Questions relating to various versions of Koszul duality, including Koszul duality between algebras and Koszul duality for operads.

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6
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1answer
310 views

Bar/Cobar Adjunction Between Modules and Comodules

There is a pretty well known, and widely written about, adjunction between augmented algebras and coaugmented coalgebras given by taking the bar construction on algebras and the cobar construction on ...
10
votes
0answers
308 views

What is Koszul dual of a curve?

Let $X$ be a curve embedded into a projective space $\mathbb P$ such that it is cut out (scheme-theoretically or ideal-theoretically) by quadrics. What is known about the Koszul dual of the ...
9
votes
0answers
291 views

What does convergence of a Bousfield-Kan spectral sequence say about the homotopy type of the totalization?

Given a cosimplicial space or spectrum $X^\bullet$, there is an associated Bousfield-Kan spectral sequence. This starts out as the bigraded object obtained by taking homotopy groups of each $X^n$ and ...
11
votes
0answers
462 views

Bar construction vs. twisted tensor product

One may study the cohomology of a space $E$ expressed as a homotopy pullback of $X$ and $Y$ over $Z$ using either the Eilenberg-Moore spectral sequence or the Serre spectral sequence for the fibration ...
5
votes
2answers
396 views

Koszul (Exterior/Symmetric) duality for a 1-dim vector space

The simplest example of Koszul duality (see introduction of http://www.ams.org/journals/jams/1996-9-02/S0894-0347-96-00192-0/) Let $V = \mathbb{C}x$ be a $1$ dimensional vector space. Then the ...
4
votes
1answer
419 views

“as close to being semisimple as it can possibly be.”

I had originally asked this question on math stack exchange but I think maybe it's more appropriate to ask it here. In the paper of Beilinson, Ginzburg and Soergel entitled "Koszul Duality Patterns......
5
votes
1answer
500 views

Computing Ext in Exterior algebra (related to Koszul duality)

Let $V = \mathbb{C}^n$, $A = \Lambda^{\bullet}(\mathbb{C}^n)$ is a graded algebra (with $A_0 = \mathbb{C}, A_1 = V$, etc). Consider $A_0$ as a left $A$-module, how do we compute the graded ring $\...
8
votes
1answer
293 views

Is the operadic butterfly symmetric?

The operadic butterfly is a diagram in the category of operads in vector spaces. It extends the short exact sequence relating commutative, associative and Lie operads. $$\begin{array}{ccccc} & ...
3
votes
0answers
195 views

Koszul duality, and coherent sheaves on $pt/G \times_{\mathfrak{g}/G} pt/G$

My questions are the following (from this paper of Arinkin-Gaitsgory): Q1 Let $P \subset G$ be algebraic groups (in my case, $P$ being a parabolic subgroup of a reductive group $G$, but the following ...
10
votes
1answer
346 views

When is the derived category of representations of a finite poset equivalent to its opposite?

If I have a finite partially ordered set $K$, I can look at its derived category of finite dimensional representations $D(K)$. Note that $D(K^{op}) \simeq D(K)^{op}$ by linear duality. But when do ...
7
votes
3answers
391 views

Koszul duality for modular operads

Has anyone defined what it means for a modular operad to be Koszul, or what the Koszul dual of a modular operad is? In particular, is it meaningful to say that a modular operad is quadratic? Merkulov, ...
10
votes
1answer
351 views

Koszulness of the cohomology ring of moduli of stable genus zero curves

Let $n \geq 3$. The ring $H^\bullet(\overline{M}_{0,n},\mathbf Q)$ was determined by Sean Keel. It is generated by the cohomology classes of boundary divisors $D_{A,B}$ corresponding to partitions $A \...
4
votes
2answers
638 views

How is this observation related to Koszul duality?

Let $X$ be a smooth variety, $\mathcal D$ the sheaf of algebraic differentail operators, $\Omega$ the algebraic deRham complex and $\mathcal M$ a quasi coherent $\mathcal O_X$-module. Now there is a ...
13
votes
1answer
1k views

Koszul duality between Weyl and Clifford algebras?

Koszul duality Given a finite-dimensional $k$-vector space $V$ (I am happy taking $k = \mathbb{C}$ anywhere in the following if it makes a difference) and a subspace $R \subseteq V \otimes V$, we can ...
7
votes
1answer
681 views

What extra conditions are necessary for the following version of Koszul duality?

Conventions: So that I don't have to worry about, fix a field $k$ of characteristic zero, and always work over it. Categories of modules, etc., are always $\infty$-categories of dg modules. ...
2
votes
1answer
276 views

What is a left dual up to homotopy?

My question is prompted by 57589 If $X$ is an object in a monoidal category with unit $I$ then $Y$ is a left dual if we have $I\rightarrow Y\otimes X$ and $X\otimes Y\rightarrow I$ which satisfy the ...
6
votes
2answers
847 views

What is the (Koszul? derived?) interpretation of a pair of Lie algebras with the same cohomology?

There are many words and sentences in mathematics that I basically completely don't understand, including the words "Koszul" and "derived". But rather than ask for a complete description of such ...
12
votes
1answer
997 views

koszul duality and algebras over operads

Given a pair of Koszul dual algebras, say $S^*(V)$ and $\bigwedge^*(V^*)$ for some vector space $V$, one obtains a triangulated equivalence between their bounded derived categories of finitely-...
41
votes
3answers
9k views

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 ...
7
votes
3answers
2k views

Beilinson-Bernstein and Koszul duality

For geometric representation theorists down here. Consider the Beilinson-Bernstein theorem: Functor of global sections establishes the correspondence between twisted D-modules with fixed ...