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In mathematics, Koszul duality, named after the French mathematician Jean-Louis Koszul, is any of various kinds of dualities found in representation theory of Lie algebras, abstract algebras (semisimple algebra)[1] as well as topology (e.g., equivariant cohomology[2]). The prototype example, due to Bernstein, Gelfand and Gelfand,[3] is the rough duality between the derived category of a symmetric algebra and that of an exterior algebra. The importance of the notion rests on the suspicion that Koszul duality seems quite ubiquitous in nature.

Koszul duality for modules over Koszul algebras
Koszul dual of a Koszul algebra

Koszul duality, as treated by Beilinson, Ginzburg, and Soergel[4] can be formulated using the notion of a Koszul algebra. An example of such a Koszul algebra A is the symmetric algebra S(V) on a finite-dimensional vector space. More generally, any Koszul algebra can be shown to be a quadratic ring, i.e., of the form

A = T(V) / R,

where T(V) is the tensor algebra on a finite-dimensional vector space, and R is a submodule of \( V \otimes V (=T^2(V)) \). The Koszul dual is then defined as

\( A^! := T(V^*) / R' \)

where \( V^* is the (k-linear) dual and\( R' \subset V^* \otimes V^* \) consists of those elements on which the elements of R (i.e., the relations in A) vanish. The Koszul dual of A=S(V) is given by \(A^! = \Lambda(V^*) \), the exterior algebra on the dual of V. In general, the dual of a Koszul algebra is again a Koszul algebra, as can be shown. Its opposite ring is given by the graded ring of self-extensions of k as an A-module:

\( (A^!)^{\text{opp}} = \operatorname{Ext}^*(k, k). \)

Koszul duality

Koszul duality states an equivalence of derived categories of graded A- and \( A^! \)-modules, respectively

\(D(A\text{-}\mathbf{Mod}) \rightleftarrows D(A^!\text{-}\mathbf{Mod}) \)

(More precisely, certain boundedness conditions on the grading vs. the cohomological degree of a complex of these modules has to be imposed.)


Variants

If properly formulated (i.e., with certain variants of the derived category), the above-mentioned boundedness conditions can be dropped.

An extension of Koszul duality to D-modules states a similar equivalence of derived categories between dg-modules over the dg-algebra\( \Omega_X of Kähler differentials on a smooth algebraic variety X and the D_X-modules. [5][6]


Koszul duality for operads

An extension of the above concept of Koszul duality was formulated by Ginzburg and Kapranov who introduced the notion of a quadratic operad and defined the quadratic dual of such an operad.[7] Very roughly, an operad is an algebraic structure consisting of an object of n-ary operations for all n. An algebra over an operad is an object on which these n-ary operations act. For example, there is an operad called the associative operad whose algebras are associative algebras, i.e., depending on the precise context, non-commutative rings (or, depending on the context, non-commutative graded rings, differential graded rings). Algebras over the so-called commutative operad are commutative algebras, i.e., commutative (possibly graded, differential graded) rings. Yet another example is the Lie operad whose algebras are Lie algebras. The quadratic duality mentioned above is such that the associative operad is self-dual, while the commutative and the Lie operad correspond to each other under this duality.

Koszul duality for operads states an equivalence between algebras over dual operads. The special case of associative algebras gives back the functor A \mapsto A^! mentioned above.
See also

Koszul algebra

Notes

Ben Webster, Koszul algebras and Koszul duality. November 1, 2007
M. Goresky, R. Kottwitz, and R. MacPherson. Equivariant cohomology, Koszul duality, and the localization theorem. Invent. Math. 131 (1998).
I. Bernstein, I. Gelfand, S. Gelfand. Algebraic bundles over P^n and problems of linear algebra. Funkts. Anal. Prilozh. 12 (1978); English translation in Functional Analysis and its Applications 12 (1978), 212-214
A. Beilinson, V. Ginzburg, W. Soergel. Koszul duality patterns in representation theory. Journal of the AMS. 9 (1996), no. 2, 473-527.
Kapranov, M. M. On DG-modules over the de Rham complex and the vanishing cycles functor. Algebraic geometry (Chicago, IL, 1989), 57–86, Lecture Notes in Math., 1479, Springer, Berlin, 1991.
Positselski, Leonid: arXiv:0905.2621 Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence.], Mem. Amer. Math. Soc. 212 (2011), no. 996, vi+133 pp. ISBN 978-0-8218-5296-5, see Appendix B

Ginzburg, Victor; Kapranov, Mikhail. Koszul duality for operads. Duke Math. J. 76 (1994), no. 1, 203–272.

References

Francis, John; Gaitsgory, Dennis. Chiral Koszul duality. Selecta Math. (N.S.) 18 (2012), no. 1, 27–87.
Priddy, Stewart B. Koszul resolutions. Trans. Amer. Math. Soc. 152 1970 39–60.

External links

http://www.math.harvard.edu/~lurie/282ynotes/LectureXXIII-Koszul.pdf
http://people.mpim-bonn.mpg.de/geordie/Soergel.pdf
http://arxiv.org/pdf/1109.6117v1.pdf

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