18
votes

Accepted

### 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 ...

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 ...

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 ...

11
votes

Accepted

### 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 ...

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 $\nabla \colon \mathcal{\Gamma}^\infty(TM) \to \mathcal{\Gamma}^\infty(T^*M\otimes TM)$ ...

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(...

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

9
votes

Accepted

### 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 ...

9
votes

Accepted

### 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 ...

8
votes

Accepted

### 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....

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 ...

7
votes

Accepted

### 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.

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 = \...

7
votes

Accepted

### 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 ...

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 ...

6
votes

Accepted

### 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 ...

6
votes

Accepted

### 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 ...

6
votes

Accepted

### 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) ...

6
votes

Accepted

### 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\...

6
votes

Accepted

### 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 $\...

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 ...

5
votes

Accepted

### 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 ...

5
votes

Accepted

### 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 ...

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 ...

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 ...

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, ...

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 ...

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. ...

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 ...

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 ...

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