Traced monoidal category: Difference between revisions

Revision as of 04:30, 20 October 2013

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:

\dots \to R (- i)^{b_{i}} \to \dots \to R (- 2)^{b_{2}} \to R (- 1)^{b_{1}} \to R \to k \to 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, $R = k [x, y] / (x y)$

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.

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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.&nbsp;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}}

Traced monoidal category: Difference between revisions

Revision as of 04:30, 20 October 2013

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