All Questions
Tagged with derived-categories differential-graded-algebras
25 questions
39
votes
9
answers
5k
views
What is a deformation of a category?
I have several naive and possibly stupid questions about deformations of categories. I hope that someone can at least point me to some appropriate references.
What is a deformation of a (linear, dg, ...
- 21k
12
votes
1
answer
2k
views
Is the derived category of $A$-dg-modules as a dg-category coincide with the ordinary definition of derived category?
Let $A$ be a unital dg-algebra over a base field $k$. We consider the category of (unbounded) right $A$-dg-modules with morphisms closed degree $0$ maps. We denote this category by dg-mod-$A$. We ...
- 9,019
11
votes
1
answer
1k
views
Do I need to know what an infinity-Gerstenhaber algebra is, and if so, what is it?
I am in the following situation. I have two (rather explicit and specific) dg commutative algebras $R,S$ over a field of characteristic $0$. In fact, $S$ is an $R$-algebra, in that I have a map $R \...
- 54.6k
10
votes
1
answer
791
views
Why does passage to DG categories cure non-locality of derived categories?
In the famous book 'Residues and duality', the author notes that one of the principal difficulties in constructing the exceptional inverse image functor $f^{!}$ is that the derived category of ...
user74900
10
votes
0
answers
488
views
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 ...
- 889
10
votes
0
answers
241
views
Has anyone seen this construction of dg algebras?
Let $A$ be an associative algebra, $M$ a right $A$-module. Suppose we are given an $A$-module homomorphism $M \to A$. Then we can make $M$ itself into an associative algebra via the multiplication
$$ ...
- 40.2k
7
votes
1
answer
288
views
Skew differential graded algebra
A sigma, or skew, derivation is a natural generalisation of the
notion of derivation depending on an algebra automorphism $\sigma$ which
when equal to $id = \sigma$ reduces to the usual notion of a
...
- 1,144
6
votes
1
answer
472
views
When may "summand of" be dropped from the definition of perfect dg module?
Let $\mathcal{A}$ be a small dg category. In Section 1 of Lunts-Orlov http://arxiv.org/pdf/0908.4187v5.pdf, $Perf(\mathcal{A})$ is defined to be the full DG subcategory of $\mathcal{S}\mathcal{F}(\...
- 1,474
6
votes
1
answer
253
views
Is the hom in derived category of a dg-algebra compatible with base field extension?
Let $k$ be a field and $A$ be an ordinary $k$-algebra. Let $M$ and $N$ be two left $A$-modules then we could consider the Ext group $\mathrm{Ext}^i_A(M,N)$ which is actually a $k$-vector space. Let $l/...
- 9,019
5
votes
1
answer
895
views
Massey products and $A_{\infty}$ structures
I know the general theorem of Kadeishvili which says that, for a DGA $C$, when $H^{i}(C)$, $i\geq 0$, is free, $H(C)$ can be made into an $A_{\infty}$ algebra. If my understanding is correct, the ...
- 928
5
votes
1
answer
404
views
Equivariant Formality
Let $G$ be a finite group and $\mathcal{A}$ be a $dg$-algebra. Assume $G$ acts on $\mathcal{A}$, i.e. there exists a homomorphism $G\to {\rm Aut}_{dg}(\mathcal{A})$.
Assume further there exists a $dg$...
- 2,554
4
votes
1
answer
731
views
Graded quivers vs "ordinary" quivers and derived categories
I have heard the "slogan" that graded quivers are (derived) equivalent to ordinary quivers (with this "result" being attributed to Keller) and am looking for a precise statement and a reference.
By a ...
- 1,211
4
votes
0
answers
240
views
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, ...
- 471
4
votes
0
answers
168
views
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 ...
- 7,300
4
votes
0
answers
348
views
Does derived equivalence imply dg Morita equivalence between DG algebras over field with char$=0$?
Let $A$, $B$ be two DG algebras and $D(A)$, $D(B)$ be derived categories of DG-modules of $A$, $B$, respectively. We call $A$ and $B$ are dg Morita equivalent if there is an $A$-$B$ bimodule $T$ with ...
- 9,019
4
votes
0
answers
252
views
Formal DG-algebras
Sorry for this question but I really have difficulties with model categories.
Usually a $dg$-algebra $A$ is called formal, if there exists a $dg$-algebra $B$ and quasi-isomorphisms $$A\leftarrow B\to ...
- 2,554
3
votes
1
answer
285
views
Is there a notion of injective, projective, flat, dimension for a differential graded algebra?
Given a differential graded algebra $(A_\bullet,d)$, is there a well-defined notion of a K-injective, K-projective, K-flat dimension of a differential graded module, or even of the category of ...
- 1,716
3
votes
0
answers
76
views
Natural transformation and Hochschild cohomology
I am reading the lecture note by Căldăraru:https://arxiv.org/pdf/math/0501094, in the last chapter of this note, he said that we should consider dg category instead of the derived category of coherent ...
- 395
3
votes
0
answers
139
views
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:\...
- 349
2
votes
1
answer
293
views
Formality of classifying spaces (for not necessarily connected groups)
As should be evident from the title this question has a similar flavor to:
Formality of classifying spaces
However, unlike Geordie's question, I will be working with torsion free coefficients (say ...
- 1,595
2
votes
0
answers
119
views
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 ...
anon
2
votes
0
answers
135
views
When is $C\text-\mathsf{dg\text-mod}$ determined by the connective base changes?
I'm using cohomological gradings.
For $C\in k\text-\mathsf{cdga}$ (where $k$ can be taken of characteristic 0), a morphism $C\to A$ to a connective dg-algebra $A\in k\text-\mathsf{cdga}_{\leq0}$ ...
- 283
1
vote
0
answers
72
views
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 $\...
- 213
1
vote
0
answers
46
views
Does a homologically bounded dg A-module admit a "locally finite" semi-free resolution
Let $A$ be a bounded dg-algebra whose underlying algebra is Noetherian and such that $H^*(A)$ is Noetherian. Let $M$ be a cohomologically bounded dg-module over $A$, whose cohomology groups are ...
- 101
0
votes
0
answers
179
views
Is the inverse of the criterion of fully-faithfulness of derived tensor product also true?
Let $A$ and $B$ be differential graded algebras over a field $k$ and $D(A)$ and $D(B)$ be the derived categories of differential graded modules.
Let $N$ be an $A$-$B$ bimodule. Then $(-)\otimes^{\...
- 9,019