All Questions
Tagged with derived-categories differential-graded-algebras
24
questions
3
votes
0
answers
121
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
10
votes
0
answers
485
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 ...
- 869
2
votes
0
answers
113
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
4
votes
0
answers
211
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, ...
- 421
1
vote
0
answers
71
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
4
votes
0
answers
161
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 ...
- 7,250
7
votes
1
answer
287
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
...
- 1,154
1
vote
0
answers
44
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
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
10
votes
1
answer
745
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
239
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
$$ ...
- 39.5k
12
votes
1
answer
1k
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 ...
- 8,767
4
votes
1
answer
692
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 ...
- 1,181
3
votes
1
answer
279
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 ...
- 1,706
6
votes
1
answer
246
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/...
- 8,767