Njia za utafitiCategory TheoryImepangwa
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
Somo la mada hii bado halijaandikwa. Muhtasari ulio hapa chini ni sehemu ya ramani ya mtaala na haujakaguliwa.
Inajengwa juu ya
Masharti ya awali ya lazima 40 (hadi chini kabisa)
Utajifunza
- 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.
Fasili
- 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.
Nadharia
- 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.
Mawazo potofu ya kawaida
- 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.
Inaelekea kwa
Ugunduzi wa hiari
- 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
Njia za kujifunza
Usomaji zaidi
- 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)