Questions tagged [differential-graded-algebras]

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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 \...
3
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0answers
51 views

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 ...
3
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0answers
40 views

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 ...
2
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1answer
32 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}...
2
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0answers
33 views

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 ...
2
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0answers
92 views

$A$ - 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 ...
1
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1answer
49 views

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 $$(\...
1
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1answer
28 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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39 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 ...
0
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1answer
44 views

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 ...
0
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51 views

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, ...