---
tag: 2VP2D51KB8
kind: concept
name: Derived category of coherent sheaves
msc: [14F08, 18G80]
specializes: [5ST7NVJFSM]
sources:
  - url: https://en.wikipedia.org/wiki/Derived_category
---

## Summary

Let $X$ be a Noetherian scheme.
Its derived category of coherent sheaves $D^b(X)$ is the bounded [[derived category|5ST7NVJFSM]] of the abelian category of [[coherent sheaves|56RXEJ8Z6S]] on $X$.
It is a [[triangulated category|T2V7GJJ2EN]].
For $X$ smooth and projective over a field, the derived functors of pushforward, pullback, tensor product and Hom act on these categories.
It is the natural home of Serre and Grothendieck duality and of Fourier–Mukai transforms.
Two varieties with equivalent derived categories share many invariants, so $D^b(X)$ is studied as an invariant of $X$ in its own right.

## Explanations

### L1 human

Many constructions with sheaves, such as cohomology, do not produce a single sheaf but a whole complex of sheaves.
The derived category keeps these complexes and treats two complexes as the same when a map between them is an isomorphism on cohomology.
For a point, an object is just a list of finite-dimensional vector spaces, one in each degree.
For the projective line, every object is a direct sum of shifts of the line bundles $\mathcal{O}(n)$ and of shifts of torsion sheaves.
Some varieties that are not isomorphic have equivalent derived categories: by a theorem of Mukai, an abelian variety and its dual always do, and they need not be isomorphic.