Koszul algebra
In abstract algebra, a Koszul algebra is a graded -algebra over which the ground field has a linear minimal graded free resolution, i.e., there exists an exact sequence:
It is named after the French mathematician Jean-Louis Koszul.
Here, is the graded algebra with grading shifted up by, i.e.. The exponents refer to the -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,
