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) as well as topology (e.g., equivariant cohomology). The prototype example, due to Bernstein, Gelfand and Gelfand, 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 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 then coincides with the quadratic dual

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

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 the underlying field k, thought of as an A-module:

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

Koszul duality

If $A$ is Koszul, there is an equivalence between certain subcategories of the derived categories of graded $A$- and $A^!$-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 $A$ and $A^!$ to obtain equivalences, it is possible instead to obtain equivalences between certain quotients of the homotopy categories. 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 $A$ and the 'coderived' category of the coalgebra $(A^!)^*$.

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.

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. 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

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