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:

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

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. 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, ${\displaystyle R=k[x,y]/(xy)}$.

The concept is named after the French mathematician Jean-Louis Koszul.

See also

• Koszul duality
• Complete intersection ring

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.