# Questions tagged [differential-graded-algebras]

The differential-graded-algebras tag has no usage guidance.

202
questions

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### When the dg cluster category of a quiver is saturated?

Let $Q$ be a finite quiver without oriented cycles. In https://arxiv.org/abs/0807.1960 , Keller defines the dg cluster category $C_Q$ of $Q$.
When is $C_Q$ smooth proper dg-category?
If $Q$ is a ...

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### Simplicial enrichment on unbounded algebras over an operad

In his paper "Homological Algebra of Homotopy Algebras" V.Hinich introduced a simplicial structure on algebras in unbounded chain complexes over arbitrary $\Sigma$-split operads. Not to get ...

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### dg-natural transformation between FM functors and Hom between kernels

The question is related to Morphism between Fourier-Mukai functors implies the morphism between kernels?
Consider a complex smooth projective variety $X$ and the bounded derived category $D^b(X)$, it ...

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### Lifting theorem for modules over a DGA

In the survey paper "Cartan's Constructions, the homology of $K(\pi,n)$'s, and some later developments" by J. C. Moore ( http://www.numdam.org/item/AST_1976__32-33__173_0/ ) he states a ...

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### Model structure for dga of (endormorphism) vector bundle valued differential forms

I was browsing and came across this discussion on the model structure for a dga. They mostly explain the commutative case but then say some things about the non-commutative case.
Context Consider a ...

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### Grothendieck group of coconnective dg-algebra

Is there an example of an $E_{2}$-coconnective differential graded algebra $A$ (with unit) such that $K_{0}(A)=0$ ?

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### Chekanov-Eliashberg Legendrian DGA with positive grading?

I was just looking back to some notes that I took a few years ago, when I was reading Etnyre's notes on Legendrian Contact Homology in $\mathbb R^3$ and I happened upon the following question that I ...

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### Lifting dg-algebras to characteristic zero

For a smooth algebra $A$ over a finite field $k$, by a Theorem R. Elkik, there always exists a lift to characteristic zero (apologies for the previous mistake). My question is how the analogous ...

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139
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### "Approximating" functors by Hom/Tensor product

Consider two dg-algebras $A,B$ and their respective derived categories $D(A),D(B)$. A natural way to give a covariant functor is to take an $(A,B)$-bimodule $X$ and to tensor with it, that is
$$D(A)\...

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### Derived category of dg modules vs. graded modules over a formal dg-algebra

Let $R = \oplus_{i\geq0} R_i$, $R_0 = k$ ($k$ a field) be a positively-graded commutative noetherian algebra, regarded as an augmented dg-algebra with zero differential.
Depending on one's interest, ...

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### Universal property of dg-algebras

Let $k$ be a field. Does the fully faithful inclusion from $k$-algebras to dg-$k$-agebras concentrated in cohomological degrees $\leq 0$
$$\operatorname{Alg}_k\hookrightarrow\operatorname{dgAlg}_k^{\...

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### Homotopy equivalent dg modules

$\newcommand\modl[1]{#1\text{-mod}}$Let $A:=(A,d_A)$ be a dg algebra. I would like to ask about isomorphisms in the homotopy category $H(\modl A)$ of the dg category $\modl A$ of dg modules over $A$.
...

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208
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### Two equivalent definitions of differential graded algebras

There are two equivalent definitions of differential graded algebras with different point of view. The first one is that it is a sequence $A=(A^n)_{n\in \mathbb{Z}}$ of vector spaces together with a ...

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### Conditions for a minimal derived $A_\infty$-algebra to be bounded

I was looking for some examples of derived $A_\infty$-algebras (or $dA_\infty$-algebras) in the original reference by Sagave, DG-algebras and derived A-infinity algebras, where some examples obtained ...

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### Exponential of a sum in a non-commutative graded algebra

Let $a,b$ be two elements of a graded algebra $A$ such that $\deg(a)=1$, $\deg(b)=0$ and $[a,b]\neq 0$.
I would like to know whether there exits an explicit expression for the degree 1 component
$$\...

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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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### Reference request: Étale base change of differential-graded algebras

I've asked this question on Math.StackExchange, but didn't receive an answer, so I'd like to try my luck here.
I'm looking for a reference for the following fact, which I've recently stumbled upon:
...

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### Symmetric algebra over a realization of Coxeter System is a dgg algebra

I have been reading a paper of Achar, Makisumi, Riche and Williamson. In the chapter 3, the authors talk us of bigraded modules and dgg modules and I'm stuck here.
Let $(W, S)$ be a Coxeter system, ...

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### Perfect dg-modules under faithfully flat extension

Let $k \subseteq \bar{k}$ be an extension of fields (Orlov in the reference below seems to indicate the same thing will hold for faithfully flat maps but the case of fields is enough for me).
On page ...

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### $A$ is a commutative connective dg-algebra satisfying $H^0(A)=k$. Is it true that a dg ideal generated by elements of negative degree acyclic?

Let $k$ be a field of characteristic zero and $A$ be a graded commutative dg-algebra over $k$ with differential of degree $+1$ satisfying $H^0(A)=k, H^i(A)=0$ for $i<0$. Denote by $\mathcal J$ a dg ...

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### CE(g) for g infinite dimensional

On the nlab page for Chevalley–Eilenberg algebras, it defines $\operatorname{CE}(\mathfrak g)$ for $\mathfrak g$ finite dimensional, and then says "This has a more or less evident generalization ...

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### $\mathbb Z$-formality of spheres

A topological space $X$ is $\mathbb Z$-formal, if the singular cochain complex $C^*(X,\mathbb Z)$ is
quasi-isomorphic to $H^*(X, \mathbb Z)$ as an augmented differential graded ring.
It's quite ...

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### Homotopy pullback in the category of DG algebras

I was wondering if somebody could tell me the definition of homotopy pullback. More precisely, is there a description of the homotopy pullback in the category of (graded-commutative) DG algebras over ...

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### Socle of a quotient of the ring of differential operators of a polynomial ring

I have been reading the following paper:
https://www.sciencedirect.com/science/article/pii/S002240491000263X
Proposition 2.4(ii) shows that if $\mathfrak D$ is a ring of $k$-linear differential ...

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272
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### Morphisms of Hochschild (or cyclic) homology induced by homotopic maps

Let $f$ and $g$ be two maps between DG algebras $A$ and $B$, and assume that $f$ and $g$ are homotopic as chain maps, hence they induce the same map on the level of homology. Moreover, $f$ and $g$ ...

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### Can any $E_1$ algebra over $\mathbb{F}_p$ be modeled as a dg algebra?

I'm not very familiar with dg algebras (not necessarily commutative) and I'm wondering if any $E_1$ algebra in the sense of infinity categories (i.e. monoid in the stable category of $R-Mod$) over $R$ ...

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347
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### K-theory of a coconnective dga

I have seen somewhere that if a differential graded algebra $A$ is connective (homologically graded), then the Grothendieck group $K_{0}(A)=K_{0}(H_{0}(A))$.
Suppose that $A$ is a differential graded ...

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### Properties of filtrations preserved by a DG-algebra homomorphism

Suppose we have a homomorphism $f : A^{\bullet} \longrightarrow B^{\bullet}$ of differential graded algebras over a field $k$, and consider the filtration
\begin{align*}
A^{\bullet} \supseteq F^0A^{\...

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### Derived quot schemes and the derived linearity locus

I am studying derived quot schemes in the paper Ciocan-Fontanine and Kapranov, “Derived Quot schemes” .
On page 36 ~ 37, the derived linearity locus is defined.
Let $S$ be a $\mathbb{Z}_-$-graded dg-...

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### Bound on Hochschild dimension of a dg-algebra

Consider a dg-algebra $A$, is there any way I can estimate the Hochschild dimension, or global dimension of $A$?
More precisely the algebra that I am considering is the Endomorphism dg-algebra $\...

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### Construction of derived Quot schemes

I am studying the construction of derived Quot schemes in the paper Borisov, Katzarkov, and Sheshmani - “Shifted symplectic structures on derived Quot-stacks”.
Derived quot stacks are constructed from ...

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### Structure sheaf of derived intersection

Everything is over a field $k$ of characteristic $0$.
Let $X$ and $Y$ two closed dg subschemes over a dg scheme $Z$. I am trying to understand the structure sheaf of the derived intersection of $X$ ...

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### Differentials in spectral sequences and Massey products

Consider a multiplicative spectral sequence such as the cohomological Serre spectral. It is known that differentials will satisfy a Leibniz rule. Is there a clean statement involving differentials and ...

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### Künneth formula for de Rham cohomology with respect to an integrable connection

I am reading through https://stacks.math.columbia.edu/tag/0FM9 which proves that for $X,Y$ schemes over some base $S$ and $X \times _S Y \overset{p}{\rightarrow} X$ resp. $X \times _S Y \overset{q}{\...

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### Degree shift of multilinear maps

Let $V$ be a graded vector space over $\mathbb{k}$ and $V[1]$ its odd degree shift.
Given $k$, $l\in \mathbb{N}_0$, is there a natural way to define the following map,
$$
\psi: \hom_{\mathbb{k}}(V^{\...

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### When is cohomology of a finitely presented dg-algebra computable?

Given a smooth affine variety $X$ defined over $\mathbb{Q}$, its singular cohomology is isomorphic to the algebraic de Rham cohomology, which is the cohomology of the complex $\Omega_X^0\to\Omega_X^1\...

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### Who introduced the abstract definition of a DGA?

Differential graded algebras, or DGAs, are a basic object of study in many areas of modern mathematics. While they were present (implicitly at least) since the start of modern differential geometry, I ...

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### Conceptual explanation for the sign in front of some binary operations

In several situations, I've seen that given a binary operation on a graded module $m:A\otimes A\to A$, a new operation $M(x,y)=(-1)^{|x|}m(x,y)$ is defined so that it satisfies some properties.
One ...

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### About Homotopy Transfer Lemma

If M, A are two differential graded complexes over a commutative ring R with the following data,
$$(M,d_M) \overset{\Delta}{\longrightarrow} (A,d_A)$$ $$(A,d_A) \overset{f}{\longrightarrow} (M,d_M)$$ ...

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### detecting a semi-free module from its bar-resolution

Let $A$ be a DG-algebra over a field (say $k$). A DG-module $M$ over $A$ is said to be semi-free if it admits an exhaustive filtration $0 = M_0 \subset M_1 \subset \ldots \subset M_p = M$ such that ...

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### Is the free algebra over an operad an algebra over that operad?

I'm asking here this question I asked on MSE that got no answers.
Let $V$ be a dg-module and $P$ an operad. The free $P$-algebra on $V$ is defined by $P(V)=\bigoplus_{r=0}^\infty (P(r)\otimes V^{\...

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### How does the knot contact homology augmentation polynomial change under a surgery on the base manifold?

So for every knot $K \in S^3$, there is a knot contact homology of $K$, and we can find the augmentation variety for this homology. The defining polynomial is known as the augmentation polynomial $A(K)...

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### The correct homotopically relevant notion of ideals of dg-algebras (or $\mathbb E_1$-rings)

I'm trying to figure out what an ideal of a, say, dg-algebra (or, if you prefer, $\mathbb E_1$-ring) $R$ is in a homotopically relevant fashion, but I can't actually figure it out. I can assume that $...

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### Saturated differentially closed field

What means "saturated" in "saturated differentially closed field" ?

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### Curvature of nonsymmetric metric tensors?

Consider a smooth manifold $M$ of arbitrary dimension. We have notions of psuedo-Riemannian or Riemannian metrics on a manifold, and they differ in the slightest way of being positive-definite or not. ...

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### Finite CDGA model for a compact manifold

Is it true that a compact smooth manifold always has a finite-dimension commutative dg algebra model?
Same question can be asked about compact CW complexes.
More generally, is it true that a CW ...

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### When is the module of Kahler differentials free?

As the title says, when is the module of Kahler differentials a free module? In particular, are there known conditions or criterions that could be met that ensures that it will be free?
For example, ...

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### Simple, explicit, functorial cylinder object in CDGA

In the model category of graded commutative dg-algebras CDGA over $\mathbb{Q}$ (with the projective model structure) there is a simple, functorial construction of a path object given by tensoring with ...

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### Shklyarov's Euler class in geometrical setting

In paper HIRZEBRUCH-RIEMANN-ROCH THEOREM FOR DG ALGEBRAS Shklyarov provides construction which for a given dg-algebra (over field $k$, proper, $k$-smooth) $A$ and perfect $A$-module $M$ associate en ...

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### Linearity of a dg category $C$ over $HH^0(C)$

Let $C$ be a pre-triangulated dg-category over a field $k$ whose Hochschild cohomology groups $\operatorname{HH}^*(C)$ are concentrated in non-negative degree (cohomologically). Is $C$ Morita ...