The dg-algebras tag has no wiki summary.

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### $B_\infty$ algebra structure on $C(\mathcal{A}, \mathcal{A})$

If $A$ is an associative unital dg algebra it is known ( see for example: http://arxiv.org/abs/hep-th/9403055) that its Hochschild complex has the structure of a $B_\infty$ algebra. Given a dg ...

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### 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 ...

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### 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 ...

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### 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:
...

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### 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 ...

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463 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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?

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

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

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### 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, ...

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

Hello,
I would like to experiment with AndrÃ©-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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### Constructible sheaves and dg-modules

Let $M$ be a smooth manifold, $A_M$ the de Rham algebra of $M$, $D_{A_M}$ the derived category of the category of differential graded (dg) $A_M$-modules and $D^+_c(M)$ the bounded below constructible ...

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### Do I need to know what an infinity-Gerstenhaber algebra is, and if so, what is it?

I am in the following situation. I have two (rather explicit and specific) dg commutative algebras $R,S$ over a field of characteristic $0$. In fact, $S$ is an $R$-algebra, in that I have a map $R ...

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### Is there a model category structure on non-negatively graded commutative chain algebras?

Let $\mathtt{DGA}$ be the category of non-negatively graded DG chain algebras, and $\mathtt{DGA}$* the category of non-negatively graded cochain algebras. Let $\mathtt{CommDGA}$ be the full ...

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### simplicial deRham complex and model category structure

To every simplicial manifold is associated its simplicial deRham complex.
Is there any literature that discusses explicitly to which extent this classical construction, regarded as a (contravariant) ...

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### On-the-nose commutative cup product $\Longrightarrow$ characteristic $0$?

I was discussing with a student today the nature of the non-commutativity of cup-product at the level of cochains. In trying to explain what happens, I came up with the following statement.
...

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### For which dg-algebras is a dg-module with bounded cohomology quasi-isomorphic to a bounded dg-module?

Hello!
Let S be a commutative local Noetherian base ring and A be a dg-S-algebra.
Suppose M is a bounded above dg-A-module with bounded cohomology. Does there exist a bounded dg-A-module isomorphic ...

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### Formality of classifying spaces

Let $G$ be a compact Lie group (or reductive algebraic group over $\mathbb{C}$), and let $BG$ be its classifying space. Fix a prime $p$. Let $\mathcal{A}$ denote the dg algebra of singular cochains on ...

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### Lie's third theorem via differential graded algebras?

Dennis Sullivan, "Infinitesimal computations in topology", Publ. IHES: At the end of section 8, he writes, among other things, roughly the following.
Let $\mathfrak{g}$ be a (finite-dimensional, ...

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### The growthrate of the homology of $H_*(M^{\otimes_A n})$ for a DG-bimodule $M$

Suppose you have an DG-algebra $A$, and a DG-bimodule $M$ over $A$. Under which conditions will the rank of the bimodules $H_*(M^{\otimes_A n})$ will grow exponentially in terms of $n$? Here ...

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### What is the (Koszul? derived?) interpretation of a pair of Lie algebras with the same cohomology?

There are many words and sentences in mathematics that I basically completely don't understand, including the words "Koszul" and "derived". But rather than ask for a complete description of such ...

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### Homotopy flat DG-modules

A right DG-module $F$ over an associative DG-algebra (or DG-category) $A$ is said to be homotopy flat (h-flat for brevity) if for any acyclic left DG-module $M$ over $A$ the complex of abelian groups ...

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### Derived algebraic geometry via dg rings?

Jacob Lurie's stuff seems to develop derived algebraic geometry via $E_\infty$ rings and/or maybe something like simplicial commutative rings. Ben Wieland's comment in this question indicates that ...

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### what is the tensor product of two anti-commutative DG-algebras?

I mean what the multiplication law of the tensor product is.

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### Hochschild (co)homology of A and of Mod_A

Let A be an algebra (or dg algebra). Where can I find a proof of HH_*(A) = HH_*(Mod_A) and HH^*(A) = HH^*(Mod_A)? (And does this hold for any A?) Here Mod_A is, e.g., the category of left A-modules.
...

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### Is every graded manifold affine, and is this definition of graded manifold the right one?

The following definition is from:
Dmitry Roytenberg, "AKSZ-BV formalism and Courant algebroid-induced topological field theories", Letters in Mathematical Physics, 2007 vol. 79 (2) pp. 143-159, ...

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### dg objects: Z-graded vs. Z/2Z-graded

I am wondering: Are there any general theorems or principles relating the theory of Z-graded dg objects and the theory of Z/2Z-graded dg objects? I am mainly interested in dg algebras, dg Lie ...

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### Smooth dg algebras (and perfect dg modules and compact dg modules)

Katzarkov-Kontsevich-Pantev define a smooth dg ($\mathbb{C}$-)algebra $A$ to be a dg algebra which is a perfect $A \otimes A^{op}$-module. They say that an $A$-module $M$ is perfect if the functor ...

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### What is 'formal' ?

The key step in Kontsevich's proof of deformation quantization of Poisson manifolds is the so-called formality theorem where 'a formal complex' means that it admits a certain condition. I wonder why ...

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### Do DG-algebras have any sensible notion of integral closure?

Suppose R → S is a map of commutative differential graded algebras over a field of characteristic zero. Under what conditions can we say that there is a factorization R → R' → S ...

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### What is a deformation of a category?

I have several naive and possibly stupid questions about deformations of categories. I hope that someone can at least point me to some appropriate references.
What is a deformation of a (linear, dg, ...

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### Model structure of commutative dg-algebras inside all dg-algebras

Most of the literature considers the standard model category structure on (graded) commutative differential algebras. But this generalizes to all (not-necessarily commutative) dg-algebras.
Details ...

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### Deformation theory and differential graded Lie algebras

There is supposed to be a philosophy that, at least over a field of characteristic zero, every "deformation problem" is somehow "governed" or "controlled" by a differential graded Lie algebra. See for ...