Derived Functors - Examples and Constructions

Derived functors appear throughout mathematics with important computational and theoretical applications.

ExampleSheaf Cohomology

For a topological space $X$ and sheaf $\mathcal{F}$ , the sheaf cohomology groups are: $H^n(X, \mathcal{F}) = R^n\Gamma(X, \mathcal{F})$

where $\Gamma(X, -)$ is the global sections functor. These measure obstructions to extending local sections to global ones.

ExampleGroup Cohomology

For a group $G$ and $G$ -module $M$ , the group cohomology is: $H^n(G, M) = R^n(\text{Inv}_G)(M)$

where $\text{Inv}_G$ takes $G$ -invariants. Equivalently, $H^n(G, M) = \text{Ext}^n_{\mathbb{Z}[G]}(\mathbb{Z}, M)$ where $\mathbb{Z}$ has trivial $G$ -action.

Definition4.10Tor Functor

The Tor functor is the left derived functor of the tensor product: $\text{Tor}_n^R(M, N) = L_n(M \otimes_R -)(N)$

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

ExampleTor Vanishing

For modules over a principal ideal domain $R$ : $\text{Tor}_n^R(M, N) = 0 \quad \text{for all } n \geq 2$

This follows from the fact that every module over a PID has projective dimension at most 1.

Definition4.11Ext Functor

The Ext functor is the right derived functor of Hom: $\text{Ext}^n_R(M, N) = R^n \text{Hom}_R(M, -)(N)$

It can be computed using either a projective resolution of $M$ or an injective resolution of $N$ : $\text{Ext}^n_R(M, N) = H^n(\text{Hom}_R(P_\bullet, N)) = H^n(\text{Hom}_R(M, I^\bullet))$

ExampleExt and Extensions

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

where an extension is a short exact sequence $0 \to N \to E \to M \to 0$ , and the equivalence relation is given by isomorphism over $M$ and $N$ .

Theorem4.12Balance of Tor and Ext

The functors Tor and Ext are balanced: $\text{Tor}_n^R(M, N) \cong \text{Tor}_n^R(N, M)$ $\text{Ext}^n_R(M, N) \cong H^n(\text{Hom}(P_\bullet, N)) \cong H^n(\text{Hom}(M, I^\bullet))$

This allows flexible choice of which variable to resolve.

Remark

The balance property is not obvious from the definitions but follows from careful analysis of resolutions and the universal property of derived functors.