# Questions tagged [differential-graded-algebras]

The differential-graded-algebras tag has no usage guidance.

159
questions

**5**

votes

**2**answers

133 views

### Curvature of nonsymmetric metric tensors?

Consider a smooth manifold $M$ of arbitrary dimension. We have notions of psuedo-Riemannian or Riemannian metrics on a manifold, and they differ in the slightest way of being positive-definite or not. ...

**4**

votes

**0**answers

138 views

### Finite CDGA model for a compact manifold

Is it true that a compact smooth manifold always has a finite-dimension commutative dg algebra model?
Same question can be asked about compact CW complexes.
More generally, is it true that a CW ...

**2**

votes

**1**answer

128 views

### When is the module of Kahler differentials free?

As the title says, when is the module of Kahler differentials a free module? In particular, are there known conditions or criterions that could be met that ensures that it will be free?
For example, ...

**5**

votes

**0**answers

205 views

### Kan extensions between categories of monoid objects

Let $K\colon\mathcal{A}\longrightarrow\mathcal{B}$ be
a functor between $\mathcal{V}$-enriched categories. Endowing $\mathcal{A}$ and $\mathcal{B}$ with promonoidal
structures, we obtain induced ...

**4**

votes

**0**answers

62 views

### Simple, explicit, functorial cylinder object in CDGA

In the model category of graded commutative dg-algebras CDGA over $\mathbb{Q}$ (with the projective model structure) there is a simple, functorial construction of a path object given by tensoring with ...

**2**

votes

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79 views

### Shklyarov's Euler class in geometrical setting

In paper HIRZEBRUCH-RIEMANN-ROCH THEOREM FOR DG ALGEBRAS Shklyarov provides construction which for a given dg-algebra (over field $k$, proper, $k$-smooth) $A$ and perfect $A$-module $M$ associate en ...

**2**

votes

**0**answers

65 views

### Linearity of a dg category $C$ over $HH^0(C)$

Let $C$ be a pre-triangulated dg-category over a field $k$ whose Hochschild cohomology groups $\operatorname{HH}^*(C)$ are concentrated in non-negative degree (cohomologically). Is $C$ Morita ...

**3**

votes

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60 views

### Applying a Hochschild cocycle to a Maurer-Cartan element: how one should think of this?

Let $C^{\bullet}(A,M)$ be the Hochschild cochain complex of a DG-algebra $A$ with coefficients in a DG-bimodule $M$. Let $\zeta \in C^0(A,M)$ be a cocycle. Let $a \in A$ be a Maurer-Cartan element, $d(...

**7**

votes

**1**answer

225 views

### Skew differential graded algebra

A sigma, or skew, derivation is a natural generalisation of the
notion of derivation depending on an algebra automorphism $\sigma$ which
when equal to $id = \sigma$ reduces to the usual notion of a
...

**5**

votes

**0**answers

150 views

### Sullivan minimal model in the case of $H^1(V)\neq 0$

Is there a simple construction of a Sullivan minimal model $\Lambda U \rightarrow V$ in the case that $H^1(V)\neq 0$? Do you have a reference? I envisage a degree-wise construction as in the case of $...

**2**

votes

**0**answers

103 views

### Does the functor sending a DGA to its zeroth component admit a right adjoint?

Let $A$ be a ring and write $\underline{A}^\bullet$ for the associated trivial DGA. We have a functor
$$\mathrm{ev}_0\colon\mathbf{dgAlg}_{\underline{A}^\bullet}\longrightarrow\mathbf{Alg}_A$$
sending ...

**1**

vote

**0**answers

55 views

### Functoriality of Hochschild cohomology for Drinfeld quotients

Let $C$ be a dg category and $C \to D$ a Drinfeld localization. Is there an induced pushforward map on $\operatorname{HH}^*(C) \to \operatorname{HH}^*(D)$, where $\operatorname{HH}^*$ denotes the ...

**6**

votes

**0**answers

142 views

### What is the definition of homotopy flat connections?

What is a definition of a homotopy flat connection - in the context of differential forms with values in a dg algebra

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vote

**0**answers

44 views

### Universal bimodule for homotopy biderivations

Recall the commutative story: for a commutative algebra $A$, its module of differentials $\Omega (A)$ is characterized by the universal property that any derivation $\delta \colon A \to M$ is in a ...

**2**

votes

**0**answers

87 views

### Reference request: Differential graded structures in mixed characteristic

I am looking for references/papers on differential graded structures and their applications in mixed characteristic. The following I have discuss differential graded algebras in the general, not in ...

**1**

vote

**0**answers

50 views

### Simplicial differential graded algebra and a filtration

Let $A$ be a simplicial differential algebra, i.e. for each $n \in \mathbb{N}$ a differential graded algebra $(A_n,d_n)$ and for each weakly increasing map $f \colon [n] \to [m]$ a morphism $f_* \...

**11**

votes

**1**answer

342 views

### De Rham and Koszul complexes

Consider the algebraic de Rham complex of the $n$-dimensional plane: this is merely
$$\ldots\rightarrow Sym(V^*)\otimes\bigwedge^{k}V^*\rightarrow Sym(V^*)\otimes\bigwedge^{k+1}V^*\rightarrow\ldots
$$...

**1**

vote

**0**answers

29 views

### Does a homologically bounded dg A-module admit a “locally finite” semi-free resolution

Let $A$ be a bounded dg-algebra whose underlying algebra is Noetherian and such that $H^*(A)$ is Noetherian. Let $M$ be a cohomologically bounded dg-module over $A$, whose cohomology groups are ...

**2**

votes

**0**answers

77 views

### $\mathbb{Z}_2$-grading by Hodge star operator (for signature theorem)

This question may be a bit low level for MO but I have not received any attention from the SE post.
Consider the algebra of exterior forms $\bigwedge T^*M$ on an even dimensional $n$-manifold $M$. We ...

**5**

votes

**0**answers

105 views

### The interaction between differentials on a graded ring and chain-homotopy equivalences

I am wondering about the following question:
Given a differential graded algebra $A$, how many other differentials can we put on the underlying graded ring of $A$, which are also chain-homotopy ...

**1**

vote

**2**answers

247 views

### Reason to apply the Koszul sign rule everywhere in graded contexts

The Koszul sign rule is a sign rule that arises from graded-commutative algebras. For instance, let $\bigwedge(x_1,\dots, x_n)$ be the free graded-commutative algebra generated by $n$ elements of ...

**1**

vote

**0**answers

67 views

### Construct $A_\infty$ bimodules maps from dg-maps

Let $ A $ be a dg-algebra. Let $U,V,W$ and $Z$ be dg-bimodules over $A$-$A$. Suppose I have cofibrant replacements $\pi_U : Up \rightarrow U$ (as right dg-module) and $\pi_W : pW \rightarrow W$ (as ...

**3**

votes

**0**answers

100 views

### Braided monoidal categories

I know that it has been shown that $E_2$ algebra objects in Categories are simply braided monoidal categories.
In particular, Lurie says that an $E_2$-monoidal structure on the infinity-category $N(C)$...

**5**

votes

**1**answer

375 views

### Sign in May’s General algebraic approach to Steenrod operations

In the first section of J. P. May’s General algebraic approach to Steenrod operations, May defines for $\pi\subseteq\Sigma_r$ an integer $q\in\mathbb{Z}$ and a commutative ring $\Lambda$, the $\Lambda\...

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vote

**0**answers

34 views

### Regarding linear splitting of lie algebra morphism and their CE complexes

The main question here is to ask if anyone has ever seen/researched the map $\alpha$ below and get a reference regarding it. Also, if anyone tells me the equivalent conditions to $\alpha=0$, I would ...

**4**

votes

**0**answers

121 views

### Differential graded Lie algebra over an ordinary Lie algebra

Given a dg (differential graded) Lie algebra $L$ and an ordinary Lie algebra $\mathfrak{g}$,
is there written somewhere a formal definition of $L$ as a dg Lie algebra over $\mathfrak{g}$?

**1**

vote

**1**answer

113 views

### Is a finite dimensional graded algebra isomorphic to the equivariant de Rham complex of a Lie group?

Edit: According to essential comment of YCore I revise the question.
Let $A$ be a finite dimensional graded algebra which is a unital, super commutative and associative algebra. Is there a Lie group $...

**2**

votes

**0**answers

89 views

### Free DGA given a map and cohomology groups

Why do Free DGAs on a morphism often give the same (co)homology as other (co)homologies?
Here is the example that comes to mind first:
Example: Let $R$ be a ring, let $A$ be an $R$-algebra, let $M$ ...

**4**

votes

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96 views

### Star product on functions of a Poisson-Lie group

Consider a Poisson-Lie group $G$, with whatever additional requirements (quasi-triangular, compact, simply connected).
We can consider $G$ as a Poisson Manifold and apply Kontsevich formality to ...

**2**

votes

**0**answers

121 views

### When is $C\text-\mathsf{dg\text-mod}$ determined by the connective base changes?

I'm using cohomological gradings.
For $C\in k\text-\mathsf{cdga}$ (where $k$ can be taken of characteristic 0), a morphism $C\to A$ to a connective dg-algebra $A\in k\text-\mathsf{cdga}_{\leq0}$ ...

**2**

votes

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52 views

### Homomorphism or derivation conserving irreducibility

Let $R$ be a integral domain and $\phi$ be an automorphism of $R$. For a given element $x \in R$, we consider a sequence $(\phi^n(x))_{n=0}^{\infty}$.
I wonder if there is any related theory to ...

**2**

votes

**0**answers

88 views

### On the definition of Tor over differential graded algebras

Let $R$ be differential graded algebra, $M$ a left-module over $R$ and $N$ a right-module over $R$. Further, let $P^{*}$ be a proper projective resolution of $M$. I have seen that $Tor_R(N,M)$ is ...

**1**

vote

**0**answers

70 views

### Two definitions of minimal models

Is there any relationship between both definitions of minimal models? (the couple of definitions I know are the one mentioned in Lefèvre's thesis, in the sense that the differential is zero, and the ...

**2**

votes

**0**answers

72 views

### Derivations of algebras graded by a group

Let $A$ be an algebra. A derivation of $A$ is a linear map $d:A \rightarrow A$ such that $d(ab)=d(a)b + a d(b)$ for $a, b \in A$.
If $A$ is a $\mathbb{Z}-$graded algebra, where $\mathbb{Z}$ is the ...

**6**

votes

**1**answer

153 views

### Is the existence of $A_{\infty}$-inverse a consequence of Homotopy Transfer Theorem?

Let $k$ be a field of characteristic $0$ and $(A,d_A)$, $(B,d_B)$ be two differential graded (dg) algebras over $k$. Let $f: A\to B$ be a closed degree $0$ map of dg-algebras and $g: B\to A$ be a map ...

**10**

votes

**1**answer

326 views

### If C is a cocomplete coalgebra, then $\psi:C\rightarrow B\Omega C$ is a filtered quasi-isomorphism

I am reading the PhD thesis thesis of Kenji Lefèvre-Hasegawa and the corresponding errata by Bernhard Keller, my question is about the first error found in the thesis. Lemma 1.3.2.3 c states 'the ...

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votes

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152 views

### DG-Modules over CDG-algebras in the sense of rational homotopy theory

I don't know if this question is elementary or not. Suppose we have a rationalization $X_\mathbb{Q}$ of a simply connected topological space $X$. Then we can construct a CDG-algebra - a minimal ...

**10**

votes

**1**answer

359 views

### On the coalgebraic homotopy transfer theorem

Let $A$ be a dg algebra, say over a field. The Homotopy Transfer Theorem says that $H(A)$ can noncanonically be given the structure of $A_\infty$-algebra, extending the induced multiplication on $H(A)$...

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votes

**0**answers

111 views

### Reference Request: A “Chevalley-Eilenberg”-style formulation of the $L_\infty$ algebra minimal model theorem?

The nicest definition of $L_\infty$-algebras ---which I will call a "Chevalley-Eilenberg" style definition after the obvious analogy with the Chevalley-Eilenberg differential of Lie algebras--- is the ...

**7**

votes

**1**answer

174 views

### Tensor product of a DGA and an $A_\infty$ algebra

In general there seems no way to naturally define the tensor product of two $A_\infty$ algebras $A$ and $B$. But, if $(A, m^A_1,m^A_2)$ is only a DGA(differential graded algebra) and $(B, m^B_k, k\ge ...

**4**

votes

**1**answer

226 views

### Algebras: Homology vs. Resolution as a dg-algebra

My question is what is the relation (if any) between the following two notions.
Starting from an augmented algebra $A$ over a field $k$, one way to compute the homology of $A$ is to find a projective ...

**7**

votes

**0**answers

166 views

### Is $\text{DGA}^{-}$ a monoidal model category?

Let $\text{DGA}^{-}$ denote the category of non-positively graded differential graded algebras with differentials of degree $+1$. It is well-known that $\text{DGA}^{-}$ has a model structure with
...

**4**

votes

**1**answer

170 views

### DGA for a general abelian category

Differential graded algebras dga-s are fundamental objects of study in homological algebra and category theory. On the nlab webpage, they are defined as follows:
a dga is a monoid in the symmetric ...

**3**

votes

**0**answers

178 views

### Wrong way Poincare duality for Calabi-Yau dg-algebras?

Let $A$ be a smooth compact Calabi-Yau dg $k$-algebra of dimension $w$. It is widely known (e.g. Atsusi Takahashi proposition 2.4) that in such situation we have non canonical isomorphism of $A^{en}$-...

**3**

votes

**1**answer

289 views

### Homology of bar complex vs homology of indecomposables

$\require{AMScd}$
Background: This question is about the bar and cobar constructions, and their relationship with the indecomposables of a dg-algebra. A brief summary of the bar and cobar ...

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95 views

### Explicit description of periodic map $S : HC_{i} \to HC_{i-2}$ for dg and $A_\infty$ algebras

Let $A$ some associative unital $k$-algebra, let $HC_*(A)$ is cyclic homology of $A$ and $HH_*(A)$ is hochschild homology of $A$. Then we have Connes exact sequence:
$$ ... \xrightarrow[]{} HH_n(A) \...

**9**

votes

**1**answer

323 views

### Why does passage to DG categories cure non-locality of derived categories?

In the famous book 'Residues and duality', the author notes that one of the principal difficulties in constructing the exceptional inverse image functor $f^{!}$ is that the derived category of ...

**6**

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270 views

### Correct notion of chain homotopy for linearized homology of augmented DGAs?

$\require{AMScd}$
Preliminaries: Let $(A,\partial)$ be a differential graded $k$-algebra with an augmentation $\epsilon$. That is, $\epsilon$ is a DGA map $\epsilon:(A,\partial) \to k$ where $k$ is ...

**5**

votes

**1**answer

111 views

### A condition for a dga to be minimal

I'm reading a book "Complex Geometry" by Daniel Huybrechts. In this book he says that a simply connected dga satisfying some conditions must be minimal. (p.147, Remark 3.A.13) I tried to prove this ...

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215 views

### Has anyone seen this construction of dg algebras?

Let $A$ be an associative algebra, $M$ a right $A$-module. Suppose we are given an $A$-module homomorphism $M \to A$. Then we can make $M$ itself into an associative algebra via the multiplication
$$ ...