---
tag: YREPR562ES
kind: definition
name: Koszul algebra
msc: [16S37]
defines: FTKFCCZ8YA
uses: [04C5ZBG1WT, 3CTW1P4HBP, 8KB4SQGC0C, HK2AA1YSQ2]
sources:
  - url: https://en.wikipedia.org/wiki/Koszul_algebra
---

## Definition

Let $k$ be a [[field|04C5ZBG1WT]].
Let $A = \bigoplus_{n \ge 0} A_n$ be a [[graded|8KB4SQGC0C]] [[associative algebra|3CTW1P4HBP]] over $k$ with $A_0 = k$ and each $A_n$ finite-dimensional.
Let $k$ also denote the graded $A$-module $A/A_{>0}$, where $A_{>0} = \bigoplus_{n > 0} A_n$.
Consider a graded [[projective resolution|HK2AA1YSQ2]] of $k$ by graded free $A$-modules:
$$\cdots \to P_2 \to P_1 \to P_0 \to k \to 0.$$
The algebra $A$ is *Koszul* if $k$ has such a resolution in which each $P_i$, $i \ge 0$, is generated by homogeneous elements of degree $i$.

## Remarks

Example: $A = k[x]$ with $\deg x = 1$.
The resolution $0 \to A(-1) \xrightarrow{x} A \to k \to 0$ has $P_0 = A$ generated in degree $0$ and $P_1 = A(-1)$ generated in degree $1$.
More generally, the [[Koszul complex|YZY11W08SD]] is a linear resolution of $k$ over $k[x_1, \dots, x_n]$.
Non-example: $A = k[x]/(x^3)$ with $\deg x = 1$.
The minimal resolution starts with $P_0 = A$ and $P_1 = A(-1)$, mapping the generator to $x$.
The kernel of multiplication by $x$ on $A$ is spanned by $x^2$, so $P_2$ is generated in degree $1 + 2 = 3$, not $2$.
Equivalent form: $A$ is Koszul if and only if $\operatorname{Ext}^i_A(k, k)$ is concentrated in internal degree $i$ for every $i$.
Every Koszul algebra is quadratic: it is generated by $A_1$ with relations in degree $2$.