The sheaves tag has no wiki summary.

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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
...

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### 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 $X$ ...

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### 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 ...

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### 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 ...

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### 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 ...

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### on relative divisors over artinian rings

Let $X$ a curve over $\mathbb{C}$, $D$ a divisor on $X$, $R$ a local artinian ring of residue field $\mathbb{C}$
Let $A=H^{0}(X_{R},\mathcal{O}(D_{R}))$ the scheme of sections over $Spec(R)$.
Let ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### + 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 ...

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### 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 ...

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### 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, ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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.
...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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
...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $\mathscr F^+$ ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $\mathcal{P}$ ...

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### 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 $X$:
$$
...

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### 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 ...

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### 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 ...

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### 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 $\mathcal O_X$ ...

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### 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):
...

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### 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 ...

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### 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 ...

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### 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 ...