# nLab Koszul duality

### Context

#### Higher algebra

higher algebra

universal algebra

duality

# Koszul duality

## Idea

Koszul duality (named after Jean-Louis Koszul) is a duality and phenomenon generalizing the duality between the symmetric and exterior algebra of a vector space to so-called quadratic differential graded algebras (which can be obtained as a free dga module an ideal of relations which live in degree 2). For a pair of Koszul dual algebras, there is a correspondence between certain parts of their derived categories (precise formulation involves some finiteness conditions). In a setup in which one of the algebras is replaced by a cocomplete dg coalgebra, there is a formulation free of finiteness conditions, but involving twisting cochain (see that entry).

and developed in

## For associative and dg-algebras

There are a lot of algebras whose derived categories are equivalent in surprising ways. Morita equivalences are pretty simple, especially for finite dimensional algebras; essentially the only free parameter is the dimension of the object.

Namely, if $A Mod$ and $B Mod$ are equivalent, then the image of $A$ as a module over itself is a projective generator of $B Mod$, and for a finite-dimensional algebra, essentially the only thing you can do is take several copies of the indecomposible projectives of $B$.

On the other hand, if you take the derived category of dg-modules over $A$ (the dg part of this is not a huge deal; it’s just that they’re very close to, but a bit better behaved than, actual derived/triangulated categories, which are just crude truncations of truly functorial dg/$A_\infty$ versions), this is equivalent to the category of dg-modules over the endomorphism algebra (this is in the dg sense, so it’s a dg-algebra whose cohomology is the $Ext$ algebra) of any generating object. There are a lot more generating objects than projective generators, so there are a lot of derived equivalences.

In particular, let $A$ be a finite dimensional algebra $A$, than an obvious not-very-projective generating object is the sum of all the simple modules, say $L$. As mentioned above, there’s an equivalence $A dg Mod = \mathrm{Ext}(L,L) dg Mod$, just given by taking $\mathrm{Ext}(L,-)$.

In general, $\mathrm{Ext}(L,L)$ is in a very complicated object (for example, often for group algebras over finite fields), but sometimes it turns out to be nice. For example, if $A$ is an exterior algebra, you’ll get a polynomial ring on the dual vector space. Another (closely related) example is that the cohomology of a reductive group (over $\mathbb{C}$) is Koszul dual to the cohomology of its classifying space.

One way to ensure that $\mathrm{Ext}(L,L)$ is nice is if the algebra $A$ is graded. Then $\mathrm{Ext}(L,L)$ inherits an “internal” grading in addition to its homological one. If these coincide, then $A$ is called Koszul.

In this case, $B=\mathrm{Ext}(L,L)$ is forced to be formal (if it had any interesting $A_\infty$ operations, they would break the grading), so you’re dealing with a derived equivalence between actual algebras, though you have to be a bit careful about the dg-issues. Thus the derived category of usual modules over $A$ is equivalent to dg-modules over $B$ (with its unique grading) and vice versa. This can be fixed by taking graded modules on both sides.

## Examples

• The most famous example of Koszul dual algebras are the exterior algebra $Alt^{\bullet}V[-1]$ and the polynomial algebra $Sym^\bullet V^*[-2]$.

• A regular block of category O? for any semisimple Lie algebra is a self-Koszul dual.

• More generally, a singular block of parabolic category O? is dual to a different singular block of parabolic category O where the combinatorial data determining the central character and finiteness conditions switch.
• Braden, Licata, Proudfoot and Webster gave a combinatorial construction of a large family of Koszul dual algebras in Gale duality and Koszul duality.

## References

Other historical references on Koszul duality include

Koszul duality is also discussed in

• The Everything Seminar , koszul-duality-and-lie-algebroids

• David Eisenbud, Gunnar Fløystad, Frank-Olaf Schreyer, Sheaf cohomology and free resolutions over exterior algebras, Trans. Amer. Math. Soc. 355 (2003), 4397-4426 MR1990756 doi

A “curved” generalization is discussed in

• Joseph Hirsh, Joan Millès, Curved Koszul duality theory, Max Planck preprint MPIM2010-104, pdf
• Gunnar Fløystad, Koszul duality and equivalences of categories, Trans. Amer. Math. Soc. 358 (2006), 2373-2398 math.AG/0012264 MR2204036 doi

Bernhard Keller and his student Lefèvre-Hasegawa described rather general framework for Koszul duality using dg-(co)algebras and twisting cochains: * Bernhard Keller, Koszul duality and coderived categories (after Lefèvre-Hasegawa) (2003) abstract dvi pdf ps * Kenji Lefèvre-Hasegawa, Sur les A-infini catégories, pdf * Leonid Positselski, Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence, arxiv/0905.2621 * Aaron M Royer, Generalized string topology and derived Koszul duality, arxiv/1306.6708 * M. M. Kapranov, On DG-modules over the de rham complex and the vanishing cycles functor, Algebraic Geometry, Lecture Notes in Mathematics 1479, 1991, pp 57-86

Revised on March 7, 2014 11:59:50 by Zoran Škoda (161.53.130.104)