updat
---
date: 2022-03-17 19:50
modification date: Thursday 17th March 2022 19:50:58
title: koszul duality
aliases: [koszul dual]

---


---

- Tags 
	- #todo/untagged
- Refs:
	- Brantner's Lecture notes: #resources 
		- [x] [Lecture 1](https://people.maths.ox.ac.uk/brantner/L1.pdf)  
		- [x] [Lecture 2](https://people.maths.ox.ac.uk/brantner/L2.pdf)  
		- [x] [Lecture 3](https://people.maths.ox.ac.uk/brantner/L3.pdf)  
		- [x] [Lecture 4](https://people.maths.ox.ac.uk/brantner/L4.pdf)  
		- [x] [Lecture 5](https://people.maths.ox.ac.uk/brantner/L5.pdf)  
		- [x] [Lecture 6](https://people.maths.ox.ac.uk/brantner/L6.pdf)  
		- [x] [Lecture 7](https://people.maths.ox.ac.uk/brantner/L7.pdf)  (Lecture 8 missing)
- Links:
	- [Morita equivalence](Morita%20equivalence)
	- [compact](Unsorted/compact%20object%20of%20a%20category.md)
	- [thick subcategory](Unsorted/thick%20subcategory.md)
	- [Morita equivalence](Unsorted/Morita%20equivalence.md)
	- [perfect complexes](Unsorted/perfect%20complexes.md)
	- [operad](Unsorted/operad.md)
	- [deformation](Unsorted/deformation.md)
	- [reflexive pair](reflexive%20pair.md)
	- [Barr-Beck](Barr-Beck)
	- [simplicial set](Unsorted/simplicial%20set.md)
	- [infinity category](Unsorted/infinity%20categories.md)

---

# koszul duality

Koszul duality patterns have influenced several recent developments in algebraic geometry, ranging from the classification of [formal deformations](formal%20deformations) by [dg Lie algebras](Unsorted/dg%20Lie%20algebras.md) and the unobstructedness of [Calabi-Yau](Calabi-Yau.md) varieties to purely inseparable Galois theory and derived [Galois deformation rings](Galois%20deformation%20rings).

![](attachments/Pasted%20image%2020220401022408.png)

![](attachments/Pasted%20image%2020220317195111.png)

![](attachments/Pasted%20image%2020220317200229.png)
![](attachments/Pasted%20image%2020220317202127.png)
![](attachments/Pasted%20image%2020220318111127.png)

# Definition

![](attachments/Pasted%20image%2020220317200836.png)

## For augmented k-algebras
![](attachments/Pasted%20image%2020220317200950.png)
![](attachments/Pasted%20image%2020220317201155.png)

## For modules over associative algebras

![](attachments/Pasted%20image%2020220317201405.png)

## Applications

![](attachments/Pasted%20image%2020220317195722.png)
![](attachments/Pasted%20image%2020220317201955.png)

# Setup

![](attachments/Pasted%20image%2020220317234336.png)


# Deformation Problems

![](attachments/Pasted%20image%2020220317202338.png)
![](attachments/Pasted%20image%2020220317202553.png)

#AG/deformation-theory of [CYs](Unsorted/Calabi-Yau.md) are unobstructed:

![](attachments/Pasted%20image%2020220317203212.png)

# Notes

## Koszul algebras

Defining the Adams grading:
![](attachments/Pasted%20image%2020220318121020.png)
![](attachments/Pasted%20image%2020220318121044.png)

![](attachments/Pasted%20image%2020220318121002.png)

Every Koszul algebra is a quadratic algebra:
![](attachments/Pasted%20image%2020220318121255.png)

Dualizing a Koszul algebra:
![](attachments/Pasted%20image%2020220318121437.png)
![](attachments/Pasted%20image%2020220318121517.png)

Examples
![](attachments/Pasted%20image%2020220318121204.png)
![](attachments/Pasted%20image%2020220318121219.png)

Checking if something is a Koszul algebra using the [Poincare series](Poincare%20series.md).
See [PBW basis](PBW%20basis.md).

# Examples
![](attachments/Pasted%20image%2020220317201127.png)
![](attachments/Pasted%20image%2020220317201136.png)![](attachments/Pasted%20image%2020220317201918.png)

![](attachments/Pasted%20image%2020220318120805.png)

The [BGG correspondence](BGG%20correspondence):
![](attachments/Pasted%20image%2020220318120701.png)

## In topology

![](attachments/Pasted%20image%2020220318135052.png)