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### Can we relate Cech cohomology and derived functor cohomology even when the cover we choose isn't nice?

In my algebraic geometry class this semester, we've learned about Leray's Theorem, which states that for a sheaf $\mathcal{F}$ on a topological space $X$, and $\mathcal{U}$ a countable cover of $X$, ...

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130 views

### Classification of Sheaves of Q-modules over R

Every constructible sheaf of $\mathbb{Q}-$modules over $\mathbb{R}$ is the direct sum of indecomposable sheaves, which are either sheaves with stalk $\mathbb{Q}$ at a point or constant sheaves with ...

**3**

votes

**1**answer

561 views

### Question on the interpretation of a presheaf category as a co-completion

The category of presheaves $Pre(C)$ on a small category $C$ is the category of functors $C^{op}\to Sets$. Since the category of sets is co-complete and every presheaf is a colimit of representable ...

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votes

**1**answer

333 views

### Extension of a first order deformation of a sheaf

Given a coherent and torsion free sheaf $F$ on a smooth projective scheme $S$.
Then we have a bijection between $Ext^1_S(F,F)$ and deformations of $F$ over $k[\epsilon]$, $\epsilon^2=0$.
Assume all ...

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votes

**1**answer

449 views

### Chern classes of the direct image of an ideal sheaf resp. skyscraper sheaf

Given a double cover $\pi: X\rightarrow \mathbb{P}^2$ of the projective plane by choosing a square root $S$ of $O_{\mathbb{P}^2}(Q)$, where $Q$ is a quartic in the plane.
Choose a closed point ...

**3**

votes

**4**answers

1k views

### Interesting examples of flasque sheaves?

Does anyone know any interesting examples of flasque sheaves? Ideally, I would like to see one that both arises naturally and is geometric in some sense. On the other hand, I know so few examples ...

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2k views

### On a proof of the existence of tubular neighborhoods.

Studying analysis on manifolds, I have found, in the proof of the existence of tubular neighborhoods, a reference to theorem 3.1.2 in "Topologie algebrique et theorie des faisceaux" of Godement.
...

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461 views

### If one wants to work with $Q_l$-adic sheaves, should the scheme be of finite type over a 1-dimensional one?

In section 6 of his 'Adic Formalism' T. Ekedahl states that $l$-adic sheaves 'behave nicely' for finite type separated schemes over $S$ that is regular of dimension $\le 1$. Is the dimension ...

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**1**answer

506 views

### How to compute cohomology groups of a closed subscheme Z of projective space, defined by a homogeneous polynomial of degree d?

Let $Z = \mathrm{Proj}\,k[x_{0},x_{1},\ldots,x_{r}]/f$ be a closed subscheme of degree $d$, i.e., $f$ is a homogeneous polynomial of degree $d$, and ...

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### 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 ...

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2k 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 ...

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331 views

### Chern classes in flat families

Given a smooth projective variety $X$ over an algebraically closed field $k$. Now given a another projective variety $T$ and a coherent $O_{X\times T}$-module $F$, which is flat over $T$.
Given $r,s ...

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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 ...

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**2**answers

931 views

### Under what circumstances do morphisms on the stalks of a sheaf induce a sheaf morphism

It is very well known that if $\alpha : \mathscr{F} \to \mathscr{G}$ is a morphism of sheaves, then it induces homomorphisms on the stalks. I have been wondering for a while if given a collection of ...

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1k views

### Presheaves are locally sheaves?

On nlab it says that a presheaf is locally isomorphic to a sheaf. What do they mean by locally isomorphic? Their definition of locally isomorphic is given in terms of Grothendieck topologies which i ...

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votes

**1**answer

313 views

### What kind of colimits are preserved by a certain Yoneda embedding?

(This question is related to this one)
Let $k$ be a field and consider the category $Sch/k$ of schemes over $k$, say also separable and of finite type. The Yoneda embedding
$$
Y:Sch/k \to Pre(Sch/k)
...

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327 views

### The single-plus construction is not the left adjoint of the inclusion of separated presheaves?

Convention:
Before I start, please note that it is not going to be sufficient to assume that the topology is subcanonical or that the site $C$ has finite limits, since the application I have in mind ...

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votes

**0**answers

226 views

### Topologies (and sheaves) on Cat and CAT

I've been wondering lately what sort of Grothendieck (pre)topologies there are on $Cat$ (the category of small categories) and $CAT$ (the v. large category of large categories - to forestall criticism ...

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votes

**1**answer

395 views

### About direct image of ideal sheaves

Let $\mu:X'\rightarrow X$ be a birational morphism of normal complex projective varieties.
Consider the two ideal sheaves $I_1= \mu_*\mathcal{O}_{X'}(-\sum d(E)E)$, ...

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425 views

### Why is this a local constant sheaf

If a group $G$ acts on a topological space $M$, and a representation of $G$ on a vector space $V$, why $M \times_G V$ is a local constant sheaf over $M/G$?

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330 views

### Homotopical descent information contained in the Dwyer-Kan function complexes of a presheaf category?

Recall that the category of sheaves on some site $C$ equipped with a grothendieck topology $\tau$ is equivalent to the localization of the category of presheaves $W^{-1}Psh(C)$ at $W$ where $W$ is the ...

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votes

**1**answer

2k views

### How to “globalize” the inverse function theorem?

Let $F: V \times W\rightarrow Z$, where $V,W,Z$ are finite-dimensional smooth (or analytic) manifolds and $F$ is smooth (or analytic). Assume that $\dim W=\dim Z$ and the usual inverse function ...

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513 views

### finite-dimensionality of cohomology groups on compact riemann surfaces

does the finite dimensionlity of the first cohomology group ($ H^1 $) of the sheaf of meromorphic sections of a holomorphic line bundle on a compact riemann surface follow easily from the finite ...

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245 views

### Automorphisms of constant sheaves

Let E be a Grothendieck topos, such as the category of sheaves of sets on a topological space. Then there is a unique geometric morphism $(\Delta \dashv \Gamma)\colon E\to \mathrm{Set}$, where ...

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**1**answer

546 views

### A form of cohomology and base change

Let $f \colon X \to Y$ be a proper morphism of (Noetherian) schemes, $\mathcal{F} \in \mathop{Coh}(X)$. Let $i_Z \colon Z \hookrightarrow Y$ be a closed subscheme and take the inverse image $W := X ...

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3k views

### Serre's FAC versus Hartshorne as an introduction to sheaves in algebraic geometry

I just found an English translation of Serre's FAC at Richard Borcherds' Algebraic Geometry course web page. I really want to read it sometime. I am beginner in Algebraic Geometry, just started ...

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349 views

### Another stupid question on l-adic sheaves: does a generically zero constructible sheaf vanish on an open subvariety?

Suppose that the stalk of a constructible l-adic ($\mathbb{Z}_l$-adic or $\mathbb{Q}_l$-adic) etale sheaf $S$ over a generic (Zariski) point of a variety $V$ is zero. Does this imply that $S$ vanishes ...

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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 ...

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360 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 ...

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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 ...

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votes

**1**answer

647 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?

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338 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 ...

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298 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 ...

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votes

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2k views

### 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 ...

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2k views

### 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 ...

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270 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 ...

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809 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 ...

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349 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 ...

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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 ...

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vote

**1**answer

262 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. ...

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votes

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7k 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 ...

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398 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 $\phi : \mathcal{F} \rightarrow \mathcal{G}$ ...

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**1**answer

935 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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126 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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420 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}, ...

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votes

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764 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 ...

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2k views

### 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 ...

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**1**answer

774 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 ...

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**1**answer

293 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 ...

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votes

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311 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 : ...