Koszul algebra

In abstract algebra, a Koszul algebra ${\displaystyle R}$ is a graded ${\displaystyle k}$-algebra over which the ground field ${\displaystyle k}$ has a linear minimal graded free resolution, i.e., there exists an exact sequence:

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

It is named after the French mathematician Jean-Louis Koszul.

Here, ${\displaystyle R(-j)}$ is the graded algebra ${\displaystyle R}$ with grading shifted up by ${\displaystyle j}$, i.e. ${\displaystyle R(-j)_{i}=R_{i-j}}$. The exponents ${\displaystyle b_{i}}$ refer to the ${\displaystyle b_{i}}$-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, ${\displaystyle R=k[x,y]/(xy)}$

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, arXiv:math/0603475, doi:10.1090/S0002-9947-08-04539-X, MR 2457393.