Derived Categories - Key Properties

Understanding triangulated structure and derived functors in the categorical setting is essential.

Definition8.6Triangulated Category

A triangulated category $\mathcal{D}$ consists of:

An additive category $\mathcal{D}$
An auto-equivalence $T: \mathcal{D} \to \mathcal{D}$ (translation/shift functor)
A class of distinguished triangles $X \to Y \to Z \to TX$

satisfying axioms (TR1)-(TR4) that generalize properties of mapping cones and exact sequences.

Theorem8.7Verdier's Theorem

The derived category $D(\mathcal{A})$ is a triangulated category with:

Translation functor: $T = [1]$ (shift by one)
Distinguished triangles from short exact sequences of complexes

This structure is functorial and compatible with derived functors.

Definition8.8Derived Functor in Categorical Sense

For an additive functor $F: \mathcal{A} \to \mathcal{B}$ , the right derived functor is: $RF: D^+(\mathcal{A}) \to D^+(\mathcal{B})$

defined by $RF(C) = F(I^\bullet)$ where $C \to I^\bullet$ is a quasi-isomorphism to a complex of $F$ -acyclic objects (usually injectives).

Similarly, the left derived functor uses projective resolutions.

Remark

In the derived category framework, derived functors are actual functors between derived categories, not just cohomological functors. This unifies and generalizes the classical theory.

Theorem8.9Octahedral Axiom

Given morphisms $X \xrightarrow{f} Y \xrightarrow{g} Z$ and distinguished triangles: $X \to Y \to C_f \to X[1]$ $Y \to Z \to C_g \to Y[1]$ $X \to Z \to C_{gf} \to X[1]$

there exists a morphism $C_f \to C_{gf} \to C_g \to C_f[1]$ making a commutative octahedron diagram.

Definition8.10t-Structure

A t-structure on a triangulated category $\mathcal{D}$ is a pair of full subcategories $(\mathcal{D}^{\leq 0}, \mathcal{D}^{\geq 0})$ satisfying certain axioms. The heart $\mathcal{D}^{\leq 0} \cap \mathcal{D}^{\geq 0}$ is an abelian category.

For $D(\mathcal{A})$ , the standard t-structure has heart isomorphic to $\mathcal{A}$ .

Remark

t-Structures provide a way to recover abelian categories from triangulated categories, essential for perverse sheaves and stability conditions.