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12
votes
1answer
267 views

The real numbers object in Sh(Top)

If $X$ is a sober topological space, the real numbers object in the topos $\mathrm{Sh}(X)$ is the sheaf of continuous real-valued functions on $X$. This is proven very explicitly in Theorem VI.8.2 of ...
0
votes
1answer
112 views

Is the cokernel of a map of sheaves a seperated presheaf?

The cokernel of a map of sheaves is not necessarily a sheaf until you sheafify. In every example I have seen of the cokernel failing to be a sheaf it is the glueability axiom that fails while the ...
2
votes
1answer
133 views

Nearby cycles and specialisation - properties

I am looking for reference for properties of nearby cycles - specifically, commutation with non-characteristic pull-back (good enough - commutation with pull-back to closed subvariety which is ...
2
votes
0answers
160 views

Two functorial definitions of schemes

I have been reading a bit about the "functor of points" theory for schemes. There seem to be two ways of going about defining schemes this way: Equip the category $\textbf ...
1
vote
1answer
151 views

Examples of nontrivial local systems in Decomposition Theorem

There is a proper map $f: X \rightarrow Y$ of projective varieties. The Decomposition Theorem of Beilinson–Bernstein–Deligne-Gabber states that $$Rf∗IC_X \cong \oplus_a ...
2
votes
2answers
318 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 ...
2
votes
0answers
57 views

Cohomology and quotients for the canonical topology

Recall that for any category $\mathcal C$, there is a unique finest topology, the canonical topology on $\mathcal C$ for which all representable functors are sheaves. I am interested in the example ...
4
votes
0answers
244 views

Cohomological dimension of the category of sheaves

Let $X$ be an $n$-dimensional manifold. Then for any sheaf $\mathcal{F}$ on $X$, the cohomology $H^i(X; \mathcal{F})$ vanishes for $i > n$. Let $k$ be a field, and let $\mathrm{Shv}_k(X)$ be the ...
0
votes
3answers
284 views

Higher cohomology of sheaves on a projective space

Let $S\subset\mathbb{P}^n$ be a finite set of $s$ reduced points. Let $\mathcal{I}$ be the ideal sheaf of $S$ in $\mathbb{P}^n$. We consider the sheaf ...
0
votes
1answer
221 views

Global to local for Ext groups and Sheaves

Let $X$ be a projective variety. The sheaf $\mathcal{E}xt^{1}(\Omega_{X},\mathcal{O}_{X})$ is supported on $Sing(X)$. Now, there should be a theorem (perhaps by Schlessinger) that says that if $X$ ...
4
votes
2answers
258 views

on the Zariski sheafification of Quillen's K-theory

Let $X$ be a smooth k-variety and denote by $K_n$ Quillen's K-theory sheaf, that is: the Zariski sheaf on $X$ associated to the presheaf $U \mapsto K_n(U)$. The Bloch-Quillen formula says that ...
3
votes
1answer
166 views

Given an exact category, viewed as a site, do there exist non-additive sheaves?

Suppose given an exact category $\mathcal{C}$. The following question arises while proving the Gabriel-Quillen-Laumon embedding theorem following Laumon [1]. Laumon constructs an abelian category ...
1
vote
1answer
285 views

Cohomology of tangent bundles

Let $X$ be a smooth scheme and $Z\subset X$ a smooth subscheme. Consider the blow-up $$\pi:\widetilde{X}:=Bl_{Z}X\rightarrow X$$ of $X$ along $Z$. What is the relation between the cohomology of the ...
3
votes
1answer
119 views

Does the bundle of germs of functions $f:X\to \mathbb R$ have the same sheaf of sections as $X\times \mathbb R$?

I'm just starting to learn about sheaves, and I'm confused about a certain matter: I've just learned, to my delight, that every sheaf $S$ on a space $X$ is the sheaf of sections of a particular ...
1
vote
1answer
123 views

Sheaves on the site of $\pi$-sets

Let $\pi$ be a group, and let $\mathcal{C}$ be the site whose underlying category is that of $\pi$-sets (with $\pi$-linear maps as morphisms). The covers are jointly surjective families of such ...
0
votes
0answers
96 views

local systems, duals, cohomology

Let $U=\mathbb{P}^1-\{p_1, \ldots, p_n\}$ be a Zariski open subset of the projective line. Consider a rank $r$ local system of complex vector spaces $V$ on $U$ and assume that the monodromy ...
9
votes
1answer
430 views

Why is there no stack of $\ell$-adic sheaves on a curve?

One of the main players in the categorical geometric langlands correspondence is the moduli stack of rank n integrable connections on a complex curve. The reason for considering such objects is that ...
3
votes
1answer
121 views

For which sites are all constant presheaves separated?

I'm intererested in open surjective geometric morphisms induced by fibrations of sites $S\to T$ a la Moerdijk, but as a warm-up, let's consider the case $S \to \ast$. In the case that $S$ is a poset ...
0
votes
0answers
127 views

Split and pure exact sequence of sheaves

Let $X$ be a topological space and $$\varepsilon \ :\ 0 \to A \to B \to C \to 0$$ be an exact sequence of sheaves of ${\cal O}_X$-modules. $\varepsilon$ is said to be pure if for each point $x\in ...
1
vote
1answer
286 views

+ functor (used to construct sheafification)'s property

Let $X$ be a topological space, $\mathcal{C}$ be a locally small category with "good properties" (such as having small inverse limit, small filtrant inductive limit...etc.) and $\mathcal{F}$ be a ...
4
votes
3answers
417 views

Are the global sections of a vector bundle a projective module?

Given a scheme $X$ with structure sheaf $\mathcal{O}_X$, we can associate to each $\mathcal{O}_X$-module $\mathcal{F}$ its global sections $\Gamma(\mathcal{F})$, which gets the structure of a ...
2
votes
0answers
124 views

Is there something interesting in the uniqueness condition for a sheaf?

After digesting the Presheaf definition by the very first time, one feels (at least I felt) a strange sensation noticing the existence and uniqueness conditions to graduate that Presheaf as a sheaf, ...
5
votes
1answer
284 views

fpqc sheafification and localisation

I am slightly confused about sheafification at the moment. I first learned sheaves defined as a subcategory of presheaves, then I was told that sheaves are also a localisation of presheaves, then I ...
1
vote
0answers
72 views

compactness of moduli stack of semistable sheaves

It's known that the moduli stack ${\cal M}$ of semistable sheaves on a given polarized projective variety, with a fixed Hilbert polynomial, is compact (meaning that ${\cal M}$ has an atlas of finite ...
9
votes
1answer
381 views

When do sheaves which are constant along the fibers come from the base?

Assume we are given a map $f: X \rightarrow Y$ between topological spaces which is open, surjective and has (pathwise) connected fibers. Consider categories $\text{Sh}(X)$, $\text{Sh}(Y)$ of sheaves ...
3
votes
1answer
193 views

Coverage, itself considered as a presheaf

A coverage $J$ on a category $C$ assigns to an object $U$ of $C$ a set of covering families $J(U)$. The covering families are required to be stable under pullback, which amounts to requiring that for ...
2
votes
1answer
197 views

Carving out subsheaves of local hom-sheaves of stacks of categories

Recall from my previous question the definition of a local hom-sheaf of a stack of categories. I am interested in stacks of categories such that the underlying stack of groupoids is a moduli stack. ...
0
votes
1answer
395 views

Spectral sequences in Hypercohomology of sheaves (For a complex of acyclic sheaves) - Follow-up to previous question

Alright, this is a follow-up to my previous question (Spectral sequences in Hypercohomology of sheaves), sorry I took so long to reply. Let $X$ be a topological space, let $F^\bullet$ be a cochain ...
4
votes
2answers
414 views

trying to understand the support of the sheaf of relative differentials

So I'm trying to understand a proof of Belyi's theorem from http://eprints.soton.ac.uk/29785/1/b45h1koe.pdf specifically lemma 3.4. The setup is as follows: Let $X/\mathbb{C}$ be a curve, and let $t ...
2
votes
1answer
528 views

Spectral sequences in Hypercohomology of sheaves

Alright, here I go again, don't know if I'm missing something here but let $X$ be a topological space and let $F^{\bullet}$ be a cochain complex of sheaves, I want to compute the cohomology of this ...
1
vote
2answers
409 views

Hypercohomology of a complex of sheaves that might be acyclic (or might not)

Back again, check this out, let $X$ be a topological space and let $F^{\bullet}$ be a cochain complex of sheaves, I'm trying to compute the cohomology of the complex of global sections of the sheaves ...
1
vote
2answers
264 views

Cohomology of a cochain complex of acyclic sheaves

Ok, sort of as a follow up to my previous question, let's recall the de Rham-Weil theorem: Let $F$ be a sheaf on a topological space $X$ and let $\mathcal{L}^{\bullet}$ be an acyclic resolution of ...
1
vote
0answers
142 views

Soft sheaves on indiscrete paracompact spaces

Let $X$ be some space, I have basically 2 questions: 1 - Are sheaves on paracompact but not Hausdorff spaces acyclic? I've been doing some reading and some authors say that soft sheaves on ...
4
votes
1answer
477 views

Euler characteristic and inclusion-exclusion

Define the Euler characteristic of a scheme to be the Euler characteristic of its structure sheaf. I remember being told that for curves, this invariant satisfies inclusion-exclusion. That is, if ...
10
votes
1answer
763 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 $\mathscr F^+$ ...
13
votes
2answers
570 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
votes
1answer
1k views

surjective morphism of schemes or epimorphism of sheaves?

I have a technical question coming from reading Toen's master course on stacks. If we view schemes as locally ringed spaces then there we could define a morphism to be surjective if it the underlying ...
0
votes
1answer
201 views

sequence of sheaves for studying intersection

I'm studying intersection of curves with a fixed plane cubic, the first case I consider is of course lines, in particular lines intersecting the cubic at only one point. The problem is quite easy and ...
1
vote
0answers
357 views

Fine and acyclic sheaves on locales

Hey all. Here's the thing, so I have a measure space. According to Johnstone's 'Topos theory' (page 213), let $(X,\Sigma,\mu)$ be a measure space, we can define a Grothendieck pretopology on it (and ...
0
votes
1answer
339 views

Sheaf of sections and local triviality

This is probably not a research level question, I'm sorry if it is inappropriate. I'm reasking here this question on math.se. Suppose that $\xi: E \to B$ is a bundle (by which I mean simply a ...
2
votes
2answers
599 views

Are presheaves of constant functions sheaves?

Hey there, I was reading 'An introduction to homological algebra' by Rotman, and on page 279 in the section about sheaves, example 5.64, Rotman gives an example of a constant presheaf $\mathcal{P}$ ...
1
vote
0answers
488 views

The “pullback presheaf” and the proper base change theorem in topology

Let $f:X\rightarrow Y$ be a continuous map of topological spaces and let $\mathcal{F}$ be a sheaf (say of abelian groups to fix the idea) on $Y$. Define the following rule on open sets of $X$: $$ ...
1
vote
1answer
400 views

Question about the definition of a sheaf cohomology group for a sheaf using tensor products of sheaves

In Warner's 'Foundations of differentiable manifolds and Lie groups', in the section about axiomatic sheaf theory (page 178), when establishing the conditions necessary for the existence of a ...
2
votes
1answer
323 views

Does anyone understand the notation in this equation for the sheafification of a presheaf on a site?

Hi there, I'm trying to sheafify a constant presheaf on a site, I went to http://ncatlab.org/nlab/show/sheafification, but can't understand the notation in the equation for W (in the proof for ...
6
votes
1answer
700 views

Sheaves of $\mathbb Z$-modules = sheaves of abelian groups

Hi. In his Algebraic Geometry, Hartshorne proves that for any ringed spaces $(X,\mathcal O_X)$, category $Mod(X)$ of sheaves of $\mathcal O_X$-modules has enough injectives. If we tako $\mathcal O_X$ ...
3
votes
2answers
741 views

morphisms of affine schemes question

So, in chapter 2, section 2 of Hartshorne, (prop 2.3), he describes how if $\varphi : A\rightarrow B$ is a homomorphism of rings, then you get a morphism of (affine schemes): ...
3
votes
0answers
276 views

Two definitions of Čech cohomology

Hello, I have found different definitions of Čech complex for sheaf $F$ od abelian groups on topological space $X$ with respect to the cover $\mathcal U$. One in Gelfand-Manin says to take product of ...
1
vote
1answer
559 views

when a section descends?

Let $C$ be a (reduced, possibly reducible, complex) projective singular curve. Let $\nu: C'\to C$ a finite surjective birational morphism. (For example the normalization, but could be some ...
3
votes
2answers
1k views

Modules, Sheaves and Vector bundles

Given a graded ring $S$ and a graded S-module $M$ we can carry out a construction in order to get $\tilde{M}$, which is a sheaf over the scheme $\mathrm{Proj}~ S$. With this in view, I have an ...