# 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.

Here, $R(-j)$ is the graded algebra $R$ with grading shifted up by $j$ , i.e. $R(-j)_{i}=R_{i-j}$ . The exponents $b_{i}$ refer to the $b_{i}$ -fold direct sum.

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, $R=k[x,y]/(xy)$ 