# Questions tagged [dg-algebras]

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

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### 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 ...

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### 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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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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
...

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### 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 ...

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### 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 $A^{en}$-...

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### 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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### 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) \...

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### 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 ...

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### 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 ...

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### 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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### 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
$$ ...

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### Vanishing of H-cohomology

This looks elementary, but somehow I am stuck, please bear with me:
Let $H$ be a differential 3-form, nowhere vanishing, but not necessarily closed. What is a sufficient condition that the sequence ...

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### Defining Massey products as transgressions

Let $A$ be a dg algebra, and $x, z \in A$ cocycles. Let's consider the maps
$$ A \to A \oplus A \to A$$
given by $y \mapsto (xy,yz)$ and $(u,v) \mapsto uz-xv$, respectively. We think of this as ...

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### Is the derived category of $A$-dg-modules as a dg-category coincide with the ordinary definition of derived category?

Let $A$ be a unital dg-algebra over a base field $k$. We consider the category of (unbounded) right $A$-dg-modules with morphisms closed degree $0$ maps. We denote this category by dg-mod-$A$. We ...

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### Which dg-algebras have minimal model which is $A_{fin}$?

$A_{fin}$ algebra it is $A_\infty$ algebra with $m_n = 0$ for $n >> 0$ and $A^i = 0$ for $|i| >> 0$.
Suppose that we have (compact) dg-algebra $A$, we can build $A_\infty$ minimal model ...

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### Graded quivers vs “ordinary” quivers and derived categories

I have heard the "slogan" that graded quivers are (derived) equivalent to ordinary quivers (with this "result" being attributed to Keller) and am looking for a precise statement and a reference.
By a ...

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### Is there a notion of injective, projective, flat, dimension for a differential graded algebra?

Given a differential graded algebra $(A_\bullet,d)$, is there a well-defined notion of a K-injective, K-projective, K-flat dimension of a differential graded module, or even of the category of ...

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### DGLA related to the deformation of hopf Algebras

Recently I was considering Hopf algebras and Drinfeld's twists. I stumbled upon
a certain DGLA one can associate to a Hopf algebra (unital bialgebras actually) by copying the formulas obtained by ...

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### Simple question about DG-algebras

Considering the following conditions for two DG-algebras $A$ and $B$:
1) There exists quasi-isomorphic DG-algebra morphism $A \to B$.
2) There exists a DG-algebra $C$ and two quasi-isomorphic DG-...

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### (co)chain homotopy of dg algebras ($A_{\infty}$ algebras) and other notion of homotopy

In this question I will work over a field of char. $0$. Let $f,g\: : \: A\to B$ be dg algebras morphism between two cochain (commutative) dg algebras $A,B$ (assume positively graded). Let $h$ be a ...

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### DGAs with pointwise Multiplication

The singular cochain complex of a space can be equipped with another product, the pointwise product of two cochains
\[\odot: C^n \otimes C^n\rightarrow C^n \qquad \Psi_1 \odot \Psi_2 (f:\Delta^n\...

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### Do chain homotopic maps between dg-algebras induce “same” maps on dg-modules

Let $A$ and $B$ be two dg-algebras over a field $k$. Let $f, g: A\to B$ be two maps between dg-algebras. We call $f$ and $g$ chain homotopic if there exists a degree $-1$ map $h: A\to B$ such that $f-...

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### Do we have the contravariant Hom exact sequence in a pretriangulated category?

Let $\mathcal{C}$ be a pretriangulated dg-category over $k$. By definition, we call $X\to Y\to Z\to X[1]$ an exact triangle in $\mathcal{C}$ if for any $W$, $C(W,Z)$ is homotopic to cone$(\mathcal{C}(...

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### Homotopy colimit of a simplicial DGA

It seems to be well-known that the homotopy colimit of a simplicial chain complex (unbounded) can be computed by taking the totalization of the associated (half-plane) double complex. The totalization ...

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### Where does one go to learn about DG-algebras?

The theory of differential graded algebras (in char 0) and their modules has numerous applications in rational homotopy theory as well as algebraic geometry.
I'm looking for a reasonably complete ...

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### Do quadratic DGCA's have rational homology series?

Suppose $A$ is a differential graded-commutative algebra (in non-negative degree) over a char-0 field, with $|d| = -1$, such that $A$ is finitely generated in degree $\le 1$ with relations in degree $\...

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### bound quiver of section — the dga version?

Let $X$ be a smooth projective variety, and $\mathcal{L} = \{L_0, \cdots, L_n\}$ be a list of distinct line bundles. The (complete) bound quiver of sections associated with $\mathcal{L}$ is a quiver ...

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### Induction along a quasiisomorphism of DGAs

Given a quasiisomorphism of DGAs $f:A\rightarrow B$ and a DG-module $M$ over $A$.
Is the canonical chain map
\[M\rightarrow B\otimes_A M \qquad m\mapsto 1\otimes m\]
an isomorphism on homology?

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### Is the hom in derived category of a dg-algebra compatible with base field extension?

Let $k$ be a field and $A$ be an ordinary $k$-algebra. Let $M$ and $N$ be two left $A$-modules then we could consider the Ext group $\mathrm{Ext}^i_A(M,N)$ which is actually a $k$-vector space. Let $l/...

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### Resolutions by free Differential Graded Algebras

I am struggling to find references for explicit computations of things like symmetric algebras and resolutions in the dg context. (any pointers in this direction would be highly appreciated!) I have ...

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### Massey Products on a specific space

Let $a,b$ be the canonical generators of $\pi_1(S^1\vee S^1)$ corresponding to the edges with some choice of orientation.
Are there nonzero-Massey products in the cohomology with $\mathbb{F}_2$-...

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### Does derived equivalence imply dg Morita equivalence between DG algebras over field with char$=0$?

Let $A$, $B$ be two DG algebras and $D(A)$, $D(B)$ be derived categories of DG-modules of $A$, $B$, respectively. We call $A$ and $B$ are dg Morita equivalent if there is an $A$-$B$ bimodule $T$ with ...

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### Is the inverse of the criterion of fully-faithfulness of derived tensor product also true?

Let $A$ and $B$ be differential graded algebras over a field $k$ and $D(A)$ and $D(B)$ be the derived categories of differential graded modules.
Let $N$ be an $A$-$B$ bimodule. Then $(-)\otimes^{\...

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### Acyclic extension of free DGA-modules

I want to find some method to do the following: given an DGA-module $M$
over some commutative ring $k$, positively graded ($M_i=0$ if $i<0$), where each component $M_i$ has a free action by the ...

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### Surjections to free commutative dgas

Consider the category $C$ of commutative dgas (unbounded in both degrees) over a ring $R$
(in practice the integers or the integers mod $p$).
Let $F$ be the free functor from chain complexes to $C$ ...

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### Cofibrations in the model structures for non-negative graded (commutative) DG algebras

Let $k$ be a field of characteristic 0. Let $\mathtt{DGA}_{k}^{+}$ denote the category of non-negative graded DG algebras and $\mathtt{CDGA}_{k}^{+}$ denote the category of non-negative graded ...

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### 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 ...

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### 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 ...

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### 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 $\mathcal{S}\mathcal{F}(\...

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### 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, 2)-...

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### 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 ...

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### 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 ...

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### 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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### 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 \...

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### 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 ...

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### 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 ...