---
created: 2021-10-21T18:42
updated: 2024-01-09T13:18
title: cotangent complex
aliases:
  - cotangent complex
  - Andre-Quillen homology
---

---

- Tags 
	- #higher-algebra/DAG 
- Refs:
	- [Notes - The cotangent complex and derived de Rham cohomology](Notes%20-%20The%20cotangent%20complex%20and%20derived%20de%20Rham%20cohomology.md)
	- <https://stacks.math.columbia.edu/download/cotangent.pdf> #resources/notes 
	- <https://kskedlaya.org/prismatic/sec_derived-deRham.html> #resources/notes 
- Links:
	- [topos](Unsorted/topos.md)
	- [derived stack](Unsorted/derived%20algebraic%20geometry.md)
	- [derivation](Unsorted/derivation.md)
	- [properties of morphisms](properties%20of%20morphisms.md)
	- [derived de Rham cohomology](Unsorted/Notes%20-%20The%20cotangent%20complex%20and%20derived%20de%20Rham%20cohomology.md)
	- [[tangent complex]]
	- [[excisive]]

---

# cotangent complex

# Motivation

![](attachments/2023-03-31-25.png)

Relation to [dg Lie algebras](Unsorted/dg%20Lie%20algebras.md) and [deformation theory](Unsorted/deformation%20theory.md): [deformation theory is controlled by dg Lie algebras](Unsorted/deformation%20theory%20is%20controlled%20by%20dg%20Lie%20algebras.md).
![](attachments/2023-03-31-26.png)

Recovers [algebraic differentials](Unsorted/algebraic%20de%20Rham%20cohomology.md) in the smooth case.

![](2023-03-31-29.png)
# Theorems

![](2023-12-19-4.png)

# Definitions

A [nonabelian derived functor](nonabelian%20derived%20functor.md) of [relative differentials](Unsorted/algebraic%20de%20Rham%20cohomology.md):
![](attachments/2023-03-31-27.png)

![](attachments/2023-03-31-28.png)

![](attachments/2023-03-31-29.png)
![](attachments/2023-03-31-30.png)

![2023-03-31-31.png](attachments/2023-03-31-31.png)
![](attachments/2023-03-31-32.png)

# Properties

![](attachments/2023-03-31-33.png)
![](attachments/2023-03-31-34.png)

Relation to [obstruction-deformation theory](obstruction-deformation%20theory):
![](attachments/2023-03-31-35.png)

# Examples
![](attachments/2023-03-31-36.png)
![](attachments/2023-03-31-37.png)

![](attachments/2023-03-31-38.png)
![](attachments/2023-03-31-39.png)
![](attachments/2023-03-31-40.png)
![](2023-03-31-41.png)

![](2023-03-31-42.png)

# Exact triangles
![](attachments/2023-03-31-43.png)
![](attachments/2023-03-31-44.png)
![](attachments/2023-03-31-45.png)

# In derived geometry
![](attachments/2023-03-31-46.png)

# For DGAs

See [cocone](cocone.md) and [relative differentials](Unsorted/algebraic%20de%20Rham%20cohomology.md):
![](2023-03-31-47.png)
![](2023-03-31-48.png)

# Andre-Quillin homology