One machine, measuring one kind of failure
Homological algebra begins with a single observation: many natural constructions preserve part of an exact sequence but not all of it. Hom is left exact; the tensor product is right exact; neither is exact. Rather than treat this as a defect, the subject makes the defect the object of study. Resolutions replace an object by a complex of well-behaved ones, derived functors measure exactly how much exactness was lost, and the resulting invariants — Ext and Tor — turn out to encode extensions, obstructions and topological information at once.
Learning objectives
- State precisely how Hom and tensor fail to be exact.
- Explain what a resolution is for and why the choice does not matter.
- Identify Ext and Tor as the derived functors of Hom and tensor.
- Trace how the same machinery yields group and Lie algebra cohomology.
- Locate any topic in this knowledge stream from the layer it belongs to.
Section 01The failure that starts everything
Let 0 → A → B → C → 0 be a short exact sequence of modules. Applying Hom(M, −) gives
and the last map need not be surjective: a homomorphism into C may not lift to B. Applying M ⊗ − gives right exactness instead, with injectivity on the left failing. In both cases something is missing, and the missing piece is not noise — it is an invariant.
Preserves kernels. The cokernel of the last map is measured by Ext1.
Preserves cokernels. The kernel of the first map is measured by Tor1.
A functor that is exact loses nothing and needs no repair. A functor that is only half-exact leaves a measurable residue, and that residue is a sequence of new functors — the derived functors. Homological algebra is the systematic study of those residues.
Section 02The layer stack
Every computation in the subject decomposes downward through the same layers. Knowing the stack is the fastest route to knowing where a given technique belongs.
- Stage 01Modules and exactnessProjectives, injectives, free modules, Hom and tensor. The raw material.
- Stage 02Categories and functorsAbelian categories, adjoints, limits. The language that makes the constructions apply far beyond modules.
- Stage 03Complexes and resolutionsChain complexes, homotopy, projective and injective resolutions.
- Stage 04Derived functorsExt, Tor and their long exact sequences. The central invariants.
- Stage 05SpecialisationGroup cohomology, Lie algebra cohomology, Künneth formulas, spectral sequences.
It would be possible to develop Ext and Tor for modules alone. The categorical layer earns its place because the same arguments then apply verbatim to sheaves, to representations, to chain complexes themselves — and because adjointness is what actually explains why projectives and injectives behave as they do.
Section 03What the invariants know
| Invariant | Derived from | Classifies |
|---|---|---|
| Ext1(C, A) | Hom(−, A), first derived functor | Extensions of C by A, up to equivalence — with the zero element the split extension |
| Extn(C, A) | Higher derived functors of Hom | n-fold extensions (Yoneda); obstruction groups |
| Tor1(A, B) | First derived functor of tensor | Torsion interaction; the correction term in the Künneth formula |
| Hn(G, A) | Ext over the group ring | n = 1 derivations modulo principal ones; n = 2 group extensions with given action |
| Hn(g, A) | Derived functors over the enveloping algebra | n = 2 Lie algebra extensions; vanishing gives the Whitehead lemmas |
| Projective dimension | Length of a shortest projective resolution | How far a module is from being projective; global dimension bounds it over the ring |
Ext1 is simultaneously an obstruction group, a classification of extensions and a derived functor. These are not analogies — they are the same group arrived at three ways, and the ability to move between the readings is most of the working skill in this subject.
Section 04How this stream is organised
| Stream | Covers | Depends on |
|---|---|---|
| Modules | Projectives, injectives, free and cofree modules, essential extensions | — |
| Categories and Functors | Categories, adjoints, limits, abelian categories | Modules |
| Extensions, Ext and Tor | Baer sum, Ext, tensor product, Tor | Modules, categories |
| Derived Functors | Complexes, homotopy, resolutions, Ext and Tor systematically | Extensions |
| The Künneth Formula | Double complexes, Künneth and universal coefficients | Derived functors |
| Cohomology of Groups | Group ring, low-dimensional interpretation, extensions, transfer | Derived functors |
| Cohomology of Lie Algebras | Enveloping algebra, Whitehead lemmas, syzygies | Derived functors |
| Spectral Sequences | Exact couples, filtrations, Grothendieck and LHS sequences | Derived functors |
| Satellites and Kan Extensions | Relative homological algebra, satellites, Kan extensions | Derived functors, categories |
| Applications | Topology, nilpotent groups, modular representations, derived categories | All of the above |
ReferenceFrequently asked questions
Is homological algebra a subject or a toolkit?
Both, and the distinction matters less than it appears. It arose as a toolkit abstracted from algebraic topology, and it has a substantial internal theory of its own. In practice most people meet it as machinery for another subject and only later find that the machinery has its own questions.
Do I need category theory first?
Not first, but soon. The module-theoretic development is self-contained up to derived functors, and many people find the concrete case clarifying. The categorical language becomes unavoidable once the same theorems are wanted for sheaves or for complexes themselves.
Why are there two indices, homology and cohomology?
Because functors come in covariant and contravariant flavours, and resolutions come in projective and injective flavours. Homology decreases the index and is computed from projectives with covariant right-exact functors; cohomology increases it and is computed from injectives or from contravariant left-exact functors.
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ProvenanceSources and further reading
The standard reference for this material is Hilton and Stammbach, A Course in Homological Algebra (Springer, Graduate Texts in Mathematics 4). Cartan and Eilenberg's Homological Algebra is the founding text; Weibel's An Introduction to Homological Algebra is the standard modern companion.
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