# Questions tagged [koszul-duality]

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

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### Why is an operad of associative algebras Koszul?

Let $Assoc$ be an operad of associative algebras. What does it mean for $A$ to be a Koszul operad? Is it related to standard Koszul duality for algebras? As far as I understand, if $Assoc_{\infty}$ ...

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### 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 ...

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68 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 $i$. ...

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264 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 ...

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142 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. ...

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805 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 ...

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44 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. ...

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### 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 $C$...

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165 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 ...

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632 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
$$F_{Comm_\ast}(z) = \exp(z)-1 \qquad F_{Lie}(z) = \ln(1-z)$$
...

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395 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 ...

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126 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 \...

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360 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 ...

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266 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" ...

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84 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 ...

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252 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 ...

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607 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 ...

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367 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 ...

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520 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 ...

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705 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 ...

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613 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 ...

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509 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......

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835 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 $\...

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357 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}
& ...

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240 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 ...

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446 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 ...

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512 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, ...

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529 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 \...

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878 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 ...

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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 ...

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938 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. ...

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281 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 ...

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### 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 ...

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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-...

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13k 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 ...

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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 ...