---
tag: D1MYE6WVZ9
kind: definition
name: Derived category of coherent sheaves
msc: [14F08, 18G80]
defines: 2VP2D51KB8
uses: [5A35TYZD02, ND6XEY68P4, 56RXEJ8Z6S, WQ82C0M3QB, C9JEC2WQKV, 5ST7NVJFSM, T2V7GJJ2EN]
sources:
  - url: https://en.wikipedia.org/wiki/Derived_category
---

## Definition

Let $X$ be a [[Noetherian scheme|ND6XEY68P4]], and let $\operatorname{Coh}(X)$ be the abelian category of [[coherent sheaves|56RXEJ8Z6S]] on $X$.
A [[chain complex|WQ82C0M3QB]] $F^\bullet$ of coherent sheaves is *bounded* if $F^i = 0$ for all but finitely many $i$.
Consider the category of bounded complexes in $\operatorname{Coh}(X)$ with chain maps.
The *bounded derived category of coherent sheaves* of $X$ is the category $D^b(X) = D^b(\operatorname{Coh}(X))$ obtained from it by formally inverting all [[quasi-isomorphisms|C9JEC2WQKV]].
It is the bounded [[derived category|5ST7NVJFSM]] of $\operatorname{Coh}(X)$.

## Remarks

$D^b(X)$ is a [[triangulated category|T2V7GJJ2EN]] with shift $F^\bullet \mapsto F^\bullet[1]$.
It is equivalent to the full subcategory of the derived category of all $\mathcal{O}_X$-modules whose objects are the complexes with coherent cohomology sheaves, nonzero in finitely many degrees.
Example: for $X = \operatorname{Spec} k$ with $k$ a field, coherent sheaves are finite-dimensional vector spaces, and every object of $D^b(X)$ is isomorphic to $\bigoplus_i H^i(F^\bullet)[-i]$, its cohomology placed in each degree.
Non-example: on $X = \mathbb{A}^1_k = \operatorname{Spec} k[x]$, the complex $k[x] \xrightarrow{x} k[x]$ in degrees $-1, 0$ maps to $k[x]/(x)$ in degree $0$ by a quasi-isomorphism.
In the homotopy category of complexes, before quasi-isomorphisms are inverted, these two complexes are not isomorphic, since there is no nonzero map of modules $k[x]/(x) \to k[x]$.
In $D^b(X)$ they are isomorphic.
Theorems on $D^b$: Beilinson showed that $D^b(\mathbb{P}^n)$ is generated by $\mathcal{O}, \mathcal{O}(1), \ldots, \mathcal{O}(n)$, and Mukai showed that an abelian variety and its dual have equivalent derived categories.