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18 votes
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Reason to apply the Koszul sign rule everywhere in graded contexts

A precise statement of the conventions (which Jesse is referring to) is that the authors are using the symmetric monoidal structure on graded vector spaces, where the braiding map,, $\tau: V \otimes W ...
Phil Tosteson's user avatar
13 votes

De Rham and Koszul complexes

What you have here is a graded version of an $\mathfrak{sl}_2$-triple. Compare for example the proof of the Lefschetz decomposition and hard Lefschetz theorem in Hodge theory, which is essentially the ...
R. van Dobben de Bruyn's user avatar
12 votes

Where does one go to learn about DG-algebras?

I am not aware of a single book that has all of these topics, but I can list a couple of very good books that can do the job together. First, Benoit Fresse works in this setting quite a bit. His book ...
David White - gone from MO's user avatar
11 votes
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Tensor product of a DGA and an $A_\infty$ algebra

In fact the tensor product of two $A_\infty$ algebras can be made into an $A_\infty$ algebra in an explicit way: there are two constructions, one by Saneblidze-Umble and one by Loday. See the paper ...
Dan Petersen's user avatar
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11 votes

Curvature of nonsymmetric metric tensors?

Consider a bilinear form $b \in \mathcal{C}^\infty (T^*M\otimes T^*M, \mathbb{R})$ and an affine connection $\nabla \colon \mathcal{\Gamma}^\infty(TM) \to \mathcal{\Gamma}^\infty(T^*M\otimes TM)$ ...
Vít Tuček's user avatar
  • 8,157
10 votes

Formality of classifying spaces

This is an old question. But sometimes old questions get answered! Benson, Greenlees, Formality of cochains on BG Here is the abstract: Let $G$ be a compact Lie group with maximal torus $T$. If $|N_G(...
Geordie Williamson's user avatar
9 votes

Where does one go to learn about DG-algebras?

There is a study of DG algebra (DG rings, DG modules, DG categories, DG functors) in the book below. With the related derived categories, etc. Derived Categories
Amnon Yekutieli's user avatar
9 votes
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Sign in May’s General algebraic approach to Steenrod operations

@FKranhold You mean I got it right? You had me fooled. I should apologize for leaving that detail to the reader, but let me give two excuses. First, one does not actually need that detail to prove ...
Peter May's user avatar
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9 votes
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Can any $E_1$ algebra over $\mathbb{F}_p$ be modeled as a dg algebra?

Yes, this is precisely the content of Theorem 7.11 in arXiv:1410.5675, which should be combined with §7.4 of arXiv:1510.04969. In fact, the cited results prove this for any nonsymmetric operad in ...
Dmitri Pavlov's user avatar
8 votes
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Algebras: Homology vs. Resolution as a dg-algebra

From your first description, I guess that what you call "the homology of $A$" is in fact the Hochschild homology of $A$ with constant coefficients, i.e. $HH_*(A;k)$. Please tell me if I got this wrong....
Najib Idrissi's user avatar
7 votes

On the coalgebraic homotopy transfer theorem

We worked out some answers to this question in our paper: arXiv:1904.03585 (Edit: the following answer refers to v1 of the paper on the arXiv!) Here's the short version. There are two possible ...
Dan Petersen's user avatar
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7 votes
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A condition for a dga to be minimal

You are correct and the statement, as you cite it, is wrong. The counterexample you describe is well-known, and is usually cited to demonstrate precisely this failure.
Gabriel C. Drummond-Cole's user avatar
7 votes

Curvature of nonsymmetric metric tensors?

A few remarks: First, in a sense, (special cases of) this (are) is very commonly studied. Because a bilinear differential form $g$ as you have defined it can naturally be written as a sum $g = \...
Robert Bryant's user avatar
7 votes
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Who introduced the abstract definition of a DGA?

Search for the earliest appearance of "differential graded algebra" and "DGA" on MathSciNet. The earliest hit is DGA in a review of a 1954 paper of Cartan where DGA (or more ...
KConrad's user avatar
  • 49.8k
6 votes

Formality of classifying spaces

I've only just seen this rather old thread. I've recently been computing with cochains on $BG$ for $G$ a finite group in characteristic $p$, and have some rather surprising conclusions. If $G$ has ...
Dave Benson's user avatar
  • 12.7k
6 votes
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Resolutions by free Differential Graded Algebras

Let $A=k[x_1,\dots,x_m]$ be the algebra of commutative polynomials in $m$ variables over a field (or commutative ring) $k$, of arbitrary characteristic. Denote by $B=\bigwedge(x_1^*,\dots,x_m^*)$ the ...
Leonid Positselski's user avatar
6 votes
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Massey Products on a specific space

Here a general way how to compute Massey Products on nice spaces. The first step would be to realize the space as the geometric realization of a simplicial set. I used the simplicial set given by the ...
HenrikRüping's user avatar
6 votes
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Graded quivers vs "ordinary" quivers and derived categories

I have not heard the slogan and perhaps do not understand the context, but it seems to me that this has nothing to do with the derived categories. For any graded quiver (with or without relations) ...
Leonid Positselski's user avatar
6 votes
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CE(g) for g infinite dimensional

A definition that always works and does agree with that one in the finite-dimensional case is the following: put $$ C^k(\mathfrak{g})=({\Lambda}^k\mathfrak{g})^*=\operatorname{Hom}({\Lambda}^k\...
Vladimir Dotsenko's user avatar
6 votes
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Why is the bar construction of a DG algebra a coalgebra?

This is the type of question with multiple correct answers, because, as you say, it depends very much on what you think the bar construction "is" initially. You say you want to think of $\...
Theo Johnson-Freyd's user avatar
5 votes

Reason to apply the Koszul sign rule everywhere in graded contexts

This is not a complete answer (none will be, really), but there is a definite reason for applying the specific sign convention you've described just when considering graded maps of graded vector ...
Jesse C McKeown's user avatar
5 votes
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Morphisms of Hochschild (or cyclic) homology induced by homotopic maps

$\newcommand{\dd}{\mathrm d}\DeclareMathOperator{\Sym}{Sym}\DeclareMathOperator{\id}{id}$If $f,g:A\to B$ are dga morphisms which are homotopic as chain maps, the maps induced by $f$ and $g$ on ...
Bertram Arnold's user avatar
5 votes
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Homotopy pullback in the category of DG algebras

I was wondering if somebody could tell me the definition of homotopy pullback. Suppose $\def\C{{\cal C}}\C$ is a relative category, i.e., a category equipped with a subcategory, morphisms in which ...
Dmitri Pavlov's user avatar
5 votes
Accepted

Equivalences of categories of complexes of modules

The answer is yes by the same type of Morita theory, namely $Z(Ch(R))\cong R$, where $Z(A)$ is $End(id_A)$, the ring of endomorphisms of an abelian category EDIT : sorry, I hadn't seen that you were ...
Maxime Ramzi's user avatar
4 votes

Do chain homotopic maps between dg-algebras induce "same" maps on dg-modules

If I understand your question correctly, the answer is no. Here is a counterexample. Let $A=k\langle 1,y\rangle$ with zero differential and trivial unital multiplication (put $y$ in degree $1$ for ...
Gabriel C. Drummond-Cole's user avatar
4 votes

Homotopy colimit of a simplicial DGA

Conditions under which the answer is "yes", for a general monad $T$ (not just the free monoid monad) and a general model category (not just Ch(R)) are given in Batanin-Berger Tame Polynomial Monads, ...
David White - gone from MO's user avatar
4 votes

A non-formal space with vanishing Massey products?

Many examples of closed highly-connected manifolds with all Massey products vanishing, yet which are not formal, are given in "The rational homotopy type of $(n-1)$-connected manifolds of dimension up ...
Aleksandar Milivojević's user avatar
4 votes

If a faithfully flat extension of dg/A_$\infty$-algebra is formal, is the original algebra formal (over positive characteristic)?

I'm not sure off hand what the situation is for $A_\infty$-algebras, but for $\mathbb{E}_\infty$-algebras it's worth noting that in many cases generic formality does not imply formality at each fiber. ...
Benjamin Antieau's user avatar
4 votes

Why do some literatures prefer right module to left module when dealing with DG modules?

Well, first of all, this is perhaps really a matter of taste. But there is good taste :) More seriously, the following argument in favour of right modules is not limited to the DG situation: if you ...
Stefan Waldmann's user avatar
4 votes

When may "summand of" be dropped from the definition of perfect dg module?

For a DGA $A$, all perfect modules are equivalent to semi-free modules if and only if $K(A)$ is generated by the class of the rank 1 free module. Furthermore, a particular perfect module is equivalent ...
Ben Wieland's user avatar
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