---
tag: YX0J6TVGZ8
kind: definition
name: Koszul duality
msc: [16S37]
defines: 6AW49BZ8TM
uses: [04C5ZBG1WT, P4J5JZDW0B, 07JBSZE0V4, FTKFCCZ8YA]
sources:
  - url: https://en.wikipedia.org/wiki/Koszul_duality
---

## Definition

Let $k$ be a [[field|04C5ZBG1WT]] and let $V$ be a finite-dimensional vector space over $k$, with [[dual space|07JBSZE0V4]] $V^*$.
Let $R \subseteq V \otimes V$ be a subspace.
Let $A = T(V)/(R)$ be the quotient of the [[tensor algebra|P4J5JZDW0B]] by the two-sided ideal generated by $R$; such an $A$ is a *quadratic algebra*.
Identify $V^* \otimes V^*$ with $(V \otimes V)^*$, and let $R^\perp \subseteq V^* \otimes V^*$ be the set of forms that vanish on $R$.
The *quadratic dual* of $A$ is $A^! = T(V^*)/(R^\perp)$.
*Koszul duality* is the correspondence $A \leftrightarrow A^!$ between [[Koszul algebras|FTKFCCZ8YA]], together with the induced relations between their module categories.

## Remarks

The construction is an involution: $(A^!)^! \cong A$, because $(R^\perp)^\perp = R$ in finite dimension.
If $A$ is Koszul, then $A^!$ is Koszul, and the algebra $\operatorname{Ext}^\bullet_A(k, k)$ is isomorphic to $A^!$ (some conventions give the opposite algebra).
For Koszul $A$ the Hilbert series satisfy $h_A(t) \, h_{A^!}(-t) = 1$.
Example: let $A = S(V)$ be the symmetric algebra, so $R$ is spanned by the tensors $x \otimes y - y \otimes x$.
A form on $V \otimes V$ vanishes on $R$ exactly when it is symmetric, so $R^\perp$ is the space of symmetric tensors, spanned by the tensors $\xi \otimes \xi$ and $\xi \otimes \eta + \eta \otimes \xi$.
Hence $A^! = \Lambda(V^*)$ is the exterior algebra, and $h_A(t) = (1 - t)^{-n}$, $h_{A^!}(-t) = (1 - t)^n$ for $n = \dim V$.