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In [[abstract algebra]], a '''Koszul algebra''' <math>R</math> is a [[graded algebra|graded]] <math>k</math>-[[algebra over a field|algebra]] over which the [[ground field]] <math>k</math> has a linear minimal graded free resolution, ''i.e.'', there exists an [[exact sequence]]: | |||
:<math>\cdots \rightarrow R(-i)^{b_i} \rightarrow \cdots \rightarrow R(-2)^{b_2} \rightarrow R(-1)^{b_1} \rightarrow R \rightarrow k \rightarrow 0.</math> | |||
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'', <math>R = k[x,y]/(xy) </math> | |||
== References == | |||
* R. Froberg, ''[http://altenua.udea.edu.co/heragis/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. | |||
* J.-L. Loday, B. Vallette ''[http://math.unice.fr/~brunov/Operads.pdf Algebraic Operads]'', Springer, 2012. | |||
* A. Beilinson, V. Ginzburg, W. Soergel, "[http://www.ams.org/journals/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. | |||
* V. Mazorchuk, S. Ovsienko, C. Stroppel, "[http://www.ams.org/journals/tran/2009-361-03/S0002-9947-08-04539-X/ Quadratic duals, Koszul dual functors, and applications]", ''Trans. of the AMS'' '''361''' (2009) 1129-1172. | |||
[[Category:Algebras]] | |||
{{algebra-stub}} | |||
Revision as of 05:30, 20 October 2013
In abstract algebra, a Koszul algebra is a graded -algebra over which the ground 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 ground field. There are Koszul algebras whose ground fields have infinite minimal graded free resolutions, e.g,
![{\displaystyle R=k[x,y]/(xy)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4c40e95922128c25cfbf4231ed218c413c7c05e)
References
- R. Froberg, 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.
- J.-L. Loday, B. Vallette Algebraic Operads, Springer, 2012.
- A. Beilinson, V. Ginzburg, W. Soergel, "Koszul duality patterns in representation theory", J. Amer. Math. Soc. 9 (1996) 473–527.
- V. Mazorchuk, S. Ovsienko, C. Stroppel, "Quadratic duals, Koszul dual functors, and applications", Trans. of the AMS 361 (2009) 1129-1172.