研究路径Category Theory计划中
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
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前置知识
40 项必需的前置知识（一直追溯到底）
你将学会
- 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.
定义
- 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.
定理
- 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.
常见误解
- 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.
通向
可选发现
- 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
学习路径
延伸阅读
- 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)