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58
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5answers
3k 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 ...
52
votes
8answers
4k 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 ...
43
votes
5answers
6k 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 ...
39
votes
7answers
3k 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 ...
28
votes
4answers
4k 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 ...
26
votes
8answers
2k 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 sheaf ...
22
votes
6answers
3k 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 ...
21
votes
2answers
975 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 ...
20
votes
6answers
4k 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 ...
20
votes
4answers
1k 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 ...
20
votes
3answers
1k 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 ...
20
votes
3answers
1k views

Sheaf Description of G-Bundles

Now, among algebraic geometers, at least, it is well known that there is an equivalence between locally free O_X-modules of rank n and vector bundles of rank n. So, equivalently, principal ...
19
votes
4answers
3k 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 ...
19
votes
0answers
372 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 ...
18
votes
2answers
985 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 ...
18
votes
0answers
555 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 ...
17
votes
3answers
1k 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 ...
17
votes
5answers
746 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 ...
17
votes
1answer
1k views

Cohomology of sheaves in different Grothendieck topologies

Suppose I have a sheaf $\mathcal{F}$ on the (small) étale site over $X$. By restriction, $\mathcal{F}$ is also a sheaf on $X$ (with the Zariski topology). When is it that the sheaf cohomologies (i.e. ...
17
votes
3answers
2k 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 ...
16
votes
3answers
859 views

What are the benefits of viewing a sheaf from the “espace étalé” persepctive?

I learned the definition of a sheaf from Hartshorne---that is, as a (co-)functor from the categor of open sets of a topological space (with morphisms given by inclusions) to, say, the category of ...
16
votes
3answers
2k 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 ...
16
votes
1answer
842 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 ...
14
votes
6answers
3k 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 ...
14
votes
6answers
868 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 ...
14
votes
4answers
1k 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 ...
14
votes
4answers
704 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): ...
14
votes
4answers
1k 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: ...
14
votes
4answers
1k 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. ...
14
votes
1answer
559 views

Why do rigid spaces have “not enough points”?

In Brian Conrad's notes here for the 2007 Arizona winter school, bottom of p18, he says that there is an affinoid rigid-analytic space and a sheaf of abelian groups on it equipped with a non-zero ...
13
votes
9answers
1k views

What representative examples of modules should I keep in mind?

So here's my problem: I have no intuition for how a "generic" module over a commutative ring should behave. (I think I should never have been told "modules are like vector spaces.") The only ...
13
votes
5answers
1k 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 ...
13
votes
4answers
2k 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 ...
13
votes
2answers
2k 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 ...
13
votes
4answers
831 views

Cohomology of a sheaf of functions locally constant along a foliation

Take a smooth manifold $M^n$ with a smooth foliation $F$. Consider the sheaf $\cal F$ of $C^{\infty}$ functions on $M^n$, locally constant along the foliation $F$. What is known about Chech cohomology ...
13
votes
2answers
558 views

Which topological spaces have the property that their sheaves of continuous functions are determined by their global sections?

I hope I'm using the terminology correctly. What I mean is this: fix $K = \mathbb{R}$ or $\mathbb{C}$ (I'm interested in both cases). Which topological spaces $X$ have the property that for every ...
12
votes
5answers
1k 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 ...
12
votes
1answer
279 views

References regarding a connection between recursion theory and sheaves

In Manin's A Course in Mathematical Logic for Mathematicians, he defines (p.201) a structure $(\mathcal{E},R)$ given an enumerable set $E \subset (\mathbb{Z}^+)^n$ by: $\mathcal{E}$ is the set of ...
12
votes
0answers
1k 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 ...
11
votes
2answers
2k views

Wikipedia's definition of 'locally free sheaf'

Let $R$ be a, say, noetherian ring and $M$ an $R$-module. The Wikipedia article on 'locally free sheaf' tells me that the following two statements are equivalent: The module $M$ is locally free ...
11
votes
3answers
571 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 ...
11
votes
1answer
742 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 ...
11
votes
2answers
554 views

Cosheafification

Hello all. I have a pre-cosheaf in the category of vector spaces. How do I cosheafify? I've failed to find literature on this topic. I'll be more specific. Let $\mathbb{X}$ be a topological ...
11
votes
2answers
603 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 ...
11
votes
2answers
2k 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 ...
11
votes
0answers
1k 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 ...
10
votes
1answer
752 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 $\mathscr F^+$ ...
10
votes
1answer
643 views

Reference request: Book of topology from “Topos” point of view

Question: Is there any book of topology in the modern language of topos theory? Motivation: In "Sheaves in Geometry and Logic" Mac Lane and Moerdijk say: "For Grothendieck, topology became the ...
10
votes
2answers
1k views

Finiteness conditions on simplicial sheaves/presheaves

Could someone give an overview, or just some examples, of "finiteness conditions" for simplicial sheaves/presheaves and/or simplicial schemes? Any answer or comment about this would be interesting, ...
10
votes
2answers
1k 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 ...