# Koszul algebra

In abstract algebra, a Koszul algebra $R$ is a graded $k$-algebra over which the ground field $k$ has a linear minimal graded free resolution, i.e., there exists an exact sequence:

$\cdots \rightarrow R(-i)^{b_i} \rightarrow \cdots \rightarrow R(-2)^{b_2} \rightarrow R(-1)^{b_1} \rightarrow R \rightarrow k \rightarrow 0.$

It is named after the French mathematician Jean-Louis Koszul.

We can choose bases for the free modules in the resolution; then the maps can be written as matrices. For a Koszul algebra, the entries in the matrices are zero or linear forms.

An example of a Koszul algebra is a polynomial ring over a field, for which the Koszul complex is the minimal graded free resolution of the ground field. There are Koszul algebras whose ground fields have infinite minimal graded free resolutions, e.g, $R = k[x,y]/(xy)$