All Questions
Tagged with ag.algebraic-geometry derived-categories
33
questions
40
votes
7
answers
9k
views
Heuristic behind the Fourier-Mukai transform
What is the heuristic idea behind the Fourier-Mukai transform? What is the connection to the classical Fourier transform?
Moreover, could someone recommend a concise introduction to the subject?
- 2,112
63
votes
5
answers
9k
views
Intuition about the cotangent complex?
Does anyone have an answer to the question "What does the cotangent complex measure?"
Algebraic intuitions (like "homology measures how far a sequence is from being exact") are as welcome as ...
- 12.1k
21
votes
2
answers
2k
views
Applications of derived categories to "Traditional Algebraic Geometry"
I would like to know how derived categories (in particular, derived categories of coherent sheaves) can give results about "Traditional Algebraic Geometry". I am mostly interested in classical ...
15
votes
4
answers
2k
views
What is the relation between the Lie bracket on $TX$ as commutator and that coming from the Atiyah class?
Let X be a complex manifold and $TX$ its tangent bundle. The Atiyah class $\alpha(E)\in \text{Ext}^1(E\otimes TX, E)$ for a vector bundle $E$ is defined to be the obstruction to the global existence ...
- 8,767
35
votes
3
answers
6k
views
What is a triangle?
So I've been reading about derived categories recently (mostly via Hartshorne's Residues and Duality and some online notes), and while talking with some other people, I've realized that I'm finding it ...
- 10k
22
votes
4
answers
4k
views
Examples for Decomposition Theorem
There's an important piece of geometric knowledge usually quoted as Beilinson-Bernstein-Deligne.
Here's a refresher: by $IC$ one means the intersection complex, which is just $\mathbb Q$ for a smooth ...
- 15k
19
votes
2
answers
3k
views
Derived functors vs universal delta functors
I would like to understand the relationship between the derived category definition of a right derived functor $Rf$ (which involves an initial natural transformation $n: Qf \rightarrow (Rf)Q$, where $...
- 11.1k
17
votes
1
answer
821
views
Determinantal identities for perfect complexes
Let $S$ be a noetherian scheme. Let $V,W$ be vector bundles on $S$. There is a canonical isomorphism of line bundles
$$
{\rm det}(V\otimes W)\cong{\rm det}(V)^{\otimes{\rm rk}(W)}\otimes{\rm det}(W)^{\...
- 3,066
15
votes
5
answers
7k
views
Sheaves without global sections
The line bundle $O(-1)$ on a projective space or $O(-\rho)$ on a flag variety has a property that all its cohomology vanish. Is there a story behind such sheaves?
Here are more precise questions. Let ...
- 12.2k
12
votes
1
answer
2k
views
Derived categories of singular varieties
Given my limited knowledge on derived categories, all the results on derived categories of complex of bounded sheaves are build upon smooth varieties, and people literally avoid singular case (as in ...
- 3,392
10
votes
1
answer
847
views
$\infty$-categorical understanding of Bridgeland stability?
On triangulated categories we have a notion of Bridgeland stability conditions.
Is there any known notion of "derived stability conditions" on a stable $\infty$-category $C$ such that they become ...
- 103
10
votes
1
answer
1k
views
Characterization of schemes whose dualizing complex is perfect
I'm wondering if there is a characterization of schemes over a a field $k$ whose dualizing complex is a perfect complex in terms of singularities. E.g. on a proper Cohen-Macauley scheme over a field, ...
- 5,002
9
votes
1
answer
1k
views
How to write down the determinant of a quasi-isomorphism?
This question about the determinant of a perfect complex reminded me of an old question that I had.
The construction of the determinant (as in MR1914072 or MR0437541) is a difficult piece of ...
- 3,284
8
votes
1
answer
1k
views
Progress on Bondal–Orlov derived equivalence conjecture
In their 1995 paper, Bondal and Orlov posed the following conjecture:
If two smooth $n$-dimensional varieties $X$ and $Y$ are related by a flop, then their bounded derived categories of coherent ...
- 305
8
votes
1
answer
807
views
A Naive Question on Mixed Motives and Mixed Hodge Structures
As a physicist, I have some naive questions about mixed motives and its mixed Hodge structure (MHS) realization. Any references, comments, answers will be appreciated!
The category of mixed motives ...
- 2,971