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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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
14 votes
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784 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 ...
Jonathan Beardsley's user avatar
11 votes
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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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7 votes
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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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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
7 votes
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399 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 ...
Mark.Neuhaus's user avatar
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5 votes
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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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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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5 votes
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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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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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4 votes
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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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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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4 votes
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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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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
3 votes
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249 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 ...
Tommaso Rossi's user avatar
3 votes
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153 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 \...
Hadrian Heine's user avatar
3 votes
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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
2 votes
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62 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 ...
Grisha Taroyan's user avatar
2 votes
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133 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 $C$...
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1 vote
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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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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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