullet \otimes_R N)$

where $P_\bullet \otimes_R N$ is the chain complex with differentials $\partial_n \otimes \text{id}_N$ .

ExampleTor Measures Torsion

For $M = \mathbb{Z}/m\mathbb{Z}$ and $N = \mathbb{Z}/n\mathbb{Z}$ : $\text{Tor}_1^{\mathbb{Z}}(\mathbb{Z}/m\mathbb{Z}, \mathbb{Z}/n\mathbb{Z}) \cong \mathbb{Z}/\gcd(m,n)\mathbb{Z}$

This group measures the torsion obstruction to tensoring being exact.

Theorem5.3Ext Vanishing Criterion

For an $R$ -module $M$ , the following are equivalent:

$M$ is projective
$\text{Ext}^1_R(M, N) = 0$ for all $R$ -modules $N$
$\text{Ext}^n_R(M, N) = 0$ for all $n \geq 1$ and all $R$ -modules $N$

Theorem5.4Tor Vanishing Criterion

For an $R$ -module $M$ , the following are equivalent:

$M$ is flat
$\text{Tor}_1^R(M, N) = 0$ for all $R$ -modules $N$
$\text{Tor}_n^R(M, N) = 0$ for all $n \geq 1$ and all $R$ -modules $N$

Remark

These vanishing criteria show that Ext and Tor precisely measure the failure of modules to be projective or flat. They transform categorical properties into computable homological invariants.