Skip to main content

Questions tagged [differential-graded-algebras]

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

Filter by
Sorted by
Tagged with
1 vote
1 answer
18 views

Confusion about representing elements as pure tensors in the definition of the tensor product of dg-algebras

I'm refamiliarizing myself with the tensor product of dg-algebras and struggling to reconcile some of the basic definitions. Let $(A, d_A)$ and $(B, d_B)$ be dg-algebras over a field $k$, and assume ...
SeraPhim's user avatar
  • 1,190
0 votes
0 answers
16 views

Cohomology of a quotient of a commutative graded differential algebra over GF(2)

Let $P$ be a graded commutative differential algebra over the field $GF(2)$ (hence also actually commutative), and let $d$ be the differential. Let $x$ be a homogeneous element of $P$, which is also a ...
vginhi's user avatar
  • 159
0 votes
0 answers
62 views

Gröbner basis and dg structures

Gröbner basis is typically defined for ideals of polynomial rings over a field and there are several generalizations/extensions of this notion for non-commutative structures or differential algebras. ...
mari's user avatar
  • 23
0 votes
0 answers
50 views

Terminology question: "Module derivations" of the form $\mu \colon M \rightarrow M$

Let $R$ be a graded ring endowed with a graded derivation $d \colon R \rightarrow R$ of degree $k$. Let $M$ be a graded $R$-module. Is there a standard name for degree $k$ maps $\mu \colon M \...
dejavu's user avatar
  • 403
0 votes
0 answers
37 views

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 ...
amd1234's user avatar
  • 349
0 votes
0 answers
90 views

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 $p$ $k$-...
Margaret's user avatar
  • 1,769
3 votes
0 answers
47 views

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(\...
Hodge's user avatar
  • 55
0 votes
0 answers
163 views

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 ...
user avatar
3 votes
1 answer
109 views

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 ...
user avatar
1 vote
0 answers
69 views

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 $$ ...
Albert's user avatar
  • 3,052
0 votes
0 answers
138 views

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 ...
Harold Finch's user avatar
2 votes
1 answer
45 views

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=...
Harold Finch's user avatar
1 vote
0 answers
85 views

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 ...
Patrick Nicodemus's user avatar
1 vote
1 answer
83 views

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 ...
Lilolance's user avatar
  • 413
5 votes
1 answer
210 views

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 ...
user7090's user avatar
  • 5,493
4 votes
0 answers
220 views

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! ...
Nary's user avatar
  • 73
2 votes
0 answers
212 views

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 $H^i =$ ??
Jim Stasheff's user avatar
1 vote
0 answers
175 views

What is the proof of graded Jacobi identity?

$\newcommand{\parcir}[2]{\frac{\partial^R #1}{\partial #2}}$ $\newcommand{\parcil}[2]{\frac{\partial^L #1}{\partial #2}}$ $\newcommand{\vprcir}[2]{\frac{\overleftarrow{\partial} #1}{\partial #2}}$ $\...
Angel Octavio Parada Flores's user avatar
1 vote
1 answer
77 views

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 ...
Javi's user avatar
  • 6,323
1 vote
3 answers
332 views

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 ...
SeraPhim's user avatar
  • 1,190
3 votes
1 answer
102 views

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 ...
Jake Wetlock's user avatar
2 votes
0 answers
103 views

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 ...
Jim Stasheff's user avatar
1 vote
0 answers
297 views

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\...
Bubaya's user avatar
  • 2,254
1 vote
0 answers
101 views

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 $\...
Ivan Burbano's user avatar
  • 1,258
1 vote
1 answer
74 views

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 ...
Bubaya's user avatar
  • 2,254
0 votes
0 answers
40 views

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 ...
Haldot's user avatar
  • 830
5 votes
1 answer
422 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 ...
Markus Zetto's user avatar
0 votes
1 answer
40 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 ...
Javi's user avatar
  • 6,323
2 votes
1 answer
62 views

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 ...
FunctionOfX's user avatar
1 vote
1 answer
260 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 $\mathfrak{g}$-...
Praphulla Koushik's user avatar
3 votes
0 answers
130 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 ...
Victor TC's user avatar
  • 213
0 votes
0 answers
66 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, ...
Victor TC's user avatar
  • 213
3 votes
0 answers
73 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 ...
Victor TC's user avatar
  • 213
0 votes
1 answer
121 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 ...
Victor TC's user avatar
  • 213
2 votes
1 answer
101 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}...
Victor TC's user avatar
  • 213
2 votes
0 answers
339 views

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 ...
user12580's user avatar
  • 857
3 votes
0 answers
64 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 ...
Fat ninja's user avatar
  • 235
3 votes
1 answer
149 views

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 \...
ufabao's user avatar
  • 475
1 vote
0 answers
57 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 ...
Ramanasa's user avatar
  • 450
2 votes
0 answers
107 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 ...
Mykola Pochekai's user avatar
1 vote
1 answer
69 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 $$(\...
PtF's user avatar
  • 9,675