Derived Categories - Examples and Constructions

Derived categories appear throughout modern mathematics, especially in algebraic geometry.

ExampleDerived Category of Coherent Sheaves

For a scheme $X$ , $D^b(\text{Coh}(X))$ is the bounded derived category of coherent sheaves. Objects are complexes: $\cdots \to \mathcal{F}^{-1} \to \mathcal{F}^0 \to \mathcal{F}^1 \to \cdots$

with finitely many non-zero cohomology sheaves, all coherent.

Definition8.11Perfect Complexes

A complex $\mathcal{F}^\bullet$ on a scheme $X$ is perfect if locally it is quasi-isomorphic to a bounded complex of locally free sheaves of finite rank. These form $D_{\text{perf}}(X) \subset D^b(\text{Coh}(X))$ .

ExampleFourier-Mukai Transforms

For schemes $X, Y$ , a Fourier-Mukai kernel is an object $\mathcal{P} \in D^b(X \times Y)$ . It induces a functor: $\Phi_{\mathcal{P}}: D^b(X) \to D^b(Y)$ $\Phi_{\mathcal{P}}(\mathcal{F}) = R\pi_{Y*}(\mathcal{P} \otimes^L \pi_X^*\mathcal{F})$

Many geometric equivalences are Fourier-Mukai transforms.

Theorem8.12Bondal-Orlov Reconstruction

For a smooth projective variety $X$ with ample canonical or anti-canonical bundle, $X$ can be reconstructed from $D^b(\text{Coh}(X))$ as a triangulated category.

This shows the derived category contains complete geometric information.

ExampleDerived Morita Theory

Two rings $R, S$ are derived Morita equivalent if: $D^b(\text{Mod}_R) \simeq D^b(\text{Mod}_S)$

as triangulated categories. This is weaker than classical Morita equivalence but still implies many ring-theoretic properties coincide.

Definition8.13Serre Functor

A Serre functor $S: D^b(X) \to D^b(X)$ satisfies: $\text{Hom}(E, F)^\vee \cong \text{Hom}(F, S(E))$

For smooth projective varieties, $S(E) = E \otimes \omega_X[d]$ where $\omega_X$ is the canonical sheaf and $d = \dim X$ .

Remark

Serre functors encode Serre duality in the derived categorical framework, providing a categorical perspective on classical duality theorems.

ExampleDG-Enhancements

The derived category admits a DG-enhancement: a differential graded (DG) category whose homotopy category is the derived category. This resolves many set-theoretic and functoriality issues.