All Questions
Tagged with ag.algebraic-geometry derived-categories
402
questions
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
52
votes
7
answers
5k
views
What does a projective resolution mean geometrically?
For R a commutative ring and M an R-module, we can always find a projective resolution of M which replaces M by a sequence of projective R-modules. But as R is commutative, we can consider the affine ...
- 1,056
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
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, ...
- 20.9k
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
30
votes
1
answer
1k
views
Which properties of a variety are detected by its derived category of coherent sheaves?
Context: I'm giving an informal seminar/reading group collection of talks on derived categories, following on from earlier talks giving the abstract definition. I am starting to talk about $\mathcal{...
28
votes
3
answers
3k
views
Why are derived categories natural places to do deformation theory?
It seems to me that a lot of people do deformation theory (of schemes, sheaves, maps etc) in derived category (of an appropriate abelian category). For example, the cotangent complex of a morphism $f:...
- 339
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
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
22
votes
1
answer
2k
views
Generating the derived category with line bundles
The following lemma is useful and well-known:
LEMMA If $L^{\pm 1}$ is ample on proper scheme over a field $k$, then some number of powers $\mathcal{O},L,...,L^{m}$ generate the unbounded derived ...
- 1,849
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 ...
20
votes
1
answer
826
views
Why would one "attempt" to define points of a motive as $\operatorname{Ext}^1(\mathbb{Q}(0),M)$?
I'm a novice when it comes to motives. (I've read multiple introductory texts.)
I'm attempting to read Galois Theory and Diophantine geometry by Minhyong Kim. In it, he says that "One might attempt, ...
- 460
20
votes
1
answer
803
views
List of known Fourier Mukai partners?
I'm familiar with some examples of pairs of derived equivalent varieties, for example an abelian variety and its dual, a K3 surface and certain moduli schemes on it, or the Pfaffian-Grassmannian ...
- 3,007
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