Complexes first, modules second
The derived category is obtained from the category of complexes by formally inverting the quasi-isomorphisms — the maps that induce isomorphisms on homology. In it, a module and any resolution of it become isomorphic, so resolutions stop being a computational device and become the objects themselves. Exact sequences are replaced by distinguished triangles, derived functors become ordinary functors between derived categories, and Ext becomes a Hom group.
Learning objectives
- Construct the derived category by localisation.
- Explain why a module and its resolution become isomorphic.
- State the triangulated structure and how triangles replace exact sequences.
- Identify Ext as a Hom in the derived category.
Section 01Localisation
- Stage 01ComplexesStart with the abelian category of chain complexes and chain maps.
- Stage 02Homotopy categoryQuotient by chain homotopy. This is already triangulated but not yet the right object.
- Stage 03Invert quasi-isomorphismsFormally invert maps inducing isomorphisms on homology. The calculus of fractions makes this manageable.
- Stage 04Derived category D(A)Objects are complexes; morphisms are roofs. A module and any of its resolutions become isomorphic.
A projective resolution is quasi-isomorphic to the module it resolves but not homotopy equivalent to it. Inverting quasi-isomorphisms is precisely what makes the resolution and the module the same object, which is the whole design goal.
Section 02Triangles
The derived category is not abelian — it has no kernels or cokernels. Its replacement structure is a class of distinguished triangles
satisfying axioms that make them behave like short exact sequences: each triangle induces a long exact sequence on homology, and the shift functor plays the role of the connecting map.
Kernels and cokernels exist. Long exact sequences arise from the snake lemma.
No kernels or cokernels. Long exact sequences arise from the triangle axioms and the shift functor.
The third term of a triangle — the mapping cone — is not determined functorially by the map. This is the well-known defect of triangulated categories, and it is what motivates the enhancements: differential graded categories, stable ∞-categories and derivators.
Section 03Consequences and variants
Derived functors become functors
RHom and the derived tensor product are ordinary functors between derived categories. Their homology recovers Ext and Tor, so the classical theory is a shadow.
Composition is automatic
The Grothendieck spectral sequence becomes the statement that the derived functor of a composite is the composite of the derived functors, with the spectral sequence recovering the homology.
Bounded derived categories
Restricting to bounded complexes gives Db, the usual setting for coherent sheaves and for tilting theory.
Stable module category
For a finite group in modular characteristic, the module category modulo projectives is triangulated and equivalent to a quotient of the derived category.
Derived equivalence
Two rings can have equivalent derived categories without being isomorphic — Morita theory's derived analogue, central to modern representation theory.
Derived algebraic geometry
Replacing rings by simplicial or differential graded rings; the derived category becomes the natural home for intersection theory with correct multiplicities.
Classical homological algebra computes invariants of modules. The derived viewpoint treats the complexes as the objects and asks about equivalences between the resulting categories. Most current research in the area is conducted in that language.
ReferenceFrequently asked questions
Is the derived category a category in the naive sense?
Yes, but morphism classes can fail to be sets without care. Working with bounded complexes over a small category, or using a suitable universe convention, resolves this. The homotopy-category route with a calculus of fractions is the standard construction.
Why is the derived category not abelian?
Because inverting quasi-isomorphisms destroys kernels and cokernels: a map with zero kernel and cokernel in the derived sense is already an isomorphism. Triangles are the structure that survives, and they carry enough to reproduce long exact sequences.
Do I need derived categories to do homological algebra?
Not for the classical results in this collection — Ext, Tor, group and Lie algebra cohomology are all accessible without them. They become necessary when the objects of interest are complexes themselves, as in algebraic geometry and modern representation theory.
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