Koszul duality

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

D(A-Mod) \rightleftarrows D(A^!-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]

See also

Notes

  1. ↑ Ben Webster, Koszul algebras and Koszul duality. November 1, 2007
  2. ↑ M. Goresky, R. Kottwitz, and R. MacPherson. Equivariant cohomology, Koszul duality, and the localization theorem. Invent. Math. 131 (1998).
  3. ↑ 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
  4. ↑ A. Beilinson, V. Ginzburg, W. Soergel. Koszul duality patterns in representation theory. Journal of the AMS. 9 (1996), no. 2, 473-527.
  5. ↑ 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

References

External links