Koszul algebra

In abstract algebra, a Koszul algebra formula_1 is a graded formula_2-algebra over which the ground field formula_2 has a linear minimal graded free resolution, "i.e.", there exists an exact sequence:
Here, formula_5 is the graded algebra formula_1 with grading shifted up by formula_7, "i.e." formula_8. The exponents formula_9 refer to the formula_9-fold direct sum. Choosing bases for the free modules in the resolution, the chain maps are given by matrices, and the definition requires the matrix entries to be 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", formula_11.
The concept is named after the French mathematician Jean-Louis Koszul.