---
date: 2021-10-21 18:42
modification date: Thursday 21st October 2021 18:42:50
title: scheme
aliases: [scheme, "schemes", affine scheme]

---

Last modified date: <%+ tp.file.last_modified_date() %>


---

- Tags 
	- #MOC #AG 
- Refs:
	- Eisenbud and Harris, *Geometry of Schemes* #resources/books 
	- Szamuely, *Galois groups and Fundamental Groups* #resources/books 
	- Contemplating spec: 
	  http://www.neverendingbooks.org/mumfords-treasure-map #resources/notes 
	- Galois theory for schemes: <https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf#page=1>
- Links:
	- [Study project: Algebraic Geometry](Projects/2022%20Algebraic%20Geometry%20Oral%20Exam/000%20AG%20(Subject%20MOC).md)
	- [Study project: Commutative Algebra](Projects/2022%20Advanced%20Qual%20Projects/Commutative%20Algebra/000%20CA%20MOC.md)
	- [properties of morphisms](Unsorted/properties%20of%20morphisms.md)
	- [variety](variety)
	- [Unsorted/reduction mod p](Unsorted/reduction%20mod%20p.md)
	- [GAGA](Unsorted/GAGA.md)
	- [Zariski tangent space](Unsorted/Zariski%20tangent%20space.md)
	- [schemes are functors](schemes%20are%20functors.md)
	- [Unsorted/diagonal morphism](Unsorted/diagonal%20morphism.md)
	- [fiber of a vector bundle over a scheme](fiber%20of%20a%20vector%20bundle%20over%20a%20scheme.md)
	- [linear system](linear%20system.md)
	- Common properties:
		- [reduced](Unsorted/reduced.md)
		- [quasicompact](Unsorted/quasicompact.md)
		- [quasiseparated](Unsorted/separated.md)
		- [geometrically connected](geometrically%20connected)

---


# scheme


# Definitions

An **affine scheme**  $X\in \Aff\Sch$ is a locally ringed space $(X, \OO_X)\congas{\Ringedspace}(\spec R, \OO_{\spec R})$ for some $R\in \CRing$. 

![](attachments/Pasted%20image%2020220418115438.png)

![](attachments/Pasted%20image%2020220407223745.png)
![](attachments/Pasted%20image%2020220407223800.png)


## Schemes as Zariski sheaves

![](attachments/Pasted%20image%2020220320031356.png)

Can generalize to define sheaves over an arbitrary [symmetric monoidal category](Unsorted/monoidal%20category.md) $\cat C$.
Define affine objects $\Aff(\cat C) \da \opcat{\Comm\Mon(\cat C)}$ as model categories.

# Motivations

![](attachments/Pasted%20image%2020220405130709.png)
![](attachments/Pasted%20image%2020220405130720.png)

# Commutative Algebra prereqs

- [regular ring](Unsorted/regular%20ring.md)
- [local ring](local%20ring)
- [Noetherian ring](Noetherian%20ring)
- [completion](Unsorted/adic%20completion.md)
- [localization of rings](Unsorted/localization%20of%20rings.md)
- [integral closure](Unsorted/integrally%20closed.md)
- [reduced ring](reduced%20ring)
- [Dedekind domain](Unsorted/Dedekind%20domain.md)
- [DVR](Unsorted/DVR.md)

# Topics

- [smooth scheme](smooth%20scheme.md)
- [finite type](finite%20type.md)
- [sheaf](sheaf.md)
- [proper morphism](proper%20morphism.md)
- [separated](separated.md)
- [geometric point](Unsorted/geometric%20fiber.md)
- [closed point](Unsorted/special%20fiber.md)
- [reduced scheme](reduced%20scheme)
- [regular scheme](Unsorted/regular%20scheme.md)
- [geometric fiber](geometric%20fiber.md)
- [group scheme](group%20scheme)
- [geometrically connected scheme](geometrically%20connected%20scheme)
- [quasi-affine](quasi-affine)
- [locally quasi-finite](locally%20quasi-finite.md)
- [singular support](singular%20support)
- [characteristic cycle](characteristic%20cycle)

Some other important ideas:

- [derivations](derivations) and  [cotangent complexes](cotangent%20complexes) representing them, 
- [formally étale morphisms](formally%20étale%20morphisms), 
- [flat morphisms](flat%20morphisms), 
- (Zariski) [open immersions](open%20immersions), 
- formally [unramified morphisms](unramified%20morphisms), 
- [finitely presented morphisms](finitely%20presented%20morphisms) of commutative monoids and modules, 
- [projective modules](projective%20modules) and [flat modules](flat%20modules), 
- [Hochschild cohomology](Hochschild%20cohomology)


# Notes

- Gabriel-Rosenberg reconstruction theorem: $X$ can be recovered from $\QCoh(X)$, the category of [quasicoherent sheaves](Unsorted/quasicoherent%20sheaf.md).on $X$.
![](attachments/Pasted%20image%2020220407234500.png)

# Analytic space

![attachments/Pasted image 20210731182717.png](attachments/Pasted%20image%2020210731182717.png)

# Morphisms of schemes

![](attachments/Pasted%20image%2020220420095442.png)