# What is Koszul duality?

Okay, let's make sure I'm on the same page with those who know homological algebra.

What is Koszul duality in general?

What does it mean that categories are Koszul dual (I guess representations of Koszul dual algebras are the examples?) What are examples of "categories which seem to a priori have no good reason to be Koszul dual actually are" [Koszul dual] other than (g, R)-admissible modules?

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Let me try to give a more down to earth answer:

First, it's important to understand there are a lot of algebras whose derived categories are equivalent in surprising ways. Morita equivalences (equivalences between the abelian categories of modules) are kind of boring, especially for finite dimensional algebras; essentially the only thing you can do is change the dimensions of objects.

The way you see this is that 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 I consider something of a historical mistake, which should be replaced with dg/A-infinity 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, you can take your favorite finite dimensional algebra A, and the most obvious not-very-projective generating object: the sum of all the simples. Call this L. As I mentioned, there's an equivalence $A-dg-mod = \mathrm{Ext}(L,L)-dg-mod$, just given by taking $\mathrm{Ext}(L,-)$.

Now, in general, $\mathrm{Ext}(L,L)$ is an absolutely horrible object (ask Mikael Vejdemo-Johansson about doing this for group algebras over finite fields some time), but sometimes it turns out to be nice. For example, if you start with A being the 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 C) is Koszul dual to the cohomology of its classifying space (here you see a hint of this delooping mentioned in Scott's answer).

One thing that could help you make sure that $\mathrm{Ext}(L,L)$ is nice is if your algebra is graded. Then $\mathrm{Ext}(L,L)$ inherits an "internal" grading in addition to its homological one. If these coincide, then $B=\mathrm{Ext}(L,L)$ is forced to be formal (if it had any interesting A-infinity 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. You've found that the derived category of usual modules over A is equivalent to dg-modules over B (with its unique grading) and vice versa. You can fix this by taking graded modules on both sides.

As for more examples...well, some collaborators and I found some cool examples coming from the combinatorics of hyperplane arrangements.

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Well... There is more to Ext(L,L) than just its A-infinity structure, and there is more to Koszul duality being nice than that it forces a trivial (formal) A-infinity structure for relatively simple degree reasons. As I view it, the A-infinity side of Koszulity is mostly Yet Another Good Thing it brings. I should write up an answer of my own here. One of these days. :-) Probably talking at length about operads, and about how to find Nice resolutions of things. –  Mikael Vejdemo-Johansson Oct 13 '09 at 3:56
That's fair. I'm just trying to bring in the perspective that tends to get used in geometric representation theory, since that's what ilya had originally been asking about. Of course, it would be good to hear your spiel too. –  Ben Webster Oct 13 '09 at 4:58
@Mikael Vejdemo-Johansson: Yes! Please do write up an answer of your own. Absolutely would be great talk about operads and I think this is how Koszulity is related to derived alg-geom? –  Ilya Nikokoshev Oct 13 '09 at 19:57
"Another (closely related) example is that the cohomology of a reductive group (over C) is Koszul dual to the cohomology of its classifying space (here you see a hint of this delooping mentioned in Scott's answer)." -- Does anyone know of a reference for this? It sounds really interesting and a good motivation to learn more about Koszul duality. –  skupers Sep 22 '10 at 13:10
In an effort to understand this answer I worked out some detais, gathered references, gave a seminar talk and puta all of this on my webpage: heinrich-hartmann.net/wiki/index.php/Koszul_Duality. –  Heinrich Hartmann Jun 15 '12 at 20:24

I'd really like to hear a complete answer to this question. I have my own idea and although I don't think it's a perfect answer to your question, perhaps it will help someone.

There's a way of defining cofibrant replacements in certain categories call the Boardman-Vogt W-construction. I know it from the work of Berger and Moerdijk on operads. I believe it was first defined topologically, but I only know it algebraically and for algebras it goes something like this:

The unit interval could be described by the ring I=kt ⊕ ks ⊕ ke, where e is concentrated in degree 1, and d(e)=t-s. t is the identity and s is idempotent, also se=e. There's a unique augmentation. So this is a chain complex modelling the unit interval. The multiplication could be described as taking the maximal value, this is how it's defined topologically. t corresponds to the maximal element 1, s to the element 0.

Now let A be an associative algebra, the W-construction W(A) looks very much like the free product A*I (perhaps it's the same I don't remember), with the induced differential. W(A) is a cofibrant replacement for A.

This contains a subalgebra quasi-isomorphic to W(A), which may be given by the bar-cobar construction. Remember that the bar complex BA is given by taking the free coalgebra on A[1], with a square-zero coderivative induced by the multiplication map. The cobar complex ΩC on C is the free algebra on C[-1]. This carries the square-zero derivative defined by the coalgebra structure.

That's a mouthful, but all you need to remember is that the bar and cobar functors define an adjunction, actually a Quillen equivalence of model categories. And the unit and counit of this look very much like the W-construction.

In Berger and Moerdijk they do this not for associative algebras but for operads. And there are similar constructions for any operad (generalising the case for the associative operad above). In one example the bar complex is based on the free colie algebra and the cobar complex is based on the free commutative algebra.

Onto the categories of modules: there is an adjunction between the category of modules of A and the category of comodules for BA. This is actually a Quillen equivalence.

Now back to the Koszul property. An algebra is Koszul when the bar complex is quasi-isomorphic as coalgebras to something very small, in many cases quasi-isomorphic to its homology. Call this nice coalgebra C.

Then there is an adjunction between the category of modules for A and the category of comodules for C.

So to me at least the Koszul property concerns situations when large resolutions of things, say ΩBA, may be replaced by much smaller things. And this means a lot for the categories of (co)modules for the objects concerned. But that's just my personal impression.

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I've spent many years researching Koszul duality in its various versions. To me, Koszul duality is a fundamental homological phenomenon which has many manifestations, e.g.

1. the relation between the homotopy groups of a topological space and its (co)homology groups;
2. the relation between an augmented algebra A and its Ext-algebra Ext_A(k,k) (and between modules over these);
3. the relation between the ring of differential operators and the de Rham DG-algebra of differential forms (and between modules over these).

Version 1. is obviously complicated and one can approach it from various angles, like rational homotopy theory or stable homotopy theory. The former is related to the duality between commutative and Lie DG-algebras (Quillen), while the latter is more akin to module duality, in that it is additive. Morphisms from the sphere spectrum to the Eilenberg-MacLane spectrum are very simple to describe, while endomorphisms of the latter spectrum are more complicated and endomorphisms of the former one are much more complicated. One can view the sphere spectrum as a kind of projective generator and the Eilenberg-MacLane spectrum as a kind of irreducible module.

Version 2. is better to approach by splitting it in two branches, namely 2a. the relation between a conilpotent coalgebra C and its Ext-algebra Ext_C(k,k); and 2b. the relation between an augmented algebra A and its Tor-coalgebra Tor^A(k,k). Then one can generalize 2a. to DG-coalgebras, and 2b. to DG-algebras, which makes these two correspondences inverse to each other. Subsequently one can notice that the augmentation assumption or structure is largely irrelevant for 2b., and generalize this duality even further, obtaining a correspondence between conilpotent CDG- (curved DG-) coalgebras and (nonaugmented) DG-algebras.

Version 3. is a relative one, with the ring of functions playing the role of the basic field. The functions are not central in the differential operators and the de Rham differential is not linear over the functions, which makes such a relativization of Koszul duality highly nontrivial and interesting.

With both 2. and 3., there is a major problem that the duality functors do not preserve acyclicity of complexes. For example, a nonacyclic DG-algebra is sometimes assigned to an acyclic DG-coalgebra, and a nonacyclic complex of D-modules is assigned to an acyclic DG-module over the de Rham complex. And for curved DG-structures, the notion of quasi-isomorphism does not even make sense. The general solution is that one has to introduce an equivalence relation more delicate than quasi-isomorphism, particularly on the coalgebra side of the story. The relevant references include Hinich's paper, Lefevre-Hasegawa's thesis and Keller's exposition of some results contained therein, and my recent preprint, which is supposed to contain state of the art. Concerning DG-modules over the de Rham complex, there were earlier approaches by Kapranov and Beilinson-Drinfeld.

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