12.1 Derived categories and triangulated categories
2 min read•august 7, 2024
Derived categories and triangulated categories are powerful tools in homological algebra. They provide a framework for studying chain complexes and their relationships, allowing us to work with quasi-isomorphisms as actual isomorphisms.
These structures generalize ideas from homological algebra to a broader setting. They're crucial for understanding advanced topics like derived functors, t-structures, and applications in and topology.
Derived Categories
Constructing the Derived Category
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D(A) formed by localizing the category of chain complexes Ch(A) with respect to the class of quasi-isomorphisms
process adds formal inverses to quasi-isomorphisms, allowing them to become isomorphisms in the derived category
a chain map f:X∙→Y∙ inducing isomorphisms on all Hn(f):Hn(X∙)→Hn(Y∙) for all n∈Z
Objects in the derived category are chain complexes, but morphisms are obtained by inverting quasi-isomorphisms (homotopy classes of chain maps)
Properties and Applications
Derived functors (Tor, Ext) arise naturally in the derived category framework, as they are obtained by applying the localization functor to the original functor
construction used to form the derived category, where the localization is performed with respect to the multiplicative system of quasi-isomorphisms
Derived category D(A) has a , with distinguished triangles corresponding to short exact sequences of chain complexes (up to quasi-isomorphism)
Derived categories play a central role in homological algebra and algebraic geometry, providing a powerful tool for studying homological invariants and derived functors
Triangulated Categories
Axioms and Structure
Triangulated category (T,Σ) consists of an additive category T and an autoequivalence Σ:T→T called the shift or
Distinguished triangles in T are sequences of the form X→Y→Z→ΣX satisfying certain axioms (rotation, morphism, octahedral)
relates distinguished triangles and ensures the existence of a commutative diagram (octahedron) involving compositions of morphisms in distinguished triangles
Triangulated categories axiomatize the properties of derived categories and stable homotopy categories in algebraic topology
t-Structures and Applications
(T≤0,T≥0) on a triangulated category T consists of two full subcategories satisfying certain orthogonality and stability conditions
A=T≤0∩T≥0 is an abelian category, allowing the study of homological algebra within the triangulated framework
Examples of t-structures include the on the derived category D(A) (with heart A) and the on the derived category of sheaves (with heart the category of perverse sheaves)
t-structures provide a way to construct abelian categories from triangulated categories and are used in the study of perverse sheaves, intersection cohomology, and the Riemann-Hilbert correspondence