Questions tagged [differential-graded-algebras]

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0answers
75 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 $M$ ...
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63 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
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0answers
109 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 $A\in k\text-\mathsf{cdga}_{\leq0}$ ...
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46 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 ...
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83 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 ...
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58 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
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47 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 ...
4
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1answer
123 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
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1answer
244 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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145 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 ...
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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 ...
7
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1answer
144 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
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1answer
201 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
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161 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
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1answer
147 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 ...
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167 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
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1answer
248 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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92 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) \...
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1answer
308 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 ...
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227 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
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1answer
102 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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206 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 $$ ...
8
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1answer
447 views

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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216 views

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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1answer
483 views

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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108 views

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 ...
4
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1answer
301 views

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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1answer
169 views

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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171 views

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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231 views

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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223 views

(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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99 views

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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1answer
182 views

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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63 views

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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1answer
298 views

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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3answers
939 views

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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137 views

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 ...
3
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1answer
109 views

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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1answer
183 views

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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1answer
418 views

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 ...
5
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1answer
185 views

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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0answers
251 views

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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72 views

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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0answers
84 views

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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0answers
72 views

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 $C$ ...
7
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1answer
232 views

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 ...
8
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1answer
276 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 ...
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1answer
115 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
282 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 $\mathcal{S}\mathcal{F}(\...