All Questions
Tagged with sheaf-theory ag.algebraic-geometry
493 questions
98
votes
10
answers
14k
views
equivalence of Grothendieck-style versus Cech-style sheaf cohomology
Given a topological space $X$, we can define the sheaf cohomology of $X$ in
I. the Grothendieck style (as the right derived functor of the global sections functor $\Gamma(X,-)$)
or
II. the Čech ...
- 1,871
66
votes
4
answers
11k
views
Is there a good way to think of vanishing cycles and nearby cycles?
Once in a while I run into literature that invokes vanishing cycle machinery with a cryptic sentence like, "this follows from a standard vanishing cycle argument." Is there a good way to look at ...
- 45.7k
62
votes
8
answers
14k
views
Sheaf cohomology and injective resolutions
In defining sheaf cohomology (say in Hartshorne), a common approach seems to be defining the cohomology functors as derived functors. Is there any conceptual reason for injective resolution to come ...
user709
57
votes
3
answers
10k
views
What are the benefits of viewing a sheaf from the "espace étalé" perspective?
I learned the definition of a sheaf from Hartshorne—that is, as a (co-)functor from the category of open sets of a topological space (with morphisms given by inclusions) to, say, the category of sets. ...
- 6,290
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
48
votes
8
answers
8k
views
When are there enough projective sheaves on a space X?
This question is being asked on behalf of a colleague of mine.
Let $X$ be a topological space. It is well known that the abelian category of sheaves on $X$ has enough injectives: that is, every ...
- 65.4k
45
votes
8
answers
14k
views
How should one think about sheafification and the difference between a sheaf and a presheaf
The first time I got in touch with the abstract notion of a sheaf on a topological space $X$, I thought of it as something which assigns to an open set $U$ of $X$ something like the ring of continuous ...
- 2,782
39
votes
6
answers
9k
views
What is the inverse image sheaf necessary for in algebraic geometry?
Given a continuous map $f \colon X \to Y$ of topological spaces, and a sheaf $\mathcal{F}$ on $Y$, the inverse image sheaf $f^{-1}\mathcal{F}$ on $X$ is the sheafification of the presheaf
$$U \mapsto \...
- 7,338
36
votes
6
answers
9k
views
What is the Zariski topology good/bad for?
In a comment to this question the quotation "The Zariski Topology is the 'Wrong' topology for Algebraic Geometry" appears.
Well, so some spontaneous questions arise:
1) What is Zariski topology ...
36
votes
3
answers
4k
views
What is the right version of "partitions of unity implies vanishing sheaf cohomology"
There are several theorems I know of the form "Let $X$ be a locally ringed space obeying some condition like existence of partitions of unity. Let $E$ be a sheaf of $\mathcal{O}_X$ modules obeying ...
- 156k
35
votes
5
answers
4k
views
Heuristic explanation of why we lose projectives in sheaves.
We know that presheaves of any category have enough projectives and that sheaves do not, why is this, and how does it effect our thinking?
This question was asked(and I found it very helpful) but I ...
- 4,842
34
votes
4
answers
15k
views
When will the pushforward of a structure sheaf still be a structure sheaf?
Let $f:X\rightarrow Y$ be a morphism of schemes.
When $PicY\rightarrow PicX$ is an embedding and $f_{*}\mathscr{O}_{X}$ is invertible, it is the structure sheaf of $Y$.
In the proof of Zariski's Main ...
- 563
31
votes
3
answers
4k
views
Sheaf description of $G$-bundles
Now, among algebraic geometers, at least, it is well known that there is an equivalence between locally free $\mathcal{O}_X$-modules of rank $n$ and vector bundles of rank $n$. So, equivalently, ...
- 16k
26
votes
1
answer
4k
views
Affine scheme on spec(A) of a ring A as the sheafification of a pre-sheave on spec(A)?
It is obvious that there is a parallel between the definition of structure sheaf of $\operatorname{Spec}(A)$
versus the sheafification of a pre-sheaf.
The definition of the sheaf $\mathscr F^+$ ...
- 363
26
votes
1
answer
1k
views
Why there is a Quot-scheme, not a Sub-scheme?
Let $X$ be a projective variety, and $E$ be a coherent sheaf on $X$. Grothendieck has proven that there is a scheme $\mathrm{Quot}_X(E)$ parametrizing arbitrary quotient sheaves of $E$. It is probably ...
- 1,980
25
votes
5
answers
3k
views
Applications of the "other" definition of sheaves
In most literature, when you try to look for the definition of sheaves you will see the usual definition for presheaves as a functor from a topological space (or from a Grothendieck topology) to some ...
- 2,275
25
votes
3
answers
5k
views
Stacks and sheaves
I'm a bit confused by the double role which sheaves play in the theory of stacks.
On the one hand, sheaves on a site are the obvious generalization of a sheaf on a topological space. On the other ...
- 14.7k
24
votes
4
answers
6k
views
What is a section?
This question comes out of the answers to Ho Chung Siu's question about vector bundles. Based on my reading, it seems that the definition of the term "section" went through several phases of ...
- 118k
24
votes
1
answer
837
views
Is there a useful theory of D-modules on smooth (non-analytic) manifolds?
D-modules are related to flat connections on vector bundles, end hence local systems. The theory of D-modules (and related notions such as crystals etc.) seems to be very popular in complex analytic ...
- 6,025
22
votes
5
answers
6k
views
Cohomology of Structure Sheaves: Algebraic, Constructible and more
I am not an algebraic geometer, but I am a topologist who uses sheaves. I have studied some algebraic geometry and am interested in what happens as I reduce the amount of rigidity in the structure ...
- 2,684
21
votes
2
answers
11k
views
Elementary short exact sequence of sheaves
This question arised when I was trying to use this answer to understand Reid's "Young Person's guide to Canonical Singularities". In particular page 352 when computing the blow-up $Y\rightarrow A^2/\...
- 2,215
21
votes
2
answers
2k
views
Naive question about constructing constructible sheaves.
In algebraic geometry, an etale sheaf on a Noetherian scheme is called constructible if the scheme has a finite stratification by locally closed subschemes such that the pullback of the sheaf to each ...
- 41.4k
20
votes
1
answer
2k
views
Do I know what "coherent sheaf" means if I know what it means on locally Noetherian schemes?
I've been trying to convince myself that "coherent sheaf" is a natural definition. One way I might be satisfied is the following: for modules over a Noetherian ring $A$, coherent and finitely ...
- 118k
20
votes
1
answer
2k
views
Functorial characterization of open subschemes?
Given a morphism of schemes f: U → X, can one determine when f is an isomorphism of U onto an open subscheme of X in terms of some induced functors between the categories of quasicoherent modules ...
- 5,407
20
votes
0
answers
3k
views
Idea of presheaf cohomology vs. sheaf cohomology
Let $X$ be a topological space and $U$ an open cover of $X$.
In this thread Angelo explained beautifully how presheaf cohomology (Cech cohomology) relates to sheaf cohomology:
The zeroth Cech ...
- 319
19
votes
6
answers
4k
views
Understanding Adjointness of Sheaves in Algebraic Geometry
Pushforward and pullback are very basic operations in algebraic geometry, as is the adjointness between them. I worked out a very careful of adjointness of sheaves (below) when I was working out of ...
- 3,555
19
votes
2
answers
3k
views
Explaining Mukai-Fourier transforms physically
A core concept in mathematics, engineering, and physics is the Fourier Transform (FT) and its many variants (Generalized Fourier Series, Green's Function, Pontryagin duality).
The basic algorithm is ...
- 10.5k
18
votes
4
answers
6k
views
Derived categories of coherent sheaves: suggested references?
I am interested in learning about the derived categories of coherent sheaves, the work of Bondal/Orlov and T. Bridgeland. Can someone suggest a reference for this, very introductory one with least ...
- 3,218
18
votes
2
answers
4k
views
Locally constant sheaves for the étale topology, lack of intuition about "étale-localness"
I have started studying some étale cohomology and I am trying to build up some intuition about the concept of local for the étale topology. I can understand some nice examples (like Kummer exact ...
- 291
18
votes
3
answers
2k
views
Can $\mathcal O_X$ be recognized abstract-nonsensically?
This question has been asked by Teimuraz Pirashvili many years ago. I forgot about it after a while and remembered only now by accident. He probably knows the answer by now, but I still don't.
In the ...
- 18.4k
18
votes
0
answers
548
views
Donaldson-Thomas Theory and "Quantum Foam" for Mathematicians
Let $X$ be a smooth, projective Calabi-Yau threefold. From an algebro-geometric perspective, the Donaldson-Thomas invariants $\text{DT}_{\beta, n}(X)$ are virtual counts of ideal sheaves on $X$ with ...
- 1,701
17
votes
1
answer
1k
views
Is a direct sum of flabby sheaves flabby?
Consider a family of flabby (= flasque) sheaves $(\mathcal F_i)_{i\in I}$ of abelian groups on the topological space $X$.
My question : is their direct sum sheaf $\mathcal F=\oplus _{i\in I} \mathcal ...
- 47.5k
17
votes
2
answers
1k
views
What are the merits of the different finiteness conditions on quasi-coherent sheaves?
It's my understanding that there's no disagreement about the right way to define a quasi-coherence for a sheaf $F$ of $O_X$-algebras (over a scheme, locally ringed space, or even locally ringed topos)....
- 9,408
16
votes
3
answers
5k
views
Do we have non-abelian sheaf cohomology?
Lets $X$ be a complex manifold (algebraic variety), $N$ an integer, and consider the sheaf $F$ defined by:
$F(U)$ ={ holomorphic maps $f: U\rightarrow GL(N,\mathbb{C})$ } with multiplicative ...
16
votes
4
answers
2k
views
Coboundaries and Gluing in Cech Cohomology - Intuition?
I'm trying to develop an intuition for Cech cohomology geometrically, but am currently failing. A lot of people seem to say that the groups $H^n$ measure obstructions to gluing local sections to get ...
- 827
16
votes
2
answers
9k
views
Canonical Sheaf of Projective Space
I am stuck on one step that occurs without explanation in several Algebraic geometry books.
Starting from the exact sequence
$$0\rightarrow \Omega_{\mathbb{P}^n}\rightarrow \mathcal{O}_{\mathbb{P}^...
- 953
16
votes
6
answers
14k
views
What books should I read before beginning Masaki Kashiwara and Pierre Schapira's "Sheaves on Manifolds"
I am a beginner trying to learn about sheaves. I am trying to read Masaki Kashiwara and Pierre Schapira's book "Sheaves on Manifolds", but I find it is not easy for me to understand.
What ...
16
votes
3
answers
3k
views
Physical interpretations/meanings of the notion of a sheaf?
I fairly understand the fiber bundles, both the mathematical concept of fiber bundles and the physics use of fiber bundles. Because the fiber bundles are tightly connected to the gauge field theory in ...
- 10.5k
16
votes
3
answers
5k
views
Stalks of sheaf-hom
Let $F$ and $G$ be sheaves on $X$. Under what conditions is the natural map from the stalk at $p$ of $\mathcal{H}\kern{-1pt}\mathit{om}(F,G)$ to $\mathrm{Hom}(F_p, G_p)$ an isomorphism?
- 161
16
votes
1
answer
448
views
Zorn's lemma for Grothendieck sites
In every treatment of Grothendieck sites I can find, flasque sheaves are not defined in the way one would naïvely expect from ordinary sheaf cohomology; namely instead of saying that "restriction ...
- 163
16
votes
0
answers
4k
views
Sheaf cohomology and inverse limits
In proving the formal function theorem, Grothendieck uses a rather technical lemma in EGA 0-III.13:
Lemma: Let $\mathcal{F}_n$ be an inverse system of sheaves on a space $X$ with surjective ...
- 25.6k
15
votes
1
answer
2k
views
How to motivate constructible sheaves
I'm writing some notes for some students which just finished a first course in scheme theory. There I would like to talk about constructible sheaves, but I found it hard to give a compelling ...
- 711
15
votes
2
answers
616
views
Is it true that $\operatorname{2-colim}_U \textsf{QCoh}(U) = \textsf{Vect}(K_X)$, as $U$ shrinks to the generic point?
Let $X$ be an integral scheme with function field $K$. If $U\subset X$ is an open subscheme, we may consider the restriction functor
$$\textsf{QCoh}(X) \to \textsf{QCoh}(U).$$
I don't know much about ...
- 711
15
votes
2
answers
2k
views
Meaning of the determinant of cohomology
The Arakelov intersection number on arithmetic surfaces is defined as an "extension" of the classical intersection number on algebraic surfaces. It was introduced to get a nice intersection theory ...
- 321
15
votes
1
answer
1k
views
Grothendieck - sheaves as meter sticks
I'm trying to read parts of McLarty's Grothendieck on Simplicity and Generality. In the article, I read Grothendieck thought of sheaves over some topological space as meter sticks measuring it.
...
- 10.5k
15
votes
2
answers
2k
views
Total Spaces of Quasicoherent Sheaves
You can construct a total space of a quasicoherent sheaf on an scheme by taking relative spec of the symmetric algebra of the dual sheaf. For locally free sheaves, you get vector bundles, and every ...
- 5,548
14
votes
2
answers
904
views
What's the easiest example of a morphism of topoi that is not from that of a site?
A topos is defined to be a category that's equivalent to the category of sheaves on a site. Morphisms between topoi is defined by a pair of adjoint functors that behave like pull-back/push-forward of ...
- 5,052
14
votes
3
answers
1k
views
Counterexamples to gluing complexes of sheaves
Note: I asked the question below last week on MathSE but received no answer.
Background:
I have read the claim that perverse sheaves behave more like sheaves than like complexes of sheaves. This ...
- 937
14
votes
1
answer
702
views
Donaldson and DT invariants
Let $X$ be a smooth projective surface. Then, using the compactified moduli space of anti self-dual connections or torsion free sheaves we can construct Donaldson invariants of $X$. Similarly, one can ...
- 587
14
votes
1
answer
2k
views
Hypercohomology of a complex via Cech cohomology
Let $X$ be a reasonable topological space. If $\mathcal{F}$ is a sheaf of abelian groups then Cech cohomology gives us a method to compute the cohomology groups $H^p(X, \mathcal{F})$ - the main input ...
- 141