The sheaf-theory tag has no wiki summary.

**14**

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### 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 ...

**3**

votes

**2**answers

308 views

### Colimits of covers

Suppose I have category $C$ equipped with a Grothendiek pretopology of covers, and let $y:C \to Sh(C)$ be the Yoneda embedding into sheaves and $y/c:C/c \to Sh(C)/y(c)\cong Sh(C/c)$. How can I show ...

**11**

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**2**answers

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 ...

**4**

votes

**1**answer

569 views

### When are non-quasi-coherent sheaves used?

Non-quasi-coherent sheaves of $\mathcal O_X$ modules on a scheme seem like a wild concept to me; are they actually used for something?

**3**

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**2**answers

320 views

### Projectivity of free O_X modules with respect to the sheafy hom?

I've heard that given a ringed topos $(X,\mathcal{O}_X)$, the functor $Hom_{\mathcal{O}_X-\operatorname{Mod}}(\mathcal{O}_X, -)$ often fails to be exact. Is this only the case for the unenriched hom ...

**2**

votes

**2**answers

289 views

### Equivalence of two definitions of sheaves on a site

I'd like to prove that two definitions of sheaves on a site are equivalent, but I'm having trouble proving one direction.
Let $C$ be a category with pullbacks. Let $(C,T)$ be a site defined through a ...

**4**

votes

**5**answers

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### sheaves and cosheaves

I am struggling hard to understand the pushforwards and pullbacks of cosheaves. Are they also cosheaves? And what are quasicoherent cosheaves? Is there anything like coquasicoherent cosheaves? Please ...

**8**

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**4**answers

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### 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 ...

**2**

votes

**2**answers

258 views

### Is a submodule of the sheaf of sections of a smooth vector bundle necessarily finitely generated?

Let $X$ be a finite-dimensional smooth manifold, $\mathcal C^\infty(X)$ its algebra of smooth functions, $V\to X$ a finite-dimensional smooth vector bundle, and $\Gamma(V)$ the space of smooth ...

**1**

vote

**1**answer

764 views

### Quasi coherent sheaf

One really stupid, trivial question: A Quasi coherent sheaf $F$ on an affine group scheme(Spec R) is simply an R-module. What happens in case R is a Hopf algebra? Will the Q.coherent sheaf $F$ be an ...

**7**

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312 views

### Applications of sheaf theory to the computation of invariants of LS-category type

I would like to know if sheaf theory can be applied to a particular class of questions in topology.
The Schwarz genus (also known as sectional category) of a continuous map $p\colon\thinspace E\to ...

**22**

votes

**6**answers

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### 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 ...

**1**

vote

**1**answer

254 views

### Realizing a restriction as direct/inverse image of sheaves

Consider the inclusion $j$ of ${\mathbb{R}}$ as the real axis of ${\mathbb{C}}$. On ${\mathbb{C}}$ I have a real polynomial algebra ${\mathbb{R}}[x,\bar{x}]$, where $\bar{x}$ denotes conjugation. ...

**42**

votes

**5**answers

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### 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 ...

**4**

votes

**0**answers

317 views

### Monomorphisms of sheaves

The following is surely pretty standard, but I have been unable to prove it or find a proof in the literature (many sources assert it without proof).
Let $\phi : \mathcal{F} \rightarrow \mathcal{G}$ ...

**1**

vote

**1**answer

753 views

### Ringed and locally ringed spaces

A pair $(X,O_X)$ is a ringed space if $X$ is a topological space and $O_X$ is a sheaf of rings. If every stalk $O_{X,x}$ is a local ring, then we say that $(X,O_X)$ is a locally ringed space.
In the ...

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115 views

### Are sieves in locally small categories still sets?

In "Sheaves in Geometry and Logic", M&M define a sieve of an object $C$ as a downward-closed set of arrows $S$ with codomain $C$. They go on to say that for a locally small category, a sieve of ...

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**2**answers

407 views

### Restriction of Ext sheaves

Let $f \colon X \to Y$ be a map of schemes, $\mathcal{F}, \mathcal{G}$ two coherent sheaves on $Y$. I'm interested in conditions which guarantee an isomorphism
$$f^{*} \mathcal{E}xt^i(\mathcal{F}, ...

**9**

votes

**4**answers

705 views

### Is there an easy way to describe the sheaf of smooth functions on a product manifold?

A smooth structure on a manifold $M$ can be given in the form of a sheaf of functions $\mathcal{F}$ such that there is an open cover $\mathcal{U}$ of $M$ with every $U\in \mathcal{U}$ isomorphic ...

**7**

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**3**answers

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### 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 ...

**10**

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**1**answer

685 views

### Understanding the etale space construction from a formal viewpoint

Suppose I have a topological space $X$. Let $\mathcal{O}(X)$ denote the poset of open subsets. There is a canonical functor $\mathcal{O}(X) \to Top/X$ which sends an open $U \in \mathcal{O}(X)$ to $U ...

**3**

votes

**1**answer

251 views

### The upper semi-continuous rank of a module sheaf

The article "Intuitionistic algebra and representations of rings" is fantastic; it develops the language, logic etc. for the topos $Sh(X)$, where $X$ is a fixed topological space, from scratch and ...

**2**

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**1**answer

307 views

### How to find the smallest flabby sheaf containing a given sheaf ?

None of the spaces C^k (\mathbb{R}^n), with 0 \leq k \leq \infty, is a flabby sheaf. However, they are respectively contained in the smallest flabby sheaves C^k_{nd} (\mathbb{R}^n) of functions f : ...

**1**

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**1**answer

641 views

### How is the right adjoint $f_*$ to the inverse image functor $f^*$ described for functor categories $Set^C$, $Set^D$ and $f : C \to D$

For $C,D$ small categories, and $f : C \to D$ a functor between them, there is a precomposition, or "inverse image", functor $f^* = (-) \circ f : Set^D \to Set^C$. It has a left and a right adjoint. ...

**9**

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**1**answer

789 views

### Mitchell's embedding theorem

Mitchell's embedding theorem http://en.wikipedia.org/wiki/Mitchell%27s_embedding_theorem tells us that every small abelian category ${\cal A}$ has a full, faithful and exact embedding $V : {\cal A} ...

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### Is the dual notion of a presheaf useful?

It seems that there is a common theme in mathematics where, if we want to find out about a category C, then we look at $\hat{C}$ (the category of contravariant functors from $C$ to $Set$). There are ...

**3**

votes

**2**answers

620 views

### Describing global sections of sheafifications

Recently on glancing through Hartshorne's description
of Cartier divisors I started pondering the definition of
sheafification which led me to a question I can't answer. Neither
can I find a ...

**1**

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**1**answer

406 views

### The fiber of the sheaf of invariants

Let us suppose the the group $G:=\mathbb{Z}/2\mathbb{Z}=(1,i)$ freely act on a smooth projective variety/$k$ $X$ and denote by $Y$ the G.I.T. quotient $X/G$. Let $\pi:X\longrightarrow Y$ the quotinet ...

**2**

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**1**answer

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### Relation between Sheaf and Group Cohomology

Let $E=\mathbb{C}/L$ be an elliptic curve. Then $\mathbb{C}$ is contractible, and $L$ is the fundamental group of $E$. What's interesting is that we can find the cohomology of $E$, which is the same ...

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### 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 ...

**3**

votes

**1**answer

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### Grothendieck spectral sequence and Mayer-Vietoris sequence

Suppose $U'\cup U''=X$ is an open cover $U$ of a topological space $X$ and $F$ is a sheaf on $X$ with values in abelian groups. There is a special instance of the Grothendieck spectral sequence ...

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203 views

### Descent of singular cohomology

When proving that singular cohomology of an appropriate space $X$ equals sheaf cohomology of $X$ with "values" (does one say that?) in the sheaf $\mathbb{Z}_X$ of locally constant functions, the ...

**16**

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**1**answer

840 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 ...

**2**

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**1**answer

338 views

### vanishing theorems

I would be glad to know about possible generalizations of the following results:
1) (Grothendieck) Let $X$ be a noetherian topological space of dimension $n$. Then for all $i>n$ and all sheaves of ...

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**2**answers

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### Sheaf cohomology question

For a topological space $X$ and a sheaf of abelian groups $F$ on it sheaf cohomology $H^n(X,F)$ is defined.
Singular cohomology of $X$ can be expressed as sheaf cohomology if $X$ is locally ...

**4**

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**1**answer

359 views

### Does the concept of a basis for a topology on a category exist?

If we want to define a sheaf F on a topological space X and we have a basis B for the topology of X, what we can do is to define objects and restrictions for guys in B, check that they satisfy the ...

**3**

votes

**1**answer

377 views

### Functoriality of base change

Let $a:W\rightarrow X$, $c:X\rightarrow Z$, $b:W\rightarrow Y$ and $d:Y\rightarrow Z$ be a pull-back diagram in the category of topological spaces. Then one can construct a natural isomorphism ...

**5**

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**1**answer

756 views

### Natural morphism appearing in Grothendieck spectral sequence

Assume we are in the setting of the Grothendieck spectral sequence (Weibel, 5.8): $G : A \to B, F : B \to C$ are left exact functors such that $G$ sends injective objects to $F$-acyclic objects. Now ...

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**2**answers

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### 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 ...

**2**

votes

**3**answers

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### Is Bredon's Topology a sufficient prelude to Bredon's Sheaf Theory?

I intend to try working through Bredon's seminal sheaf theory text prior to graduating (I am currently a second year undergraduate), but it is at a level which is far beyond my own (friends of mine ...

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### Sheaf Cohomology on a Stone Space

Let $X$ be a stone space, i.e. a compact, totally disconnected hausdorff space. Then $H^1(X,\mathbb{Z}/2)=0$. Here's one way of proving this: $X$ with $\mathbb{Z}/2$ (the constant sheaf) is an affine ...

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### sheaves on thickened nodal cubics

Suppose F is an algebraically closed field (of any characteristic) and that h in F[x,y,z]
is an irreducible cubic form defining a plane curve C with a node. A lot is known about
sheaves on C; for ...

**12**

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**1**answer

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### 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 ...

**2**

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**3**answers

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### Why is continuity required for sheaf-theoretic definitions of a structure on a space

For example, I take differentiability, analyticity, and algebraicity(of a function).
All(more or less) imply continuity. So when we define a differentiable function on $\mathbb R^n$ or an analytic ...

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**0**answers

279 views

### A question on a proof that fine sheaves are soft

Let's open R.O.Wells "Differential Analysis on Complex Manifolds" p. 53 and have a look at the Proposition 3.5 stating that all fine sheaves are soft (over a paracompact Hausdorff $X$). In the proof ...

**2**

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### Are two sheaves that are locally isomorphic globally isomorphic ?

Let $X$ be a topological space and let $\mathcal{F}$ and $\mathcal{G}$ be two sheaves over $X$.
Of course, if one has a morphism $f : \mathcal{F} \to \mathcal{G}$ such that for all $x\in X$, $f_x : ...

**9**

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**2**answers

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### 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 ...

**5**

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**5**answers

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### When is the push-forward of the structure sheaf locally free

Let $f:X\longrightarrow Y$ be a morphism of noetherian schemes. Under what conditions is $f_\ast \mathcal{O}_X$ a locally free $\mathcal{O}_Y$-module?
Example 1. Suppose that $f$ is affine. Then ...

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**0**answers

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### Wikipedia's definition of constant sheaf is wrong [closed]

According to wikipedia (constant sheaf) the constant sheaf $\underline{S}$ for an object $S$ is given by defining $\underline{S}(U)$ to be the set of functions $U \to S$, which are constant on each ...

**4**

votes

**1**answer

432 views

### Morphisms between pure complexes of sheaves

I would like to understand the theory of pure complexes of (etale?) sheaves (of geometric origin?). In particular, I would like to understand which conditions are realy necessary in (part 1 of) ...