Koszul algebra

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

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, ${\displaystyle R=k[x,y]/(xy)}$

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