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:

$\cdots \rightarrow R(-i)^{{b_{i}}}\rightarrow \cdots \rightarrow R(-2)^{{b_{2}}}\rightarrow R(-1)^{{b_{1}}}\rightarrow R\rightarrow k\rightarrow 0.$

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 ground field. There are Koszul algebras whose ground fields have infinite minimal graded free resolutions, e.g, $R=k[x,y]/(xy)$

See also

• Koszul duality

References

• Fröberg, R. (1999), "Koszul algebras", Advances in commutative ring theory (Fez, 1997), Lecture Notes in Pure and Applied Mathematics, 205, New York: Marcel Dekker, pp. 337–350, MR 1767430.
• Loday, Jean-Louis; Vallette, Bruno (2012), Algebraic operads (PDF), Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 346, Heidelberg: Springer, doi:10.1007/978-3-642-30362-3, ISBN 978-3-642-30361-6, MR 2954392.
• Beilinson, Alexander; Ginzburg, Victor; Soergel, Wolfgang (1996), "Koszul duality patterns in representation theory", Journal of the American Mathematical Society, 9 (2): 473–527, doi:10.1090/S0894-0347-96-00192-0, MR 1322847.
• Mazorchuk, Volodymyr; Ovsienko, Serge; Stroppel, Catharina (2009), "Quadratic duals, Koszul dual functors, and applications", Transactions of the American Mathematical Society, 361 (3): 1129–1172, doi:10.1090/S0002-9947-08-04539-X, MR 2457393.