# 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 Joseph Bernstein, Israel Gelfand, and Sergei 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.[citation needed]

## Koszul Duality for modules over Koszul algebras

### Koszul dual of a Koszul algebra

Koszul duality, as treated by Alexander Beilinson, Victor Ginzburg, and Wolfgang 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

${\displaystyle A=T(V)/R,}$

where ${\displaystyle T(V)}$ is the tensor algebra on a finite-dimensional vector space, and R is a submodule of ${\displaystyle V\otimes V(=T^{2}(V))}$. The Koszul dual then coincides with the quadratic dual

${\displaystyle A^{!}:=T(V^{*})/R'}$

where ${\displaystyle V^{*}}$ is the (k-linear) dual and ${\displaystyle R'\subset V^{*}\otimes V^{*}}$ consists of those elements on which the elements of R (i.e., the relations in A) vanish. The Koszul dual of ${\displaystyle A=S(V)}$ is given by ${\displaystyle A^{!}=\Lambda (V^{*})}$, 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 the underlying field k, thought of as an A-module:

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

### Koszul duality

If ${\displaystyle A}$ is Koszul, there is an equivalence between certain subcategories of the derived categories of graded ${\displaystyle A}$- and ${\displaystyle A^{!}}$-modules. These subcategories are defined by certain boundedness conditions on the grading vs. the cohomological degree of a complex.

### Variants

As an alternative to passing to certain subcategories of the derived categories of ${\displaystyle A}$ and ${\displaystyle A^{!}}$ to obtain equivalences, it is possible instead to obtain equivalences between certain quotients of the homotopy categories.[5] Usually these quotients are larger than the derived category, as they are obtained by factoring out some subcategory of the category of acyclic complexes, but they have the advantage that every complex of modules determines some element of the category, without needing to impose boundedness conditions. A different reformulation gives an equivalence between the derived category of ${\displaystyle A}$ and the 'coderived' category of the coalgebra ${\displaystyle (A^{!})^{*}}$.

An extension of Koszul duality to D-modules states a similar equivalence of derived categories between dg-modules over the dg-algebra ${\displaystyle \Omega _{X}}$ of Kähler differentials on a smooth algebraic variety X and the ${\displaystyle D_{X}}$-modules. [6][7][8]

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

## Notes

1. ^ Ben Webster, Koszul algebras and Koszul duality. November 1, 2007
2. ^ Mark Goresky, Robert Kottwitz, and Robert MacPherson. Equivariant cohomology, Koszul duality, and the localization theorem. Inventiones Mathematicae 131 (1998).
3. ^ Joseph Bernstein, Israel Gelfand, and Sergei Gelfand. Algebraic bundles over ${\displaystyle P^{n}}$ and problems of linear algebra. Funkts. Anal. Prilozh. 12 (1978); English translation in Functional Analysis and its Applications 12 (1978), 212-214
4. ^ Alexander Beilinson, Victor Ginzburg, Wolfgang Soergel. Koszul duality patterns in representation theory, Journal of the American Mathematical Society 9 (1996), no. 2, 473-527.
5. ^ Fløystad, Gunnar (2006-01-01). "Koszul duality and equivalences of categories". Transactions of the American Mathematical Society. 358 (6): 2373–2398. doi:10.1090/S0002-9947-05-04035-3. ISSN 0002-9947.
6. ^ 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.
7. ^ 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
8. ^ Faltings, Gerd; Chai, Ching-Li. Degeneration of abelian varieties. With an appendix by David Mumford. Springer-Verlag, Berlin, 1990. xii+316 pp. ISBN 3-540-52015-5. Section VI.3
9. ^ 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.