All Questions
Tagged with derived-categories ra.rings-and-algebras
18 questions
3
votes
1
answer
387
views
Concrete examples of derived categories
What examples of abelian categories $\mathcal{A}$ are there such that the derived category $\mathcal{D}(\mathcal{A})$ can be described concretely? For example, is there a concrete way of describing $\...
- 30.3k
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
4
votes
1
answer
477
views
Are eigenvalues preserved under derived equivalence?
Let $A$ and $B$ be finite dimensional algebras such that $A$ and $B$ are derived equivalent.
Denote by $C_A$ (resp. $C_B$) the Cartan matrix of $A$ (resp. $B$).
Then does the set of eigenvalues of $...
- 12.9k
1
vote
0
answers
156
views
Pseudo-coherent complexes over sheaves of non-commutative rings
I am posing a question on derived categories to which I was not able to find an answer anywhere in the literature. I would appreciate any answer, hint or suggestion.
Assume that $\mathcal{R}_X$ is a ...
- 609
6
votes
1
answer
215
views
What is the smallest group for which Broué's abelian defect group conjecture has not yet been verified?
Let $G$ be a finite group. Let $p$ be a prime dividing $|G|$. Let $k:=\overline{\mathbb{F}_p}$.
Let $b$ be a $p$-block of $kG$ with abelian defect group $D$. Let $H:=N_G(D)$. Let $c$ be the Brauer ...
- 35.2k
6
votes
1
answer
294
views
Rickard's strengthening of Broué's abelian defect group conjecture and the lifting of some equivalences up to splendid derived equivalences
Let $G$ be a finite group. Let $p$ be a prime dividing $|G|$. Let $K:=\overline{\mathbb{F}_p}$.
Let $b$ be a $p$-block of $G$ with abelian defect group $D$. Let $H:=N_G(D)$. Let $c$ be the Brauer ...
- 35.2k
3
votes
1
answer
125
views
Smallness condition for augmented algebras
I'm not sure this question is research level question. Sorry in advance.
Hypothesis
$k$ is a commutative ring.
$A$ is an augmented $k$-algebra.
$A^e$ is defined as the $k$-algebra $A\otimes_{k}A^{op}$...
- 701
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
...
- 784
3
votes
2
answers
603
views
Balanced dualizing complexes according to A. Yekutieli
I am reading A. Yekutieli's original article on dualizing complexes for noncommutative algebras and I found a problem I cannot solve.
First, some background. We start with a field $k$ and a ...
- 610
1
vote
1
answer
160
views
Example of tensor category with non-simple unit $J\to \mathbb{1} \to Q$ and suitably extension $Q\to M\to J$
Edit: Thanx very much to Neil Strickland for quickly explaining to us that the following cannot be realized over finite commutative $\mathbb{C}$-algebras, as I had originally asked.
I know that there ...
- 1,846
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
$$ ...
- 40.3k
4
votes
0
answers
235
views
Serre duality graded singularity category
Let $R$ be a local Gorenstein ring of Krull dimension $d$ with an isolated singularity. Defined $D_{sing}(R)$ as the Verdier quotient $D^b(R)/Perf(R)$). Then, a famous result of Auslander says that ...
- 7,320
2
votes
0
answers
169
views
Restriction of scalars from an Azumaya algebra and preservation of perfect/compact objects of the derived categories
An Azumaya variety over a field is by definition a pair $(X,\mathcal A_X)$, where $X$ is an algebraic variety of finite type over that field and $\mathcal A_X$ is a sheaf of Azumaya algebras, namely ...
- 2,079
8
votes
0
answers
337
views
flatness and derived completion
Let $A$ be a local ring of maximal ideal $\mathfrak{m}$. Let $\hat{A}$ be its completion.
If $A$ is noetherian , then we know that $A\rightarrow\hat{A}$ is faithfully flat.
If $A$ is not noetherian, ...
- 3,472
5
votes
1
answer
404
views
Equivariant Formality
Let $G$ be a finite group and $\mathcal{A}$ be a $dg$-algebra. Assume $G$ acts on $\mathcal{A}$, i.e. there exists a homomorphism $G\to {\rm Aut}_{dg}(\mathcal{A})$.
Assume further there exists a $dg$...
- 35.9k
4
votes
0
answers
252
views
Formal DG-algebras
Sorry for this question but I really have difficulties with model categories.
Usually a $dg$-algebra $A$ is called formal, if there exists a $dg$-algebra $B$ and quasi-isomorphisms $$A\leftarrow B\to ...
- 2,554
2
votes
0
answers
81
views
injective dimension of dualizing complex
Let $R$ be a balanced dualizing complex of a Noetherian connected graded algebras $A$. Dose one always have $\text{id}_A R = \text{id}_{A^{op}} R$?
Thanks a lot.
- 141
6
votes
1
answer
896
views
Generators of the derived category
For a ring $R$, which is a finite-dimensional algebra over a field, the category of finite-dimensional, projective, right $R$-modules, $\mathcal{P}_R$ is generated by the indecomposable projective ...
- 44.7k