Questions tagged [derived-categories]
For questions about the derived categories of various abelian categories and questions regarding the derived category construction itself.
324
questions with no upvoted or accepted answers
24
votes
0
answers
709
views
What is the status of a result of Kontsevich and Rosenberg?
In their influential paper Noncommutative Smooth Spaces (https://arxiv.org/abs/math/9812158), Kontsevich and Rosenberg define the notion of a noncommutative projective space. In Section 3.3 they ...
- 331
17
votes
0
answers
637
views
Is there an Infinite dimensional sheaf theory for analysis on manifolds?
I apologize if this question is slightly vague but I don't know how to ask it non-vaguely. Moreover, my question is about an ideal situation. If there's a close answer which doesn't satisfy all the ...
- 7,689
15
votes
0
answers
627
views
Full Exceptional Collection of Vector Bundles for Toric Varieties
Kawamata showed the derived category of coherent sheaves on a smooth projective toric variety has a full exceptional collection consisting of sheaves. I was wondering if it is know whether every ...
- 291
13
votes
0
answers
715
views
Why do people study unbounded derived category of quasi-coherent sheaves rather than focus on bounded derived category of coherent sheaves?
Let $X$ be a scheme and let $D_{qoch}(X)$ and $D^b_{coh}(X)$ be the unbounded derived category of quasi-coherent sheaves and bounded derived category of coherent sheaves on $X$, respectively.
$D^b_{...
- 8,767
13
votes
0
answers
613
views
The derived category of integral representations of a Dynkin quiver
Let $Q$ be a Dynkin quiver. Let $\mathbb CQ$ be its complex path algebra. It is defined in a way such that modules over $\mathbb CQ$ are the same as representations of the quiver $Q$. Let's write $\...
- 3,154
12
votes
0
answers
295
views
Is there a bestiary of "derived 2-vector spaces"?
The appendix "A bestiary of 2–vector spaces" of Bartlett, Douglas, Schommer-Pries, Vicary, "Modular categories as representations of the 3-dimensional bordism 2-category" analyzes ...
- 53.5k
12
votes
0
answers
314
views
Comparing derived categories of quasi-coherent sheaves in the lisse-etale and the big etale toplogy on an algebraic stack
I am trying to understand the proof of Proposition 1.4.2. in "A study of derived algebraic geometry Volume 1" by Gaitsgory-Rozenblyum. http://www.math.harvard.edu/~gaitsgde/GL/QCohBook.pdf, page 8.
...
- 595
12
votes
0
answers
550
views
References for a certain generalization of Hochschild cohomology?
Let $C$ be an algebra. Let $E = C^{\otimes 2n}$ be the tensor product (over the ground field) of $2n$ copies of $C$. [EDIT: Or better, $E = C\otimes C^{op}\otimes C\otimes C^{op}\cdots\otimes C \...
- 12.7k
11
votes
0
answers
443
views
K-stability is invariant under D-equivalency
Kawamata conjectured that
Let $X$ and $Y$ be birationally equivalent smooth
projective varieties. Then the following are equivalent. We denote by
$D^b(Coh(X))$ the derived category of bounded ...
user21574
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
10
votes
0
answers
308
views
Triangle $X'\to X\to X''\to\Sigma X'$ splits if $X\simeq X'\oplus X''$?
Given a commutative ring $R$ and a distinguished triangle $X'\to X\to X''\xrightarrow e\Sigma X'$ in the derived category $D(R)$, where $X',X,X''$ are perfect complexes. If we have an equivalence $X\...
user20948
10
votes
0
answers
183
views
Derived equivalences preserved by blow-ups
Let $X$ and $Y$ be smooth projective varieties over $\mathbb{C}$. Assume that $X$ and $Y$ are derived equivalent. Let $\pi : \tilde{X} \longrightarrow X$ be a blow-up of $X$ along a smooth center. Can ...
- 7,250
10
votes
0
answers
525
views
What is the mirror of an algebraic group?
Background: Kontsevich's homological mirror symmetry conjecture posits the existence of pairs $(X,\check X)$ with an equivalence of dg/$A_\infty$-categories
$$\mathcal F(X)=\mathcal D^b(\check X)$$
...
- 18.5k
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
10
votes
0
answers
428
views
McKay correspondence and tensor products
The theorem of Bridgeland-King-Reid says that if $M$ is a smooth quasi-projective complex variety of dimension at most $3$ on which a finite group $G$ acts such that the canonical sheaf $\omega_M$ is $...
- 5,022