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) and topology (e.g., equivariant cohomology). The prototype example, due to Joseph Bernstein, Israel Gelfand, and Sergei 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.
The simplest, and in a sense prototypical case of Koszul duality arises as follows: for a 1-dimensional vector space "V" over a field "k", with dual vector space formula_1, the exterior algebra of "V" has two non-trivial components, namely
This exterior algebra and the symmetric algebra of formula_1, formula_4, serve to build a two-step chain complex
whose differential is induced by natural evaluation map
Choosing a basis of "V", formula_4 can be identified with the polynomial ring in one variable, formula_8, and the previous chain complex becomes isomorphic to the complex
whose differential is multiplication by "t". This computation shows that the cohomology of the above complex is 0 at the left hand term, and is "k" at the right hand term. In other words, "k" (regarded as a chain complex concentrated in a single degree) is quasi-isomorphic to the above complex, which provides a close link between the exterior algebra of "V" and the symmetric algebra of its dual.
Koszul dual of a Koszul algebra.
Koszul duality, as treated by Alexander Beilinson, Victor Ginzburg, and Wolfgang Soergel can be formulated using the notion of Koszul algebra. An example of such a Koszul algebra "A" is the symmetric algebra formula_10 on a finite-dimensional vector space. More generally, any Koszul algebra can be shown to be a quadratic algebra, i.e., of the form
where formula_12 is the tensor algebra on a finite-dimensional vector space, and formula_13 is a submodule of formula_14. The "Koszul dual" then coincides with the quadratic dual
where formula_1 is the ("k"-linear) dual and formula_17 consists of those elements on which the elements of "R" (i.e., the relations in "A") vanish. The Koszul dual of formula_18 is given by formula_19, the exterior algebra on the dual of "V". In general, the dual of a Koszul algebra is again a Koszul algebra. Its opposite ring is given by the graded ring of self-extensions of the underlying field "k," thought of as an "A"-module:
Koszul duality.
If an algebra formula_21 is Koszul, there is an equivalence between certain subcategories of the derived categories of graded formula_21- and formula_23-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 formula_21 and formula_23 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 formula_21 and the 'coderived' category of the coalgebra formula_27.
An extension of Koszul duality to "D"-modules states a similar equivalence of derived categories between dg-modules over the dg-algebra formula_28 of Kähler differentials on a smooth algebraic variety "X" and the formula_29-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 formula_30 mentioned above.