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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Does Manin's construction of non-commutative endomorphism algebra $\mathrm{End}(A)$ produce Koszul algebra, if $A$ is Koszul?

$\newcommand{\dual}{\mathrm{dual}}\DeclareMathOperator\End{End}\DeclareMathOperator\Fun{Fun}\DeclareMathOperator\Spec{Spec}\DeclareMathOperator\GL{GL}$Around 1986–7 Yu.Manin proposed natural and ...
Alexander Chervov's user avatar
13 votes
1 answer
551 views

Is algebra: ac=ca, bd = db , ad - da = cb - bc ("Manin matrix algebra") - a Koszul algebra?

Question: Consider quadratic algebra with four generators $a,b,c,d,d$ and three relations $ac=ca,bd = db, ad-da = cb - bc$ . Is it a Koszul algebra ? (i.e. Koszul complex is resolution of ground field ...
Alexander Chervov's user avatar
4 votes
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Nice proof that de Rham complex computes Lie algebra cohomology?

If $G$ is a nice enough group acting on a nice enough space $X$, then the relative de Rham complex $$\Omega^\bullet_{X/(X/G)}\ \simeq\ \mathcal{O}_X\otimes\text{Sym}\,\mathfrak{g}^*[-1]$$ is given by (...
Pulcinella's user avatar
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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 ...
Pulcinella's user avatar
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6 votes
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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 $n$-...
skr's user avatar
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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 ...
Tim Campion's user avatar
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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 ...
Lilolance's user avatar
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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, ...
DKS's user avatar
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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 ...
Grisha Taroyan's user avatar
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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 ...
Mare's user avatar
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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 ...
Mare's user avatar
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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 ...
Lilolance's user avatar
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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 ...
Chris H's user avatar
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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 ...
Tommaso Rossi's user avatar
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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 ...
Tim Campion's user avatar
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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/...
Mare's user avatar
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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 ...
Fernando Muro's user avatar
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1 answer
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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 ...
Yining Zhang's user avatar
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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 $i$. ...
Federico Barbacovi's user avatar
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1 answer
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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 ...
Tim Campion's user avatar
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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. ...
Saal Hardali's user avatar
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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 ...
James Cheung's user avatar
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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. ...
Najib Idrissi's user avatar
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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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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 ...
Chris Kuo's user avatar
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19 votes
2 answers
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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 $$F_{Comm_\ast}(z) = \exp(z)-1 \qquad F_{Lie}(z) = \ln(1-z)$$ ...
Tim Campion's user avatar
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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 ...
Pedro's user avatar
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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 \...
Hadrian Heine's user avatar
10 votes
1 answer
623 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 ...
Dan Petersen's user avatar
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7 votes
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383 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" ...
Marc Besson's user avatar
1 vote
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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 ...
Mark.Neuhaus's user avatar
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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 ...
Mark.Neuhaus's user avatar
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9 votes
1 answer
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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 ...
Jonathan Beardsley's user avatar
11 votes
0 answers
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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 ...
Lev Borisov's user avatar
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14 votes
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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 ...
Jonathan Beardsley's user avatar
22 votes
0 answers
844 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 ...
Ben Knudsen's user avatar
7 votes
2 answers
849 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 $1$ ...
Puraṭci Vinnani's user avatar
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"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......
Anette's user avatar
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1 answer
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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 $\...
Puraṭci Vinnani's user avatar
8 votes
1 answer
401 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} & ...
Ben Cooper's user avatar
3 votes
0 answers
246 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 ...
Puraṭci Vinnani's user avatar
12 votes
1 answer
529 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 ...
Sam Gunningham's user avatar
7 votes
3 answers
669 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, ...
Dan Petersen's user avatar
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14 votes
2 answers
666 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 \...
Dan Petersen's user avatar
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5 votes
2 answers
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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 ...
Jan Weidner's user avatar
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23 votes
2 answers
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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 ...
MTS's user avatar
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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. ...
Theo Johnson-Freyd's user avatar
3 votes
1 answer
293 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 ...
Bruce Westbury's user avatar
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 ...
Theo Johnson-Freyd's user avatar
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-...
Harold Williams's user avatar