---
created: 2022-02-23T18:45
updated: 2023-04-02T13:45
title: quasicoherent sheaf
aliases: [quasicoherent sheaf, coherent sheaf, quasicoherent, coherent, QCoh, Coh]
---


- Tags 
	- #AG/sheaves
- Refs:
	- #todo/add-references
- Links:
	- [sheaf associated to a module](Unsorted/sheaf%20associated%20to%20a%20module.md)

---

# quasicoherent sheaf

## Definition

![](2023-04-02-18.png)

- A sheaf $\mcf \in \Sh(X; \mods{\OO_X})$ is **quasicoherent** if $X$ is covered by open affine subsets $U_{i}=\operatorname{Spec} A_{i}$ such that for each $i$ there is an $A_{i}$-module $M_{i}$ with $\left.\mathcal{F}\right|_{U_{i}} \cong \widetilde{M}_{i}$, the [sheaf associated to a module](Unsorted/sheaf%20associated%20to%20a%20module.md) for $M_i$.

- $\mathcal{F}$ is **coherent** if it is quasicoherent and each $M_{i}$ is [finitely generated](finitely%20generated.md).

![](attachments/Pasted%20image%2020220914183651.png)

![](attachments/Pasted%20image%2020220315100612.png)

![](attachments/Pasted%20image%2020220315100837.png)

![](attachments/Pasted%20image%2020221106015542.png)
![](attachments/Pasted%20image%2020221106015559.png)

# Notes
> Reference: <https://arxiv.org/pdf/1410.1716.pdf> #resources/papers 

- A Noetherian [scheme](scheme.md) $X$ can be reconstructed from $X$ can be reconstructed from $\QCoh(X)$, see Gabriel 1962.
	- Idea: associate to an [ringed space](ringed%20space) $\spec \cat{A}$ and show 
	$$X \underset{\Ringedspace}{\mapsvia{\cong}} \spec \cat{\QCoh(A)}$$
- A smooth variety $X$ can *not* generally be reconstructed from its [ample bundle](ample%20bundle).
- In 2002 Balmer reconstructs a [Noetherian scheme](Noetherian%20scheme.md) from its [perfect complexes](perfect%20complexes.md).
	- Lurie 2006ish: some more general result?
	
![Pasted image 20211105015441.png](Pasted%20image%2020211105015441.png)

![](attachments/Pasted%20image%2020220209093221.png)