**5**

votes

**0**answers

65 views

### $E_{\infty}$ $R$-algebras vs commutative DG $R$-algebras vs simplicial commutative $R$-algebras

I've been trying to understand better the relation between the basic blocks of derived algebraic geometry. More precisely, I'm trying to understand the relation between the DG approach, the spectral ...

**8**

votes

**1**answer

210 views

### Why do the model structures on dg-algebras and on dg-categories are not compatible?

First we talk about dg-algebras. According to this n-lab page, we write $dgAlg$ for the category of cochain dg-algebras in non-negative degree over a field $k$ of characteristic $0$. Write ...

**7**

votes

**1**answer

178 views

### Frobenius $A_{\infty}$-bialgebras?

Recall that a finite dimensional associative algebra $A$ over a field $k$ is called a symmetric Frobenius algebra (sometimes called "closed" Frobenius algebra) if its equipped with a symmetric non ...

**3**

votes

**1**answer

202 views

### Model structure on non-negative differential graded algebras with homological grading

I was wondering if there exists a model structure on the category of non-negative differential graded algebras with homological grading. To be more precise: Let $Ch_{k}$ the model category of ...

**5**

votes

**0**answers

120 views

### Matrix factorizations as a dg-category?

Matrix factorizations (in the graded case) give a triangulated category. I would imagine that there should be an underlying dg-category. Is there such a definition, and if so, where can I find it in ...

**4**

votes

**1**answer

255 views

### Massey products and $A_{\infty}$ structures

I know the general theorem of Kadeishvili which says that, for a DGA $C$, when $H^{i}(C)$, $i\geq 0$, is free, $H(C)$ can be made into an $A_{\infty}$ algebra. If my understanding is correct, the ...

**1**

vote

**0**answers

52 views

### Turning left modules into right modules over a homotopy Gerstenhaber algebra

For simplicity's sake, let $A$ be a dg-algebra over $\mathbb{Z}/2\mathbb{Z}$.
In the case when $A$ is a commutative algebra, we can turn a left $A$ module into a right $A$ module trivially. Of course ...

**5**

votes

**0**answers

100 views

### Hochschild cohains-type models for smooth functions on shifted cotangent bundles

Let $M$ be a smooth manifold. Then, by the well known Hochschild-Kostant-Rosenberg theorem, the cochain complex $C^*(C^\infty(M))$ of Hochshild cochains on the algebra $C^\infty(M)$ of smooth ...

**3**

votes

**0**answers

100 views

### model structure of noncommutative non-negatively graded DGAs

Take non-negatively graded differential graded algebras, which are vector spaces over some field (the last assumption just to simplify things). We have the usual $d^2=0$, but the product is not ...

**15**

votes

**2**answers

810 views

### Is homology finitely generated as an algebra?

If a differential graded algebra is finitely generated as an algebra, is its homology finitely generated as an algebra?
Is it easier if we impose any of the three conditions: characteristic zero; ...

**3**

votes

**0**answers

108 views

### When does a commutative DGA have a finitely generated quasi-free resolution?

Suppose that $A$ is a commutative dg-algebra (say over base $k$) which is bounded in non-positive degree (with cochain complex conventions). There exists a quasi-free resolution of $A$. My question ...

**2**

votes

**2**answers

142 views

### Does semi-free behave well under totalization

Suppose I have a dg algebra $(A,d)$ and a chain complex $M^\bullet$ of semi-free $(A,d)$ modules. I am hoping it is true that $ Tot^\coprod (M^\bullet)$ is again a semi-free $(A,d)$ module. Is this ...

**7**

votes

**1**answer

405 views

### DG enhancements of $\ell$-adic derived categories

This question is similar in flavor to Existence of dg realization for 6 functors
Let $X$ be a complex variety and $D(X)$ the bounded derived category of constructible sheaves (the Euclidean topology ...

**10**

votes

**1**answer

244 views

### A cdga for compactly supported cohomology (à la Sullivan's algebra of polynomial forms)

Let $M$ be a smooth manifold, and let $\Omega^\bullet(M)$ be the commutative dg-algebra of differential forms on $M$. It is quasi-isomorphic to the dg-algebra of singular cochains on $M$. If $M$ is no ...

**4**

votes

**0**answers

104 views

### Definition of modules over $C_\infty$-algebras (“commutative $A_\infty$-algebras”)

Let $\Lambda$ be a finite-dimensional associative algebra. We can think of this as an $A_\infty$-algebra with vanishing $m_i$ for $i \ne 2$ and consider $A_\infty$-modules $M$ over it. A list of ...

**9**

votes

**0**answers

158 views

### When were bordered Heegaard Floer homology's DA bimodules invented?

This is less of a strictly mathematical question and more of a reference request.
In their paper "Bordered Heegaard Floer Homology (http://arxiv.org/pdf/0810.0687.pdf)," Lipshitz-Ozsvath-Thurston ...

**8**

votes

**3**answers

612 views

### The cohomology plus what characterizes the rational homotopy type?

For simplicity let me work only with connected and simply connected spaces. "Space" will mean a space of this type.
A space is rational if its homotopy groups are rational vector spaces ...

**5**

votes

**1**answer

212 views

### Equivariant Formality

Let $G$ be a finite group and $\mathcal{A}$ be a $dg$-algebra. Assume $G$ acts on $\mathcal{A}$, i.e. there exists a homomorphism $G\to {\rm Aut}_{dg}(\mathcal{A})$.
Assume further there exists a ...

**4**

votes

**0**answers

125 views

### Formal DG-algebras

Sorry for this question but I really have difficulties with model categories.
Usually a $dg$-algebra $A$ is called formal, if there exists a $dg$-algebra $B$ and quasi-isomorphisms $$A\leftarrow B\to ...

**3**

votes

**0**answers

66 views

### Quasi-isomorphisms and Subalgebras

Let $A$ and $B$ $dg$-algebras over $\mathbb{C}$. If there exists an isomorphism $f:A\to B$, then every subalgebra $A'$ of $A$ is isomorphic to the subalgebra $f(A')$ of $B$.
What is if $f$ is ...

**2**

votes

**0**answers

253 views

### Gauss Manin connection in algebraic geometry and DG setting

E.Geltzler defined Gauss Manin connection on periodic cyclic homology of smooth proper DG algebra.I just began to learn his definition.But it seems that his construction is very different to the Gauss ...

**4**

votes

**1**answer

179 views

### Complexes of sheaves with locally constant cohomology versus $C_{*}(\Omega M)$-modules

Let $M$ be a nice, connected topological space. Assume it is a manifold, if you like.
There are two rather similar looking differential-graded (dg) categories that one can associate to $M$ that ...

**1**

vote

**0**answers

266 views

### Tensor product of d.g-algebras

I'd like to prove that the tensor product functor $- \otimes Y$, where $Y$ is a d.g-algebra over a field of characteristic 0, preserves finite products of d.g-algebras. This statement is in a paper by ...

**2**

votes

**1**answer

300 views

### Semi-free resolutions

Let $\mathscr{C}$ be a DG category (not much will be lost if you assume that $\mathscr{C}$ has one object, i.e. is a DG algebra). One way to construct the unbounded derived category of ...

**3**

votes

**2**answers

320 views

### What is the correct definition of the (derived) tensor product over a dg-algebra?

Let $A_\bullet$ be a dg-algebra over a field $k$. Let $M_\bullet$ (resp. $N_\bullet$) be a right (resp. left) $A_\bullet$-module. There is then a notion of the derived tensor product:
...

**6**

votes

**2**answers

493 views

### Higher commutators in E_n algebras and the Maurer--Cartan equation

Let $A$ be an associative algebra in $dgVect_k$. Then the commutator $[\cdot,\cdot]:A\otimes A\to A$ defined by $[x,y]=xy-(-1)^{|x||y|}yx$ gives $A$ the structure of a (dg-)Lie algebra. The ...

**2**

votes

**2**answers

539 views

### Homological smoothness implies projectivity?

Let $A$ be a unital associative algebra over a commutative noetherian ring $R$. Assume that $A$ is homologically smooth, which means that $A\in D_{perf}(A\otimes_R^L A^{op})$, which also means that ...

**2**

votes

**1**answer

202 views

### Formality of classifying spaces (for not necessarily connected groups)

As should be evident from the title this question has a similar flavor to:
Formality of classifying spaces
However, unlike Geordie's question, I will be working with torsion free coefficients (say ...

**4**

votes

**2**answers

171 views

### Natural Isomorphism of $S(V[1])$ and $(\bigwedge V)[n]$

Let $V:=\oplus_{j\in\mathbb{Z}}V_j$ be a graded $\mathbb{F}$-vector space over
the field $\mathbb{F}$. The graded tensor product of graded vector spaces is given
by
$V \otimes W:= \oplus_{j\in ...

**0**

votes

**0**answers

111 views

### Decalage isomorphism and algebra structure

Consider the symmetric monoidal category of graded vector spaces in which the symmetric structure is given by the Koszul sign rule. Assume if necessary that the ground field is of characteristic zero.
...

**4**

votes

**0**answers

193 views

### Coordinate free Koszul-Tate resolution

Tate's original construction of the Koszul-Tate resolution involved choosing cocycles representing the cohomology to be killed. Where is it written in a coordinate free treatment, perhaps via a ...

**1**

vote

**1**answer

283 views

### Coequalizer in category of dg-algebras

It is known that there is a model structure on category of dg algebras (non-commutative over arbitrary commutative ring). In particular it is complete and co-complete category. My question is how to ...

**1**

vote

**1**answer

159 views

### Contraction of graded vector fields on de Rham complex

Given a commutative algebra $A$ smooth over a field $k$ of characteristic zero, the module of K\"ahler differentials $\Omega^{1}$ is projective of finite rank and so the sum of all wedge powers ...

**1**

vote

**0**answers

97 views

### bar construction for algebras with unusual grading of d

The bar construction is usually applied to differential graded algebras with differential $d$ of degree +1 or -1. Using multiple (de)suspensions, it also works for $d$ of any degree $\neq 0$. Is this ...

**3**

votes

**1**answer

362 views

### Do the solutions of the Maurer--Cartan equation form a simplicial set?

The Maurer--Cartan equation is the equation:
$$d\gamma+\frac 12[\gamma,\gamma]=0$$
where $\gamma$ represents a degree one element in a differential graded Lie algebra $\mathfrak g^\ast$. Let's denote ...

**5**

votes

**2**answers

596 views

### Homotopy Transfer Theorem for Differential Graded Associative Algebras

As in Algebra+Homotopy=Operad by Bruno Vallette, let $A$ with multiplication $\nu$ be a differential graded associative algebra equipped with degree +1 map $h$ and let $H$ be a chain complex such that ...

**7**

votes

**1**answer

211 views

### Morphisms between formal dg-algebras

Suppose we are given a map $f:A \rightarrow B$ between two dg-algebras which are formal.
Is the map $f$ also "formal" in some sense?
More precisely can we find isomorphisms $\phi_A:A\rightarrow ...

**2**

votes

**1**answer

238 views

### Negative and periodic cyclic homology of a semi-free cdga

Let $A$ be a semi-free commutative differential graded algebra in non-negative degrees with differential $d$ of degree $-1$, over a field of characteristic zero. Recall that semi-free means that if ...

**13**

votes

**2**answers

344 views

### Exact DG Poisson algebra

A symplectic manifold gives rise to a Poisson algebra. If the symplectic form is exact,
how is this revealed in the algebra?

**5**

votes

**1**answer

710 views

### What is knot contact homology?

Recently, it was conjectured by the paper of Aganagic and Vafa that the $Q$-deformed $A$-polynomials can be identified with the augmentation polynomials of the knot contact homology. The $Q$-deformed ...

**1**

vote

**1**answer

364 views

### Associated graded of a filtration of a tensor product

I'm trying to understand a part of the PhD thesis of Kenji Lefèvre-Hasegawa (e.g. available here). My question is about the proof of Lemma 1.3.2.3b stating:
Remarquons que nous avons un ...

**3**

votes

**2**answers

665 views

### How to define the equivalence of Maurer-Cartan elements in an $L_{\infty}$-algebra?

First let $L^{\bullet}$ be a pro-nilpotent differential graded Lie algebra (dgla). We have the set of Maurer-Cartan elements in $L^{\bullet}$ ($MC(L^{\bullet})$) which are $\alpha \in L^1$ such that ...

**9**

votes

**0**answers

296 views

### Algebras Morita equivalent to their centers

Hi,
I wonder if there is a name for:
1) Algebras which are Morita equivalent to their centers, or
2) dg-algebras which are derived Morita equivalent to their Hochshild cohomology?
For instance, ...

**7**

votes

**0**answers

242 views

### Computer Algebra solution for simplicial resolutions for André-Quillen cohomology

Hello,
I would like to experiment with André-Quillen (co)homology. Especially for singular rings.
A key problem is that the construction of a simplicial resolution of a ring seems to require a rather ...

**2**

votes

**2**answers

642 views

### Origin of the sign convention in the Tensor product of graded vector spaces

Suppose $V := \bigoplus_{i \in \mathbb{N}}V_i$ and $W := \bigoplus_{i \in \mathbb{N}}W_i$ are $\mathbb{N}$-graded vector spaces. Then their graded tensor power is defined by
$V \bigotimes W := ...

**1**

vote

**2**answers

336 views

### Sign convention for derivations in CDGAs

I'm trying to understand K\"ahler differentials for CDGAs (commutative differential graded algebras). A few minor things have been confusing me all afternoon. Pointers and references are welcome.
Let ...

**4**

votes

**2**answers

561 views

### Reference request - CDGA vs. sAlg in char. 0

Hello,
Are the model categories of simplicial commutative algebras over $k$ and that of commutative differential graded algebras (in negative cohmological dimension) Quillen equivalent in char. 0 (or ...

**4**

votes

**2**answers

210 views

### Reference for Tensors on graded spaces needed

Is there a good introduction to
1.) Tensor (co)algebras on graded vector spaces ?
2.) Tensor (co)algebras on graded modules ?
In the research field of $L_\infty$-algebras there is some stuff, but ...

**4**

votes

**0**answers

130 views

### Have equivalent commutative dg-algebras equivalent monoidal derived categories?

Let $R,S$ be dg-algebras and $f:R \rightarrow S$ be a quasi-isomorphism. Then $f$ induces an equivalence between their derived categories of dg-modules.
If in addition $R,S$ are graded commutative ...

**6**

votes

**2**answers

419 views

### Somewhat general question that includes: “Do quasi-isomorphic cdgas have quasi-isomorphic spaces of derivations?”

Question: Given two quasi-isomorphic dg commutative algebras (over a field of characteristic zero, if you like), to what extent do their various homological geometric data agree?
Example: Given a dg ...