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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Cellular Cochains are DG algebra
Consider a finite dimensional CW complex $X$ and let $C^*(X)$ denote the cochain complex of its cellular cochains. We have the diagonal map $\Delta: X \rightarrow X^2$ sending $x \mapsto (x,x)$. This ...
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Decomposable differential of operads
To my understanding, a degree $p$ derivation (for $p\in \mathbb{N}_{>0}$) on a graded reduced (meaning $P(0)=0$) operad $P$ of vector spaces is given by a sequence of equivariant degree $p$ $k$-...
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Existence of biproducts in pretriangulated dg-categories
I'm studying dg-categories, and mostly following Bernhard Keller (https://arxiv.org/abs/math/0601185). I'm trying to understand how for a pretriangulated dg-category $\mathcal{A}$, the category $H^0(\...
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Tensor product preserves quasi-isomorphisms
Let $A, B$ be dg Lie algebras. I'm trying to prove that if $f: A \rightarrow B$ is a quasi-isomorphism then $\otimes^n f: A^{\otimes n} \rightarrow B^{\otimes n}$ is also a quasi-isomorphism. I'm ...
user632057
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Do Chevalley-Eilenberg homology functor and taking the cohomology commute?
I've stumbled upon some ideas from homological algebra that I'm trying to piece together from a talk I heard. I don't have much background in this area, so I'm not sure if this is a reasonable thing ...
user632057
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Conceptual proof that Hochschild boundary is a derivation for the shuffle product
Let $k$ be a commutative ring with $1$ and $A$ a commutative unital $k$-algebra ($k$ and $A$ are assumed to be associative). Denote by $(C_\bullet(A),b)$ the Hochschild chain complex of $A$ and let
$$
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Definitions of semi-free DG modules
I saw two versions of semi-free DG modules $M$ over a small category $\mathcal{A}$, and I would like to know if they are equivalent or not.
In many Orlov's papers, a DG $\mathcal{A}$-module $M$ is ...
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What is the definition of gluing dg algebras along bimodule?
When I am reading Lunts' Categorical Resolution of Singularities, section 3.2, I found the following:
Let $A$ and $B$ be DG algebras and $N$ is a $A$-$B$-bimodule. Then we obtain a new DG algebra
$$C=...
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Why is the bar construction of a DG algebra a coalgebra?
Let $A$ be a differentially graded augmented algebra. Then $\mathbf{B}A$ can be equipped with the structure of a coalgebra. This is proved in, for example, Loday and Vallette's book on Algebraic ...
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Signs involved in suspension of algebra
I have a question that seems almost too trivial to ask, yet it confuses me often.
Suppose now that we have a dg-algebra $(A,m,d)$ with multiplication $m: A \otimes A \longrightarrow A$. If we look at ...
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Definition of module of Kähler differentials for a DG algebra
Let $R$ be a $DG$ algebra over $A$, i.e, a $\mathbb{Z}$-graded $A$- algebra with a derivation $d$. For example, if $R$ is an $A$-algebra, then any chain complex $C^{\bullet}$ of $R$-modules with a ...
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Natural transformation between sheaves in homotopy theory
Firstly a small disclaimer. I am not an expert in the theory of higher sheaves and their presentation in the model categories, so please feel free to correct all inaccuracies in the question itself!
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What is the (co)homology of a free (graded) Lie algebra?
In characteristic $0$, what is the Chevalley-Eilenberg (co)homology of a free (graded) Lie algebra?
Not the definition, but $H^i =$ ??
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What is the proof of graded Jacobi identity?
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$\newcommand{\parcil}[2]{\frac{\partial^L #1}{\partial #2}}$
$\newcommand{\vprcir}[2]{\frac{\overleftarrow{\partial} #1}{\partial #2}}$
$\...
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Show that this map is well-defined on cohomology
I'm following this paper, trying to understand Corollary 5.
Essentially, there is a Lie Bracket $[,]:C^*\otimes C^*\to C^*$ on a cochain complex $C^*$ such that the the differential $d$ of the complex ...
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Examples of DG-algebras in algebraic geometry, representation theory and abstract algebra
I'm studying DG-algebras at the moment and I'm looking for interesting examples of where they occur. I've been told that they have applications in algebraic geometry and representation theory, but ...
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Pairing higher forms of a Lie group with the universal enveloping algebra
For a (compact) Lie group $G$ we have a pairing between its cotangent space $T^*$ and its Lie algebra $\frak{g}$. What happens for higher forms? Do we have a pairing between $\Lambda(T^*)$, the ...
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What is a proof of the Bianchi identity in the framework of Cartan's dg algebra connections?
For a general principal bundle (not necessarily Riemaninian),H Cartan developed a purely dg algebra analog in which connection and curvature make sense. How can one prove the Bianchi identity in this ...
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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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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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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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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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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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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:
$$\partial_{\tau}(x\otimes y)=\partial x\otimes y+(-1)^{\lvert x\rvert}...
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quasi isomorphism of two dg algebras
I want to construct a chain of morphisms from a dg algebra $A$ to $B$. I assume that $A$ and $B$ is non positive, i.e, $A^n$ vanishes for $n$ greater than zero. What I have is that $H^*(A)$ is ...
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DG-Modules over CDG-algebras in the sense of rational homotopy theory.
Suppose we have a rationalization $\bar{X} $ of a simply connected topological space $X$. Then we can construct a CDG-algebra - a Sullivan model $S$ corresponding to $\bar{X} $.
How can we interpret ...
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Understanding the algebra structure of $HH(\mathbb{F}_p)$
As the title suggests, i'm trying to understand this calculation of the algebra structure on $HH(\mathbb{F}_p)$, which I will outline below:
We can calculate $HH(\mathbb{F}_p)$ as $$\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 $$(\...
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