# Questions tagged [differential-graded-algebras]

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### 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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### 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 ...
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### 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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### 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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### 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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### 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 ...
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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 ...
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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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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 ...
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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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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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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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