Koszul algebra

In abstract algebra, a Koszul algebra $R$ is a graded $k$-algebra over which the residue field $k$ has a linear minimal graded free resolution, "i.e.", there exists an exact sequence: :$cdots\; ightarrow\; R(-i)^\{b\_i\}\; ightarrow\; cdots\; ightarrow\; R(-2)^\{b\_2\}\; ightarrow\; R(-1)^\{b\_1\}\; ightarrow\; R\; ightarrow\; k\; ightarrow\; 0.$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", $R\; =\; k\; [x,y]\; /(xy)$

References

* R. Froberg, " [http://www.ime.usp.br/~enmarcos/Cursos/homologica/koszulalgfroberg.pdf Koszul Algebras] ", In: Advances in Commutative Ring Theory. Proceedings of the 3rd International Conference, Fez, Lect. Notes Pure Appl. Math. 205, Marcel Dekker, New York, 1999, pp.337--350.
* A. Beilinson, V. Ginzburg, W. Soergel, " [http://www.ams.org.proxy.uchicago.edu/jams/1996-9-02/S0894-0347-96-00192-0/S0894-0347-96-00192-0.pdf Koszul duality patterns in representation theory] ", "J. Amer. Math. Soc." 9 (1996) 473--527.