The derived functors of a composite, from the composites of derived functors
Given F: A → B and G: B → C left exact, with F carrying injectives to G-acyclic objects, there is a spectral sequence with E2p,q = RpG ∘ RqF converging to Rp+q(G ∘ F). Nearly every named spectral sequence in the subject is a special case.
Learning objectives
- State the theorem and its acyclicity hypothesis.
- Explain why the hypothesis is needed.
- Identify the edge homomorphisms and the five-term sequence.
- Recognise the standard specialisations.
Section 01The statement
F must send injective objects of A to G-acyclic objects of B. Without it, applying G to an injective resolution of A gives a complex whose homology is not what is wanted, and the construction collapses. The hypothesis is usually verified via an adjunction: if F has an exact left adjoint, it preserves injectives.
- Take an injective resolution I• of A in A.
- Apply F to get a complex F(I•) in B, whose terms are G-acyclic by hypothesis.
- Take a Cartan–Eilenberg resolution of that complex and apply G, giving a double complex.
- Filter the double complex in the two directions.
- One filtration collapses because the terms are G-acyclic, computing R*(G ∘ F); the other gives the E2 page above.
Section 02Edge maps and the five-term sequence
The edges of the E2 page give natural maps in low degrees, and truncating gives the five-term exact sequence
which is usable without any spectral sequence machinery and is the form in which the result is most often applied.
Section 03Specialisations
| Composite | Resulting spectral sequence |
|---|---|
| Invariants under N, then under G/N | Lyndon–Hochschild–Serre for group cohomology |
| Direct image, then global sections | Leray spectral sequence of a map of spaces |
| Restriction, then Hom over the smaller ring | Change-of-rings spectral sequence |
| Sheafification, then global sections | Čech-to-derived-functor spectral sequence |
| Invariants under a Lie subalgebra, then the quotient | Hochschild–Serre for Lie algebras |
| Tensor then Hom | One route to the balance and adjunction spectral sequences |
Recognising a construction as a composite of functors immediately supplies a spectral sequence, its E2 page and its five-term sequence, with no separate work. This is the main practical reason the Grothendieck formulation is worth knowing rather than the individual cases.
ReferenceFrequently asked questions
How do I verify the acyclicity hypothesis?
Most often by adjointness: a functor with an exact left adjoint preserves injectives, and injectives are acyclic for any left exact functor. For sheaf cohomology, flasque and injective sheaves supply the acyclic class directly.
Is there a homological version?
Yes, for right exact functors and projective resolutions, with the arrows and indices reversed. The hypothesis becomes that the first functor carries projectives to acyclics for the second.
When does the sequence collapse?
When one of the two functors is exact, so that all but one row or column of the E2 page vanishes. That case reduces to the composition isomorphism for derived functors and is how most change-of-rings isomorphisms are proved.
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