All Questions
Tagged with sheaves or sheaf-theory
979 questions
113
votes
4
answers
13k
views
Is there a sheaf theoretical characterization of a differentiable manifold?
I'm going through the crisis of being unhappy with the textbook definition of a differentiable manifold. I'm wondering whether there is a sheaf-theoretic approach which will make me happier. In a ...
- 22.1k
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
93
votes
3
answers
11k
views
What is homology anyway?
Disclaimer: I don't feel qualified to ask this question and yet it's been troubling me for some time now and I lost my patience and decided to ask to get some kind of answer. If there are any stupid ...
- 7,789
89
votes
5
answers
18k
views
What is sheaf cohomology intuitively?
What is sheaf cohomology intuitively?
For local systems it is ordinary cohomology with twisted coefficients. But what
if the sheaf in question is far from being constant?
Can one still understand ...
- 13.2k
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
54
votes
3
answers
11k
views
Sheaves and bundles in differential geometry
Because the theory of sheaves is a functorial theory, it has been adopted in algebraic geometry (both using the functor of points approach and the locally ringed space approach) as the "main theory" ...
- 19.6k
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
49
votes
4
answers
7k
views
Sheaf-theoretic approach to forcing
Inspired by the question here, I have been trying to understand the sheaf-theoretic approach to forcing, as in MacLane–Moerdijk's book "Sheaves in geometry and logic", Chapter VI.
A general ...
- 21.3k
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
33
votes
4
answers
6k
views
What (if anything) happened to Intersection Homology?
In the early 1990's, Gil Kalai introduced me to a very interesting generalization of homology theory called intersection homology, which existed for like 10 years back then I believe. Defined ...
- 6,734
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
29
votes
3
answers
4k
views
Is there a good general definition of "sheaves with values in a category"?
Let $\mathcal{A}$ be a category.
There is a common definition of "sheaves with values in $\mathcal{A}$", which is what one obtains by taking the Grothendieck-style definition of "sheaf ...
- 15.9k
28
votes
1
answer
3k
views
Two points of view about Borel-moore homology
They are several ways to define the Borel-Moore homology on a locally compact space $X$.
The first one is by analogy with the singular homology but instead of using finite chains, we use locally ...
- 1,017
28
votes
3
answers
4k
views
How do I compare the different notions of Fourier transform for sheaves?
There is a close but not perfect relationship between algebraic D-modules on C^n, constructible sheaves on C^n in the analytic topology, and \ell-adic sheaves on an n-dimensional vector space over a ...
- 4,949
27
votes
4
answers
3k
views
What is an example of a presheaf P where P^+ is not a sheaf, only a separated presheaf?
There is a standard way to construct the sheafification of a presheaf on a Grothendieck topology which involves matching families. Details may be found here:
http://ncatlab.org/nlab/show/matching+...
- 3,918
27
votes
1
answer
2k
views
Galois Group as a Sheaf
I've noticed that the Galois groups associated to Galois field extensions $L$ of a given field $K$ seem remarkably like a sheaf, with the field extensions taking the place of open set, and the Galois ...
- 15.4k
27
votes
0
answers
469
views
Are these comparison morphisms between Čech and Grothendieck cohomology the same?
For better or for worse, there is more than one approach to comparing Čech cohomology $\smash{\check{H}}^\bullet(\mathfrak{U},X;\mathscr{F})$ of a sheaf $\mathscr{F}$ on a space $X$ w.r.t the cover $\...
- 1,021
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
2
answers
2k
views
Loop Spaces as Generalized Smooth spaces or as Infinite dimensional Manifolds?
There are two ways to define smooth mapping spaces and I want to know how they compare.
Let's take the concrete special case of free loops spaces. I think this is the most studied example so will ...
- 27.5k
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
26
votes
1
answer
1k
views
Why are quasitopological spaces needed in sheaf theoretic approaches to the h-principle?
Recently I have been learning more about the h-principle and in particular the methods of "continuous sheaves". In many treatments of this I see people using "quasi-topological spaces" and I am trying ...
- 27.5k
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
25
votes
2
answers
3k
views
Are there (enough) injectives in condensed abelian groups?
The question is very simple : does $Cond(\mathbf{Ab})$, the category of condensed abelian groups (as defined in Scholze's Lectures in Condensed Mathematics), have enough injectives ?
Does it, in fact, ...
- 15.9k
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
24
votes
0
answers
918
views
The topologies for which a presheaf is a sheaf?
Given a set $S$, let $Top(S)$ denote the partially ordered set (poset) of topologies on $S$, ordered by fineness, so the discrete topology, $Disc(S)$, is maximal.
Suppose that $Q$ is a presheaf on $...
- 8,659
23
votes
5
answers
1k
views
Sheafification via hypercovers
The sheafification of a presheaf on a site is often constructed in a two-step process $X^{++}$, where $X^+$ consists of matching families in $X$, is always separated, and is a sheaf if $X$ is ...
- 66.8k
23
votes
4
answers
5k
views
De Rham decomposition theorem, generalisations and good references
De Rham decomposition theorem states that every simply-connected Riemannian manifold $M$
that admits complementary sub-bundles $T'(M)$ and $T''(M)$ of its tangent bundle parallel with respect to the ...
- 28.9k
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
22
votes
3
answers
3k
views
Necessity of hypercovers for sheaf condition for simplicial sheaves
I'm trying to understand where the definition of simplicial sheaf on a space/site comes from.
For a presheaf $F$ of sets on a topological space $X$, the sheaf condition can be viewed as saying that ...
- 661
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
3k
views
Sheaves of complexes and complexes of sheaves
Let A be an abelian category, and X a topological space.
There are two ways one could try to construct some oo-category of sheaves on X from this data:
Consider the category $Sh(X,A)$ of sheaves on ...
21
votes
3
answers
2k
views
Is there a "categorical" description of Grothendieck's algebra of differential operators?
First, pick a commutative ring $k$ as the "ground field". Everything I say will be $k$-linear, e.g. "algebra" means "unital associative algebra over $k$". Then recall the following construction due ...
- 54.6k
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
5
answers
2k
views
Equivalence of ordered and unordered cech cohomology.
Given a topological space X and a finite cover X = $\cup X_i$, one can define Cech cohomology of a sheaf of abelian groups F with respect to the cover $\{X_i\}$ in two different ways:
(Ordered): ...
- 10.5k
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
4
answers
4k
views
The tangent bundle to an infinite-dimensional manifold
Suppose that $A,B$ are smooth ($\mathrm C^\infty$) manifolds, and denote by $\hom(A,B)$ the set of $\mathrm C^\infty$-maps $A \to B$. It is a perfectly well-defined set, but often one wants more. ...
- 54.6k
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
3
answers
911
views
Brouwer's theorem for the Cauchy reals
Brouwer famously proved, using principles motivated by intuitionistic choice sequences, that every function $\mathbb{R}\to \mathbb{R}$ is continuous. In Sheaves in geometry and logic (section VI.9), ...
- 66.8k
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