# Questions tagged [differential-graded-algebras]

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218
questions

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### Minimal Model for $\mathbb{CP}^2 \# \mathbb{CP}^2 \# \mathbb{CP}^2$

I'm a grad student studying non-negative curvature on simply-connected manifolds and the conjectured relationship with rational homotopy theory.
The rational homotopy groups (and so the number of ...

5
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1
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412
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### Does coproduct preserve cohomology in differential graded algebra category

Consider two cochain DGA (differential graded algebras) named $A$ and $B$. By "coproduct" of two DGA I mean the category theory coproduct, not the coalgebra coproduct. It is defined in "...

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2
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### Is the exterior algebra intrinsically formal?

Following 4.6 and 4.7 of this paper by Seidel and Thomas, a graded algebra $A$ is called intrinsically formal if any two dgas with cohomology $A$ are quasi-isomorphic. There is a sufficient condition ...

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### Resolutions of semi free (or almost free) commutative dg algebras with finitely generated cohomology

Let $A^{\bullet}:=\{ \cdots \rightarrow A^i \overset{d^i}{\rightarrow} A^{i+1} \rightarrow \cdots \rightarrow A^{-1} \rightarrow A^0 \rightarrow 0 \rightarrow \cdots \}$ be a non-positively graded ...

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### proper smooth dg-categories and colimit

Let $I$ be a filtered category and $\{k_i\}_{i\in I}$ be a system of commutative rings over $I$. Toen proved that there is an equivalence of categories
$$
\text{Colim}-\otimes^{\mathbb{L}}_{k_i} k:\...

4
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1
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133
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### DG algebra structure on minimal free resolution of modules over regular local ring

Let $(Q, \mathfrak n, k)$ be a regular local ring. Let $I\subseteq \mathfrak n^2$ be an ideal, and fix a minimal generating set $\mathbb f= f_1,\cdots, f_n$ of $I$. The Koszul complex $E:= Q[e_1,...,...

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277
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### Serre subcategories of the category of chain complexes of modules

Let $k$ be an algebraically closed field of characteristic $0$.
Let $R$ be a commutative $k$-algebra.
We denote by $\operatorname{Mod}(R), C(R), $ and $ D(R)$ the category of $R$-modules, the category ...

5
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1
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203
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### Equivalences of categories of complexes of modules

Let $k$ be an algebraically closed field of characteristic $0$.
Let $R, S$ be two commutative $k$-algebras.
Let $\operatorname{Ch}(R), \operatorname{Ch}(S)$ be the categories of complexes of $R$-...

2
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100
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### Differential graded modules and the Serre-Swan theorem

I am thinking about how connections combine with a modification of the Serre-Swan theorem, which relates vector bundles to projective modules.
If $E \rightarrow B$ is a vector bundle, or even just any ...

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482
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### Reconstruction of commutative differential graded algebras

Let $k$ be an algebraically closed field of characteristic $0$.
Let $A,B$ be commutative differential graded algebras (cdga) over $k$ such that $H^{i}(A)=H^{i}(B) =0 \ (i>0)$.
Here, differentials ...

3
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2
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291
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### Lie's third theorem via graded geometry

Lie's third theorem : Given any finite dimensional Lie algebra $\mathfrak{g}$, there exists a Lie group $G$ whose Lie algebra is equal to $\mathfrak{g}$.
In one of the talks, speaker mentions that ...

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64
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### Twisting a graded algebra by an automorphism (Transitivity)

Definition: Let $A=A=\bigoplus_{j=0}^{\infty} A_j$ be a connected $\mathbb{N}$-graded $k$-graded algebra and let $\phi\in\text{Aut($A$)}$ be a graded automorphism of degree zero. A new graded algebra ...

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### Singularity category of a hypersurface associated to $M_{11}$

For reasons to do with classifying spaces of finite groups, I have the following algebra. Let $k$ be a field of characteristic two, and let $R = k[x,y,z]/(x^2 y + z^2)$, as a graded $k$-algebra with $|...

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### Is any deformation of an acyclic complex gauge equivalent to a trivial one?

This question is related to this question. Let $(C^{\bullet},d)$ be a cochain complex over a field $k$ with char$k$=0. We fix an Artin local $k$-algebra $(A,m)$. A deformation of $d$ is a new operator ...

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135
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### Does homotopy equivalence between deformations of cochain complexes imply gauge equivalence?

Let $(C^{\bullet},d)$ be a cochain complex over a field $k$ with char$k$=0. We fix an Artin local $k$-algebra $(A,m)$. A deformation of $d$ is a new operator $d+\epsilon: C^{\bullet}\otimes_k m\to C^{\...

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107
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### Correct notion of "connected" for dga of bundle-valued forms

Consider a vector bundle $E$ over a manifold $M$ with flat connection, $\nabla$. From this data I can form the associative/unital differential graded algebra $\mathcal{A} = \left(\Omega^{\bullet}(M, ...

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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 ...

4
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1
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104
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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 ...

2
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113
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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 ...

2
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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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155
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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 $K_{0}(A)=0$ ?

3
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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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159
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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 ...

2
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1
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163
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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)\...

4
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210
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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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245
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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 $A$.
...

3
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238
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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 ...

3
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112
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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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76
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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
$$\...

4
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1
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432
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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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1
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64
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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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1
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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 ...

4
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1
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432
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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 ...

3
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1
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278
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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 ...

5
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1
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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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### 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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1
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107
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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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226
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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 ...

6
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2
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497
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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}{\...