# Questions tagged [koszul-duality]

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

49
questions

7
votes

1
answer

373
views

### Koszul duality and $\operatorname{H}^*(BG)$ — concise proof?

If $G$ is an algebraic group, then one can show $\operatorname{H}^*(BG,k)$ and $\operatorname{H}_*(G,k)$ are Koszul dual dg algebras, e.g. Drinfeld and Gaitsgory, "Finiteness questions for ...

6
votes

1
answer

277
views

### Homology and cohomology of free loop spaces

String topology, as well as Hochschild (co)homology, give a rich perspective on the homology and cohomology of a free loop space $LM$ of a manifold $M$.
Let $k$ be a field and let $M$ be $n$-...

5
votes

0
answers

197
views

### Is Koszul duality a deformation theory when not over a field?

Let $k$ be a field. Then Thm 15.3.3.1 of Lurie's SAG says that Koszul duality, regarded as a contravariant endofunctor $\bar D$ of augmented $E_n$-algebras over $k$, is a deformation theory in the ...

1
vote

0
answers

89
views

### Koszul complex of the cobar construction is acyclic

This is a follow-up question on my question on math stackexchange (https://math.stackexchange.com/questions/4399553/proof-that-the-coaugmented-cobar-construction-of-a-cooperad-is-acyclic)
I think I ...

4
votes

0
answers

118
views

### Derived category of dg modules vs. graded modules over a formal dg-algebra

Let $R = \oplus_{i\geq0} R_i$, $R_0 = k$ ($k$ a field) be a positively-graded commutative noetherian algebra, regarded as an augmented dg-algebra with zero differential.
Depending on one's interest, ...

2
votes

0
answers

59
views

### What is the "correct" notion of a perfect graded commutative algebra?

My question is rather simplistic. While trying to dualize some statements about rational homotopy algebra of a space I got stuck with the following problem.
We have a notion of a perfect Lie algebra ...

4
votes

0
answers

175
views

### Quillen–Suslin theorem in a more general context

Let $A$ be a finite dimensional local Frobenius algebra that is Koszul.
Question: Is it true for the Koszul dual of $A$ that every finitely generated projective module is free? If not, is there a ...

9
votes

1
answer

346
views

### Which finite posets are Koszul self-dual?

Let $P$ be a finite connected poset with incidence algebra $A_P$.
For the definition and results on Koszul algebras for incidence algebras, see for example here
Question: Which posets have the ...

1
vote

1
answer

94
views

### Differential of the Twisted complex for algebraic operads

I have a question about the proof of lemma 6.4.12 in the book Algebraic Operads (Loday-Vallette) which I do not seem to be able to fully complete on my own. Hopefully, somebody here can point out what ...

6
votes

0
answers

100
views

### How are symmetric functions related to Koszul duality?

Staying within the world of linear algebra, we have the following two "dualities" between exterior powers and symmetric powers.
The first is that of Kozsul duality, so these two graded ...

3
votes

0
answers

222
views

### Ginzburg Kapranov paper on Koszul duality

I am studying the article "Koszul duality for operads" by Ginzburg and Kapranov, https://arxiv.org/pdf/0709.1228.pdf.
The problem is that this version of the paper contains empty spaces ...

24
votes

1
answer

913
views

### What are Koszul dualities?

I am bewildered by the number of things I've heard referred to as "Koszul duality", and I would like to sort it out. At various different times, I believe I've seen any of the following ...

4
votes

0
answers

111
views

### Perfect modules for the Beilinson algebra

The Beilinson algebra $A=A_n$ is a finite dimensional quiver algebra that is derived equivalent to the category of coherent sheaves of $\mathcal{P}^n$. See for example https://link.springer.com/...

11
votes

1
answer

242
views

### Infinity-homotopies

Koszul duality for operads allows for straightforward generalizations of $A$-infinity algebras and $A$-infinity morphisms for the so called Koszul operads $\mathcal{O}$, among which we find the ...

2
votes

1
answer

197
views

### An exercise from Loday and Vallette about Koszul morphism

I tried to solve the following exercise from Loday and Vallette's Algebraic Operad. The first three parts are straightforward, however I have no idea how to solve the last part. I can't find any ...

1
vote

1
answer

100
views

### Augmented algebras over semisimple ring

Let $A$ be a non-negatively graded algebra such that $A_0 = k$. We say that $A$ is Koszul if $k$ has a projective resolution by projective modules such that the i-th piece is generated in degree $i$. ...

8
votes

1
answer

402
views

### Is there something "Koszul dual" to formal groups?

The Lie operad is Koszul dual to the commutative operad. In some sense, the data of a formal group is an "elaboration" of the data of a Lie algebra. Is there some corresponding "elaboration" of the ...

5
votes

0
answers

182
views

### The notion of $\infty$-Cooperads for which Bar-Cobar duality is an equivalence

In the paper Bar-Cobar Duality by Michael Ching, he proves that the category of operads in spectra is equivalent via the Bar-Cobar adjunction to some model category of co-operads defined in the paper. ...

10
votes

2
answers

892
views

### Strange formulas that gave rise to Koszul duality

According to p.8 of the note KOSZUL DUALITY AND APPLICATIONS IN
REPRESENTATION THEORY by Geordie Williamson.
Let $M(\eta)$ be the Verma module of weight $\eta$, $L(\eta)$ be its unique simple ...

4
votes

0
answers

71
views

### Graded commutative PBW bases

A Poincaré–Birkhoff–Witt (PBW) basis is a particularly nice basis of a quadratic algebra that can be used to prove that it is Koszul (see Priddy's 1970 paper "Koszul resolutions", Trans. Amer. Math. ...

2
votes

0
answers

127
views

### Computing Hochschild Invariants of Positselski's Coderived Categories

Positselski's work allows one to frame Koszul duality very elegantly in terms of so called coderived categories of modules over coalgebras, these are somewhat exotic dg categories of comodules over $C$...

5
votes

0
answers

201
views

### Koszul duality between QLS algebras and cdg algebras

A Quadratic-Linear-Constant (QLC) algebra $U$ is an algebra which can be written as $T(V)/P$ where
$T(V) = \bigoplus_{n \in \mathbb{N}} V^{\otimes n}$ is the tensor algebra
and $P \subseteq k \oplus ...

19
votes

2
answers

879
views

### Does Koszul duality between $Comm$ and $Lie$ imply the power series identity $\exp(\ln(1-z))-1 = -z$?

To a symmetric sequence $V_\bullet$ of vector spaces, associate the generating function $F_V(z) = \sum_n \frac{\dim(V_n)}{n!} z^n$. Then
$$F_{Comm_\ast}(z) = \exp(z)-1 \qquad F_{Lie}(z) = \ln(1-z)$$
...

3
votes

1
answer

442
views

### What bigrading is used in this spectral sequence?

I am reading this paper of Positselski and Vishik, in particular the main theorem: the cohomology algebra of a conilpotent algebra (i.e. the cohomology of its cobar construction) is Koszul if it ...

3
votes

0
answers

142
views

### Which kind of functors preserve the bar-construction?

Let C, D be monoidal infinity categories that admit geometric realizations.
Let $F: C \to D$ be a monoidal functor and A an augmented associative algebra of C.
Denote $Bar(A)= \mathbb{1} \otimes_A \...

10
votes

1
answer

528
views

### Tensor products of $\infty$-algebras over operads

Let $A$ and $B$ be $A_\infty$-algebras. It's true, but it's a quite nontrivial fact, that the tensor product $A \otimes B$ can be given the structure of $A_\infty$-algebra, too. What is much easier to ...

7
votes

0
answers

351
views

### Understanding Koszul Duality in BGG and Gelfand, Manin

I'm trying to understand a particular point in the proof of Koszul duality between $D^b(\Lambda(V))$ and $D^b(S(V^*))$ as seen in "Algebraic Bundles over $\mathbb{P}^n$ and Problems of Linear Algebra" ...

1
vote

0
answers

91
views

### Proof-verification: Existence of an explicit formality morphism from the Barratt-Eccles Koszul dual cooperad

I know asking for proof-verification on MO is a tricky thing. On one hand interesting research level proofs are usually subject of articles and can not be discussed here in detail. On the other hand ...

7
votes

0
answers

361
views

### What is the endomorphism cooperad?

In Loday and Vallette's book on algebraic operads, they use the "Endomorphism cooperad $End^c_{s\mathbb{K}}$", where $s\mathbb{K}$ is the base field, shifted into (homological) degree one. This is an ...

9
votes

1
answer

761
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 ...

11
votes

0
answers

388
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 ...

14
votes

0
answers

710
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 ...

22
votes

0
answers

824
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 ...

7
votes

2
answers

820
views

### Koszul (exterior/symmetric) duality for a 1-dim vector space

The simplest example of Koszul duality (see introduction of Beilinson, Ginzburg, and Soergel - Koszul Duality Patterns in Representation Theory) is as follows.
Let $V = \mathbb{C}x$ be a $1$ ...

5
votes

1
answer

546
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......

4
votes

1
answer

1k
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

1
answer

389
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

0
answers

244
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 ...

12
votes

1
answer

514
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

3
answers

614
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, ...

14
votes

2
answers

643
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 \...

5
votes

2
answers

964
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 ...

22
votes

2
answers

2k
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 ...

8
votes

1
answer

1k
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. ...

3
votes

1
answer

289
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 ...

7
votes

2
answers

1k
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

1
answer

1k
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-...

70
votes

3
answers

16k
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

3
answers

3k
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 ...