Derived categories

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
• ← Symplectic geometry and topology
• ← Complex and Kähler geometry; Hodge theory

学习路径

• Research preparation

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)