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:

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 ] / ( x y )