A Sheaf $\mathcal F$ on a topological space $X$ captures local data $\mathcal F(U)$ given on open sets $U\subseteq X$ and how such data can be restricted to smaller open sets or glued together. In typical cases, $\mathcal F(U)$ is a set of functions defined on $U$ and an element of $\mathcal F(V)$, ...

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1answer
23 views

Universal property of sheafification

Given a presheaf $\mathcal{F}$ there is a sheaf $\mathcal{F}^+$ and a morphism $\theta: \mathcal{F}\to\mathcal{F}^+$ with the property that for any sheaf $\mathcal{G}$ and any morphism $\varphi: ...
0
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1answer
49 views

Presheaf image of a monomorphism of sheaves is a sheaf

EDIT: The original version regarding coker is false. Consider a morphism of sheaves of abelian groups $\Phi:\mathscr{F}\to\mathscr{G}$. It is not in general true that Im$_{pre}\Phi$ is a sheaf. ...
2
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1answer
50 views

First encounters with sheaves: could you tell me if my thoughts are correct?

Let $X=\mathbb R$ be the reals and $\mathbb Z$ the integers (with the discrete topology). Do I understand correctly the definition of sheaf? Here is how I understand it (illustrated by a ...
2
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0answers
58 views

Image sheaf is the sheafification of the image presheaf

This is an exercise in Vakil's notes on foundations of algebraic geometry. Suppose $\Phi:\mathscr{F}\to\mathscr{G}$ is a morphism of sheaves of abelian groups, show that the image sheaf Im $\Phi$ ...
2
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1answer
56 views

Colimit preserves monomorphisms under certain conditions

I know that colimit preserves epimorphisms. Consider the special case where The diagrams are indexed by a directed set $I$, We are in the category of certain algebraic structures, such as ...
2
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2answers
44 views

Pullback commutes with dual for locally free sheaf of finite rank

Let $ f:X\rightarrow Y$ be a morphism of ringed spaces. Let $ \mathscr{E} $ be an $\mathcal{O}_Y$ module that is locally free of finite rank. I want to show that $ (f^{*}\mathscr{E})^\vee\cong ...
3
votes
1answer
53 views

Computation of the global sections of a normal sheaf

Let $Y\subset X=\mathbb{P}^r$ be the image of the Veronese embedding $\mathbb{P}^1\rightarrow\mathbb{P}^r$. I want to calculate $dim$ $H^{0}(C,\mathcal{N}_{Y|X})$, where $\mathcal{N}_{Y|X}$ is the ...
2
votes
1answer
30 views

Sheaf hom and the adjunction of push forward and inverse image

I'm trying to show that the tensor product of sheaves commutes with inverse image. I've reduced the problem to the following isomorphism $$f_*\mathscr{H}om_X(f^*\mathcal{N},\mathcal{P}) \cong ...
0
votes
1answer
27 views

Interaction of sheaf hom and push forward

I'm trying to show the following statement from this answer $$f_* \mathscr{H}om_X(A,\;B) \cong \mathscr{H}om_Y(f_* A,\; f_* B)$$ where $ f:X\rightarrow Y$ is a map of topological spaces and $ ...
5
votes
1answer
84 views

In which cases does pullback commute with the Hom-sheaf?

Assume $f: (X,\mathcal{O}_X)\rightarrow (Y,\mathcal{O}_Y)$ is a morphism of locally ringed spaces and E and F are two locally free $\mathcal{O}_Y$-moduels of finite rank. I was wondering if we have ...
4
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0answers
76 views
+50

Regular sequence of sections of line bundles over a coherent sheaf

I am reading the first chapter from the book by Huybrechts and Lehn, where I encountered the following definition. I have the following doubts regarding this definition : What is the map ...
4
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2answers
129 views

When is the pushforward of a quasi-coherent sheaf quasi-coherent? Hartshorne proof

In the following we have $f:X \rightarrow Y$ morphism of schemes, $\mathcal{F}$ a quasi-coherent sheaf on $X$ and I am referring to proposition 5.8 page 115 in Hartshorne. To prove that the ...
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0answers
20 views

Complex cohomology $\cong$ sheaf cohomology of constant sheaf on Riemann surface?

I am currently reading in a rather down to earth book on Riemann surfaces. They define the first complex cohomology group $H^1(X, \mathbb{C})$ associated to a Riemann surface $X$ via $H^1(X, ...
1
vote
1answer
38 views

Constant presheaf is not a sheaf

I am suppose to give en example of a variety $X$ where the constant presheaf $\mathcal{F}$ is not a sheaf, this is my attempt, is it ok? Pick the constant abelian presheaf $\mathcal{F}$ with ...
3
votes
1answer
44 views

Projective dimension of a coherent sheaf in a short exact sequence

Let $X$ be a noetherian integral scheme. We define the projective/homological dimension of a torsion free coherent sheaf $E$ to be $\mathrm{dh}(E)= \sup\{dh(E_x)|x\in X\}$, here dh$(E_x)$ denotes the ...
2
votes
1answer
48 views

Sheaf module and morphism to sheaf hom

A (unital) $R$-module $M$ can also viewed as given by a (unital) ring morphism $\rho:R\to \text{End}(M)$. This point of view can be extended to many other examples of "compatible actions". The ...
2
votes
2answers
40 views

Why is dh$(k(x))=$dim $X$?

Let $X$ be an integral Noetherian scheme. Let $x\in X$ be a regular closed point of $X$. Then Huybrechts and Lehn in his book, says that dh$(k(x))=$dim $X$. Here dh$(k(x))$ refers to the ...
2
votes
1answer
59 views

Morphisms of sheaves

My question concerns direct/inverse image of sheaves and their properties. Let $\mathfrak{R}$, $\mathfrak{S}$, $\mathfrak{T}$ three sheaves of groups, over topological spaces $X$, $Y$ and $Z$ ...
0
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0answers
46 views

Sheaf of differential p-forms

Shafarevich defines the cotangent bundle at page 60 of "Basic Algebraic Geometry 2". Now he says that: 1) $\mathcal{F}_x=\mathcal{O}_x dt_1 + \dots + \mathcal{O}_x dt_n$, where $\mathcal{F}_x$ is the ...
2
votes
1answer
63 views

Sheafs of abelian groups are the same as $\underline{\mathbb{Z}}$-modules

In Vakil's notes page 76, he claims that a sheaf of abelian groups is the same as a $\underline{\mathbb{Z}}$-module, where $\underline{\mathbb{Z}}$ is the constant sheaf associated to $\mathbb{Z}$. ...
6
votes
0answers
65 views

Leray's theorem up to some degree

I am interested in the proof of Leray's theorem that relates Čech cohomology and sheaf cohomology. The theorem states that if we have a space $X$, a sheaf $\mathcal{F}$ and a covering of $X$ such ...
0
votes
0answers
9 views

Transition functions of sheaf tensor product

Suppose $\mathcal{L},\mathcal{M}$ are invertible sheafs on a scheme $X$. I've seen an abstract construction of $\mathcal{L}\otimes_X \mathcal{M}$, but I'm having trouble connecting this with a more ...
1
vote
1answer
12 views

open subset of scheme with zero section

Let's take a scheme $X$. Is it possible to have an open non-empty subset $U$ of $X$ such that $\mathcal{O}_X(U)=0$? I can't find an argument against it, since there could exist some open set ...
0
votes
1answer
116 views

Proof that structure sheaf of a variety is indeed a sheaf

I am trying to prove to myself that the structure sheaf of an irreducible variety is indeed a sheaf, where the structure sheaf for an irreducible variety $V$ is defined as $\mathcal O_V(U)=\{ f \in ...
2
votes
2answers
44 views

Sheafication of a presheaf

I have to solve exercise 2.3 by Shafarevich - Basic algebraic geometry, vol. II. Let $X$ be a topological space, $M$ an Abelian group and $\mathfrak{F}(U)$ the quotient group of all locally constant ...
1
vote
1answer
45 views

Isomorphism between stalk and stalk of the pushforward sheaf.

Let $f \colon X \to Y$ be a continuous map and $\mathcal{O}_X$ a sheaf (of sets) on $X$. Question: Is the stalk $(f_*\mathcal{O}_X)_{f(p)}$ for $P \in X$ isomorphic to the stalk $\mathcal{O}_{X,P}$ ? ...
3
votes
1answer
99 views

Maps between two line bundles versus sections of their tensor product

There is a point about maps betwen line bundles (continuous or smooth or holomorphic --- I don't think it matters for this question) that is glossed over in many texts. A map from one line bundle ...
0
votes
0answers
25 views

is sheafification morphism of presheaf image injective?

Let $\mathcal F$ be the presheaf image of some morphism of sheaves $\varphi:\mathcal G'\to\mathcal G$. Consider the sheafification morphism $\theta:\mathcal F\to\mathcal F^+=\text{im }\varphi$. Is ...
4
votes
1answer
59 views

sheaves of rings and maps to classifying topos

Let $R$ be the category of finitely presented commutative rings (but I don't know how necessary the hypothesis of finite presentation is for my question). Let $Set^R=Fun(R, Set)$ be the category of ...
2
votes
1answer
55 views

uniqueness of morphism $Spec(K) \to X$ of schemes

let $K$ be a field and $X$ a scheme. I'd like to understand the bijection $Hom_{Sch}(Spec(K), X) \cong \{x \in X | \exists \kappa(x) \to K \}$ That map is given by sending a morphism $f: Spec(K) \to ...
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vote
0answers
29 views

Sheaf-theoretic approach to Morse functions?

It is known that one can define a smooth structure on a manifold using a sheaf-theoretic formulation via defining the algebra of the (a fortiori) smooth functions on it (which satisfies the usual ...
1
vote
1answer
67 views

The $C^k$ Sheaves

In Sheaves in Geometry and Logic, the authors claim that $C^k$ are all sheaves because differentiability is local. How do I formally prove this?
3
votes
1answer
93 views

Degree of a torsion-free subsheaf

Suppose that $R$ is a torsion-free subsheaf (of positive rank) in another torsion-free sheaf $S$, on a smooth complex projective variety $X$. If $S$ is (slope) semistable, is it true that the degree ...
2
votes
1answer
61 views

direct and inverse images of sheaves and some canonical morphisms

Consider a continuous map $f\colon X\to Y$ between topological spaces. Let $\mathcal F$ be a sheaf on $X$ and $\mathcal G$ a sheaf on $Y$ (let's say of abelian groups). There exists canonical ...
2
votes
1answer
40 views

Basic question related to sheaf of a scheme

Suppose I have a scheme $X$. And some non-empty open set $U \subseteq X$. Does it then follow that $O_X(U)$ is not the trivial $0$-ring by any chance?
2
votes
3answers
91 views

Is $Spec(R_p)$ to $Spec(R)$ is an open immersion?

Let $R$ be a ring. $p$ be a prime ideal of $R$ . Let $R\rightarrow R_p$ be the canonical. Consider the map of schemes $Spec(R_p) \rightarrow Spec(R)$. Is it an open immersion. $Spec(R_p)$ is an open ...
4
votes
0answers
50 views

Help understanding the proof of a theorem about Cohomology of Vector Bundles

I am trying to understand a paper called Betti tables of graded modules and cohomology of vector bundles, but i am stuck in Proposition 6.8 which states: Let $\mathcal{E}$ be a vector bundle on ...
3
votes
2answers
261 views

Why can we use flabby sheaves to define cohomology?

In my algebraic geometry class, we defined sheaf cohomology using flabby sheaves, and the functor on the category of sheaves on a space $X$: $$ D: \mathcal F \mapsto D\mathcal F $$ where $$ ...
0
votes
1answer
17 views

How to glue local datum to get a global setion?

I want to prove for an $\mathcal{R}-$module $\mathcal{F}$ over a topological space $X$,where $R$ is a sheaf of rings,if there exist sections $s_1,\ldots,s_p\in\mathcal{F}(X)$ that generate every stalk ...
2
votes
2answers
64 views

Cohomology groups of coherent sheaves for very small and very big twists.

Let $\mathcal{F}$ be nonzero coherent sheaf over the projective space $\mathbb{P}_k^n$. The Serre vanishing Theorem says that $h^i \mathcal{F}(d)=0$ for $i>0$ and $d\gg 0$. I am wondering if it is ...
2
votes
0answers
44 views

Question about Tate resolution and cohomology groups of nonzero coherent sheaves.

Let $\mathcal{F}$ be a nonzero coherent sheaf on the projective space $\mathbb{P}_{k}^m$. I am trying to show that for every integer $d$ there is $j$ for which $h^j\mathcal{F}(d-j) \neq 0$. My ...
2
votes
0answers
38 views

Hartshorne Ex. II 1.16 b) Flasque sheaves and exact sequences

the exercise states that when we have an exact sequence $0\to\mathcal{F}'\to\mathcal{F}\to\mathcal{F}''\to 0$ of sheaves (say of Abelian groups) over a topological space $X$, and when $\mathcal{F}'$ ...
4
votes
1answer
85 views

The Theorem on Formal Funtions - Harthshorne Theorem 11.1

Let $f:X \rightarrow Y$ a projective morphism of noetherian schemes, let $\mathcal{F}$ be a coherent sheaf on $X$, and let $y\in Y$ be a point. For each $n\geq 1$ we define $X_n=X \times_Y Spec ...
1
vote
1answer
52 views

What is a natural exact sequence?

I know what an exact sequence is, but I have searched for the definition of a natural exact sequence, and could not find it. Does "natural" perhaps mean some sort of preservation of structure? I ...
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vote
0answers
45 views

on the sheafification of a presheaf

There's something bugging me on the theory of sheafification. I proved that, given $P$ a presheaf on a topological space $X$ , there exists a sheaf $P^*$ and a presheaf homomorphism $f:P\to P^*$ such ...
1
vote
2answers
51 views

Structure sheaf of $Proj \ S$ in terms of compatible stalks

Let $S$ be a graded ring. I was wondering if someone could please explain me how I can interpret structure sheaf of $Proj \ S$ in terms of compatible stalks? Thank you! Edit: This is Exercise 4.5.M. ...
2
votes
1answer
45 views

Simpleminded example: flasque sheaves

Consider the sheaf $\mathcal{F}$ of polynomial functions on $\mathbb{R}^2$ endowed with the usual topology. A sheaf is said to be "flasque" (or "flabby") if, given $V \subset U$ both open sets, the ...
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vote
0answers
16 views

Is the cokernel of the pullback-pushforward of a coherent sheaf in the image of the pushforward on the complement

Let $V$ be a projective variety. Let $W \subseteq V$ be a projective subvariety. Let $U$ denote the complement of $W$ in $V$. Denote by $i \colon W \hookrightarrow V$ and $j \colon U \hookrightarrow ...
1
vote
1answer
44 views

Pullback of the skyscraper sheaf

Let $\phi:X\longrightarrow Y$ be a morphism of schemes, and let $y\in Y$. Let $k(y)$ be the constant sheaf $k(y)$ on the closed subset $\{\bar{y}\}$. Then what is $\phi^*(k(y))$? By definition, ...
1
vote
1answer
40 views

$S$-local objects of presheaves are reflective and characterize local presentability

Let $PSh(\mathcal{C})$ be the category of presheaves on a small category $\mathcal{C}$. Let also $S$ be any fixed set of morphisms in $PSh(\mathcal{C})$. I say that an object $F\in PSh(\mathcal{C})$ ...