In abstract algebra, a Koszul algebra is a graded -algebra over which the residue 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.
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 residue field. There are Koszul algebras whose residue fields have infinite minimal graded free resolutions, e.g,
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