---
date: 2021-11-05 11:22
modification date: Friday 5th November 2021 11:22:10
title: derived scheme
aliases: [derived scheme]

---

Tags: 
#todo #todo/stub 
Refs:
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# derived scheme

Replace structure sheaf with a sheaf of [simplicial commutative rings](Unsorted/simplicial%20commutative%20ring.md) $(X, \OO_X)$ where $(X, \pi_0 \OO_X)\in \Sch$ and $\pi_i\OO_X \in \QCoh(\mods{\pi_0\OO_X})$.

Example: take a surface and a curve, the self-intersection of the curve will be a derived scheme.


![Pasted image 20211105112213.png](Pasted%20image%2020211105112213.png)
![Pasted image 20211105112228.png](Pasted%20image%2020211105112228.png)
![Pasted image 20211105112331.png](Pasted%20image%2020211105112331.png)
![Pasted image 20211105112359.png](Pasted%20image%2020211105112359.png)
![Pasted image 20211105112418.png](Pasted%20image%2020211105112418.png)
![Pasted image 20211105112445.png](Pasted%20image%2020211105112445.png)

See [cotangent complex](Unsorted/cotangent%20complex.md)

![Pasted image 20211105112628.png](Pasted%20image%2020211105112628.png)
![Pasted image 20211105112635.png](Pasted%20image%2020211105112635.png)
![Pasted image 20211105112757.png](Pasted%20image%2020211105112757.png)