12.1 Derived categories and triangulated categories

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 algebraic geometry and topology.

Derived Categories

Constructing the Derived Category

• Derived category $D(A)$ formed by localizing the category of chain complexes $Ch(A)$ with respect to the class of quasi-isomorphisms
• Localization process adds formal inverses to quasi-isomorphisms, allowing them to become isomorphisms in the derived category
• Quasi-isomorphism a chain map $f: X_\bullet \to Y_\bullet$ inducing isomorphisms on all homology groups $H_n(f): H_n(X_\bullet) \to H_n(Y_\bullet)$ for all $n \in \mathbb{Z}$
• 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
• Verdier quotient 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 triangulated structure, 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 $(\mathcal{T}, \Sigma)$ consists of an additive category $\mathcal{T}$ and an autoequivalence $\Sigma: \mathcal{T} \to \mathcal{T}$ called the shift or suspension functor
• Distinguished triangles in $\mathcal{T}$ are sequences of the form $X \to Y \to Z \to \Sigma X$ satisfying certain axioms (rotation, morphism, octahedral)
• Octahedral axiom 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-structure $(\mathcal{T}^{\leq 0}, \mathcal{T}^{\geq 0})$ on a triangulated category $\mathcal{T}$ consists of two full subcategories satisfying certain orthogonality and stability conditions
• Heart of a t-structure $\mathcal{A} = \mathcal{T}^{\leq 0} \cap \mathcal{T}^{\geq 0}$ is an abelian category, allowing the study of homological algebra within the triangulated framework
• Examples of t-structures include the standard t-structure on the derived category $D(A)$ (with heart $A$) and the perverse t-structure 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