Questions tagged [differential-graded-algebras]

A differential graded algebra is a graded algebra with an added chain complex structure that respects the algebra structure.

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When is it enough to consider roofs in the derived category?

In the derived category $D^b(\mathcal{A})$ of an abelian category $\mathcal{A}$, obtained by taking the Verdier quotient wrt. all quasi-isomorphism, a morphism is given by a roof (or span) $ X\...
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Where does the dualization of maps sign come from in graded vector spaces?

In the book Rational Homotopy Theory of Félix, Halperin and Thomas they state the following. Given linear maps $f:V\rightarrow W$ and $g:W'\rightarrow V'$ between graded vector spaces then we have $\...
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Questions wrt. definition of $L_\infty$- and dg-Lie-algebras

I am trying to understand this definition from nLab (Def. 3.2) of $L_\infty$-algebras, and the following example that is supposed to boil down to dg-Lie algebras. What is the difference between ...
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How to show that for an element $x$ in DGA cohomology it is true that $[x]=[-x]$?

Perhaps it is a really dumb question, but I want to use this property and it seems that it must hold. However, if one takes a specific homogeneous element $x$ of a differential graded algebra, then ...
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62 views

Left adjoint to dg-nerve?

In https://kerodon.net/tag/00PK, Lurie introduces the dg-nerve functor from differential graded categories to simplicial sets as a tool to translate statements/ constructions from the dg-context to ...
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Suspension Operator on Graded Algebras

Given a morphism of GAs $f:A\to B$ of degree $-k$, that is, $f(A_n)\subset B_{n-k}$, I want to understand the sign conventions of commutation with the suspension operator. That is, if $s:A\to A$ is ...
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27 views

Why $f_1$ commutes with the product up to boundaries

Let $(A,m_1)$ be a differential graded algebra, $m_2:A\otimes A\to A$ its product and $H^*A$ its homology, on which there is another product induced by $m_2$ which I'll call equally. I'm trying to ...
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Zeroth page of spectral sequence of a filtered chain complex

Let $C^{\bullet}$ be a (cohomological) chain complex of modules with coboundary map $\delta$ and filtration $$ \cdots = F_{-1}C^{\bullet} = F_{0}C^{\bullet} = C^{\bullet} \supset F_{1}C^{\bullet} \...
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30 views

DG algebra and its zeroth cohomology are derived equivalent

This is slightly related to this question: Can an algebra be morita equivalent to its dg-extension? . Suppose we have a DG-algebra $A$, such that $H^0(A)$ is Noetherian (both left and right) and $H^\...
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Can an algebra be morita equivalent to its dg-extension?

Say we have a DG algebra $A=\bigoplus_{n\geq 0}A_n$, let $B=A_0$, the 0th degree of $A$. Assume we have that the category of DG-modules over $A$ is equivalent to the category of module over $B$. Does ...
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98 views

Definition of $\mathfrak{g}$-differential graded algebra

I am reading Group actions on manifolds by Eckhard Meinrenken (Lecture Notes, University of Toronto, Spring 2003). In page $45$, definition $5.2$, author introduce the notion of $\mathfrak{g}$-...
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Explicit formula for the equalizer of coalgebras

The article Limits of Coalgebras, Bialgebras and Hopf Algebras offers two descriptions for the equalizer of two unital coassociative coalgebras over a field. The latter description (Remark 1.2) is ...
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When does a fibration $f:X\rightarrow Y$ in a model category admit a section?.

If we have a fibration $f:X\rightarrow Y$ in a model category $C$, where $Y$ is cofibrant and both $X, Y$ are fibrant. Does f admit a section (right inverse)?. If it does not work in general, ...
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Categorical product of non-unital associative differential graded coalgebras

Given two non-unital associative dg coalgebras $D$ and $C$, I want to give an explicit construction of the product $C\prod D$, this may follow from the dual construction (coproduct of non-unital ...
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if $C$ is a filtered coalgebra, does Gr($B\Omega C)\backsimeq B\Omega ($Gr $C)$ hold?

I have heard that under some assumptions, the functor 'Gr' from filtered graded objects with exhaustive filtration to graded objects $X\rightarrow$ Gr$(X)$ commutes with direct sums (this seems to be ...
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51 views

The twisted tensor product $BA\otimes_{\tau} A$ as the non-unital Hochschild complex

The twisting universal morphism $\tau: BA\rightarrow A$ induces a differential $\partial_{\tau}$ on $BA\otimes_{\tau}A$, we have: $$\partial_{\tau}(x\otimes y)=\partial x\otimes y+(-1)^{\lvert x\rvert}...
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DG-Modules over CDG-algebras in the sense of rational homotopy theory.

Suppose we have a rationalization $\bar{X} $ of a simply connected topological space $X$. Then we can construct a CDG-algebra - a Sullivan model $S$ corresponding to $\bar{X} $. How can we interpret ...
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Understanding the algebra structure of $HH(\mathbb{F}_p)$

As the title suggests, i'm trying to understand this calculation of the algebra structure on $HH(\mathbb{F}_p)$, which I will outline below: We can calculate $HH(\mathbb{F}_p)$ as $$\mathbb{F}_p \...
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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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$A$ - dga over field, then $H^i(A) = 0, i > 1$ implies $HH_i(A) = 0, i < -1$

Let $(A,d)$ - dg-algebra with unit over field $k$ such that $H^i(A) = 0$ for $i > 1$, then $HH_i(A) = 0, i < -1$. I prove that using bar resolution, fact that cohomology commutes with filtered ...
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Why aren't there any derivations of degree inferior to $-1$ of the DG-algebra $(\Omega(A), d_A, \wedge)$?

Let $A$ be a vector bundle over a manifold $M$. We can assotiate a graded algebra $(\Omega(A), \wedge)$ where $$\wedge:\Omega^i(A)\times \Omega^j(A)\longrightarrow \Omega^{i+j}(A),$$ is given by $$(\...