Research pathwaysCategory TheoryPlanned
Derived categories
Optional research direction: the homotopy category of complexes, localisation at quasi-isomorphisms, the derived category D(A) of an abelian category, total derived functors, and why derived categories have become a standard language in algebra and geometry.
math.cat.derived-categories
The lesson for this topic has not been written yet. The outline below is part of the curriculum map and has not been reviewed.
Builds on
Required prerequisites, all the way down: 40
You will learn to
- Construct K(A) and D(A), and explain why localising at quasi-isomorphisms needs care.
- Compute derived functors such as RHom and the derived tensor product in examples, and recover Ext as morphisms in D(A).
- Read about derived equivalences, such as those from tilting theory and Fourier–Mukai transforms, and separate established results from open problems.
Definition
- K(A) has chain complexes as objects and homotopy classes of chain maps as morphisms.
- D(A) is obtained from the category of complexes by formally inverting the quasi-isomorphisms.
- A total right derived functor RF extends a left exact functor F to derived categories; when A has enough injectives it is computed with injective resolutions.
Theorem
- Extⁿ_A(M, N) ≅ Hom_{D(A)}(M, N[n]).
- If A has enough injectives, D⁺(A) is equivalent to the homotopy category of bounded-below complexes of injective objects.
- Non-isomorphic algebras can have equivalent derived categories, for example via tilting theory.
Common misconceptions
- Thinking D(A) is abelian; it is triangulated and in general not abelian.
- Believing every object of D(A) is determined by its homology; this holds over fields and, more generally, for hereditary abelian categories, but fails in general.
Leads to
Optional discoveries
- Quiver representations Happel described the bounded derived category of the path algebra of a Dynkin quiver Q in terms of the repetitive quiver ℤQ.
- ← Algebraic geometry
Learning pathways
Further reading
- Charles A. Weibel, An Introduction to Homological Algebra (1994)
- Sergei I. Gelfand and Yuri I. Manin, Methods of Homological Algebra
- Daniel Huybrechts, Fourier–Mukai Transforms in Algebraic Geometry (2006)