Resolve, apply, take homology
Given an additive functor that is only half exact, the derived functors measure the failure. For a right exact covariant functor, resolve by projectives, apply the functor, take homology — the left derived functors. For a left exact one, resolve by injectives and take cohomology — the right derived functors. In degree 0 the original functor is recovered, and the whole family fits into long exact sequences. Grothendieck's axiomatic characterisation shows the construction is forced.
Learning objectives
- Construct left and right derived functors.
- State the defining properties in degree 0 and the long exact sequences.
- Explain the axiomatic characterisation by universal δ-functors.
- Recognise Ext and Tor as instances.
- Identify when a derived functor vanishes.
Section 01The construction
- Choose a projective resolution P• ↠ M.
- Delete M, leaving the deleted resolution.
- Apply T degreewise to obtain a complex T(P•). Additivity guarantees this is still a complex.
- Set LnT(M) = Hn(T(P•)).
- L0T(M) ≅ T(M), because T is right exact and the resolution is exact at the last two spots.
| Functor | Variance | Resolve by | Derived functors |
|---|---|---|---|
| Right exact | Covariant | Projectives | Left derived LnT |
| Left exact | Covariant | Injectives | Right derived RnT |
| Right exact | Contravariant | Injectives | Left derived |
| Left exact | Contravariant | Projectives | Right derived |
Torn(M, −) = Ln(M ⊗ −), and Extn(−, A) = RnHom(−, A). Everything proved about derived functors in general applies to both without restatement.
Section 02The defining properties
- Stage 01Degree 0L0T ≅ T for right exact T; R0T ≅ T for left exact T.
- Stage 02Long exact sequencesEvery short exact sequence of modules yields a long exact sequence in the derived functors, natural in the sequence.
- Stage 03Vanishing on projectivesLnT(P) = 0 for n ≥ 1 when P is projective; dually RnT(I) = 0 for I injective.
- Stage 04AdditivityEach LnT is itself an additive functor.
It is the property that makes dimension shifting work and, in the axiomatic treatment, the condition (effaceability) that forces uniqueness. A family of functors with long exact sequences but without it need not be the derived functors of anything.
Section 03The axiomatic characterisation
A δ-functor is a family of functors together with connecting maps making every short exact sequence produce a long exact sequence, naturally. It is universal when any morphism from its degree-0 part extends uniquely to the whole family.
Grothendieck's theorem: an effaceable δ-functor is universal, and the derived functors are effaceable. Hence the derived functors of a given functor are the unique universal δ-functor extending it.
It identifies derived functors without mentioning resolutions, so two constructions can be shown to agree by checking degree 0 and effaceability. This is the standard route to proving that group cohomology defined by the bar resolution agrees with Ext over the group ring, and that sheaf cohomology agrees with Čech cohomology under suitable hypotheses.
ReferenceFrequently asked questions
Why does the degree-0 term recover the original functor?
Because right exactness means the functor preserves the cokernel presentation of M by the last two terms of the resolution, and homology in degree 0 is exactly that cokernel. For left exact functors the dual argument gives the same at the start of an injective resolution.
Can a functor that is neither left nor right exact be derived?
Not by this construction, which uses exactness in degree 0. Half-exact functors can be handled by satellites, which is the relative theory covered separately in this stream.
Do derived functors depend on the ambient category?
They depend on which objects count as projective or injective, so yes. Computing Ext in the category of all modules and in a subcategory can give different answers, which is why relative homological algebra fixes a projective class explicitly.
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