A differential graded algebra is a graded algebra endowed with a differential of degree $1$ respecting the graded Leibniz rule.

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155 views

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

Proposition 6.2 of Formality of DG algebras (after Kaledin) by Lunts reads (with a few additions to clarify notation): Let $k$ be a field of characteristic 0. Let $A$ be an $A_\infty$ algebra ...
2
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1answer
84 views

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

I've been trying to read some papers on differential graded modules (for example, Keller, Deriving DG categories) In most of literature I found about dg-modules, they define them as right modules (Of ...
4
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1answer
180 views

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

Let $\mathcal{A}$ be a small dg category. In Section 1 of Lunts-Orlov http://arxiv.org/pdf/0908.4187v5.pdf, $Perf(\mathcal{A})$ is defined to be the full DG subcategory of ...
4
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0answers
100 views

Fully dualizability of dg-algebras

I am working in the context of fully extended TQFTs and, at the moment, I am trying to find fully dualizable objects in certain $(\infty, 2)-$categories. In particular I know that for the $(\infty, ...
8
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201 views

When does an $E_\infty$ algebra come from a commutative differential graded algebra?

Suppose that $K$ is an $E_\infty$-algebra on a space $X$ (more generally, any ringed topos; also, feel free to assume that $X$ is a point). That is, $K$ is a cochain complex of sheaves on $X$, endowed ...
2
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0answers
117 views

Reference request for a “truncated version” of the de Rham algebra

Let's start on the $n$-torus for sake of simplicity.$\newcommand{\T}{\mathbb T}$ If I understand the relevant definitions correctly, the usual de Rham algebra of smooth differential forms on $\T^n$ is ...
16
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2answers
1k views

Why do people say DG-algebras behave badly in positive characteristic?

It seems to be a common wisdom in derived algebraic geometry that commutative DG-algebras are not, in general, a good model for derived rings, with the stated reason that they behave badly in positive ...
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0answers
161 views

Bockstein cohomology

There is a notion of Bockstein homomorphism $\beta$. I am interested precisly in the case of sequence $$0 \rightarrow \mathbb{Z}/p \mathbb{Z} \rightarrow \mathbb{Z}/p^2 \mathbb{Z} \rightarrow ...
1
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1answer
99 views

dg-resolution of the polynomial algebra

I am intersted in constructing a cofibrant resolution of the commutative polynomial algebra in some number of variables in the category of dg-algebras(not necceserily commutative). The resolutions ...
6
votes
2answers
298 views

Seifert--van Kampen for the loop space dga

I recently noticed that I could mostly prove a special case of the following statement. I think it's true in general, though perhaps only for nice spaces. Let $X$ be a topological space, and ...
7
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0answers
173 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 ...
9
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1answer
299 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
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1answer
216 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
1answer
240 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
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145 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 ...
5
votes
1answer
296 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 ...
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57 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
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121 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
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113 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 ...
16
votes
2answers
868 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
0answers
122 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
2answers
146 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
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1answer
435 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
1answer
264 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 ...
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113 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 ...
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179 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 ...
9
votes
3answers
663 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
1answer
231 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
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136 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
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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
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279 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
1answer
196 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 ...
3
votes
1answer
396 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
2answers
371 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
2answers
518 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
2answers
605 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
1answer
219 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
2answers
182 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 ...
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votes
0answers
134 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
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225 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
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1answer
308 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
1answer
166 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 ...
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0answers
99 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
1answer
390 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
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2answers
752 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
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1answer
218 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
1answer
261 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
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2answers
355 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
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1answer
844 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 ...
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1answer
418 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 ...