---
tag: 6AW49BZ8TM
kind: concept
name: Koszul duality
msc: [16S37, "16E35"]
sources:
  - url: https://en.wikipedia.org/wiki/Koszul_duality
same_as:
  wikidata: Q16981474
  wikipedia: Koszul duality
---

## Summary

Koszul duality pairs a [[Koszul algebra|FTKFCCZ8YA]] $A$ with its quadratic dual $A^!$, which is again Koszul and computes the Ext algebra of the trivial $A$-module.
The basic pair is the symmetric algebra of a vector space and the exterior algebra of its dual.
Under suitable finiteness conditions it gives an equivalence between [[derived categories|5ST7NVJFSM]] of graded modules over $A$ and $A^!$.
The same pattern appears in representation theory, algebraic topology and the theory of operads.

## Explanations

### L1 human

Koszul duality is a way to trade one algebra for another, simpler or more familiar one, without losing homological information.
The polynomial ring $k[x_1, \dots, x_n]$, where the variables commute, is paired with the exterior algebra, where the variables anticommute.
The relations of one algebra are, roughly, the orthogonal complement of the relations of the other.
Problems about modules over one side can be translated into problems about modules over the other side.
For instance, the Bernstein–Gelfand–Gelfand correspondence relates sheaves on projective space with modules over an exterior algebra.