Derived Functors - Core Definitions

Derived functors measure the failure of a functor to be exact, providing a systematic way to extend non-exact functors to exact sequences.

Definition4.1Left Derived Functor

Let $F: \mathcal{A} \to \mathcal{B}$ be a right exact additive functor between abelian categories. For an object $M \in \mathcal{A}$ , choose a projective resolution: $\cdots \to P_2 \to P_1 \to P_0 \to M \to 0$

The left derived functors $L_nF$ are defined by: $L_nF(M) = H_n(F(P_\bullet))$

where $F(P_\bullet)$ is the chain complex obtained by applying $F$ to the resolution.

Remark

The definition is independent of the choice of projective resolution, up to canonical isomorphism. This follows from the comparison theorem for resolutions.

Definition4.2Right Derived Functor

Let $F: \mathcal{A} \to \mathcal{B}$ be a left exact additive functor. For an object $M \in \mathcal{A}$ , choose an injective resolution: $0 \to M \to I^0 \to I^1 \to I^2 \to \cdots$

The right derived functors $R^nF$ are defined by: $R^nF(M) = H^n(F(I^\bullet))$

where $H^n$ denotes cohomology (kernel mod image at degree $n$ ).

ExampleExt and Tor

The most important derived functors are:

$\text{Ext}^n_R(M, N) = R^n \text{Hom}_R(M, -)(N)$ (right derived functor of Hom)
$\text{Tor}_n^R(M, N) = L_n(M \otimes_R -)(N)$ (left derived functor of tensor product)

Theorem4.3Independence of Resolution

If $P_\bullet \to M$ and $Q_\bullet \to M$ are two projective resolutions of $M$ , then for any right exact functor $F$ : $H_n(F(P_\bullet)) \cong H_n(F(Q_\bullet))$

The isomorphism is natural and independent of the choice of comparison map.

Proof

By the comparison theorem, there exists a chain map $f: P_\bullet \to Q_\bullet$ lifting the identity on $M$ . Moreover, any two such liftings are chain homotopic. Since $F$ preserves chain homotopy equivalence (up to homotopy), the induced maps on homology are isomorphisms.

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Definition4.4Universal δ-Functor

A δ-functor is a sequence of functors $T^n: \mathcal{A} \to \mathcal{B}$ together with connecting homomorphisms $\delta: T^n(C) \to T^{n+1}(A)$ for each short exact sequence $0 \to A \to B \to C \to 0$ , satisfying naturality and exactness conditions.

A δ-functor is universal if it is the unique (up to unique isomorphism) effaceable δ-functor with the same $T^0$ .

Remark

Right derived functors form universal δ-functors. This universality property characterizes them uniquely and provides powerful computational tools.