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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16 views

sheafification construction in Hartshorne

In section II.1 of Hartshorne, the sheaf $\mathscr F^+$ associated to a presheaf $\mathscr F$ is constructed so that $\mathscr F^+(U)$ is the set of functions $$ s\colon U \to \bigcup_{p \in U} ...
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2answers
54 views

Stalks of the sheaf $\mathscr{H}om$?

The question is basically the title. What are the stalks of the sheaf $\mathscr{H}om_{\mathscr{O}_X}(\mathscr{F},\mathscr{G})$? If $X$ is a noetherian scheme and $\mathscr{F}$ is coherent, then ...
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40 views

How could we define a sheaf or presheaf of polynomials? [closed]

Good evening everyone , Is there a sheaf or presheaf whose sections are polynomials defined on opens of a topology ? . If yes , what is this topology ?. Is it the Zariski topology , and why? And how ...
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15 views

Resolution of the locally constant sheaf $ GL_n( \mathbb{C} ) $.

I would like to find one natural and nice resolution of the locally constant sheaf which contains sections with values in $ GL_n( \mathbb{C} ) $ or $ GL_n( \mathbb{R} ) $. Thanks a lot for your help.
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0answers
27 views

Question regarding a section of an open set of the form $U \cup V$

Suppose I have some scheme $(X, O_X)$. Suppose I have two open subsets of X, $U$ and $V$. I was wondering about the following: 1) Is $\Gamma (U \cup V, O_X) \cong \Gamma (U , O_X) \times_{\Gamma (U ...
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1answer
33 views

Affine line with double origin

Let $X = Spec \ k[t]$ and $Y = Spec \ k[u]$ and let $U = D(t)$ and $V = D(u)$. I construct the affine line with double origin by gluing the two affine schemes $X$ and $Y$ together along $U \cong V$ ...
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1answer
37 views

Disjoint union of two affine schemes

Say I have two commutative rings with unity, $R$ and $S$. What does the sheaf of disjoint union of $\DeclareMathOperator{Spec}{Spec}(\Spec(R), \mathscr O_{\Spec(R)})$ and $(\Spec(S), \mathscr ...
3
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1answer
45 views

Structure sheaf of $Spec \ k[x,y]$

Let $k$ be a field. We consider the affine scheme $(Spec \ k[x,y], O_{Spec \ k[x,y]})$. Let $U = D(x) \cup D(y)$. We have that $\Gamma(D(x), O_{Spec \ k[x,y]}) = A_x$ and similarly $\Gamma(D(y), ...
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0answers
18 views

Confusion in basic defintion of sheaf cohomology

I am reading the first chapter of Homological algebra by Weibel and after definition of sheaves, he remarked that there is a short exact sequence:$$ 0 \to \mathbb{Z} \xrightarrow{2\pi i} \mathcal{O} ...
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1answer
50 views

Is a coherent locally free sheaf isomorphic it's dual?

Hartshorne chapter II problem 5.1 a) is to prove that the double dual of a coherent locally free sheaf $\mathscr{E}$ over a ringed space $(X,O_X)$ is isomorphic to $\mathscr{E}$. This can be done by ...
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1answer
17 views

Adjoint functors of sheaves and stalks

Let $X$ and $Y$ be topological spaces and $F:Sh(X)\to Sh(Y)$, $G:Sh(Y)\to Sh(X)$ be functors between the categories of sheaves over the respective topological spaces. It seems like a very important ...
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1answer
66 views

Hartshorne Chapter II exercise 5.7 on Invertible sheaves

I'm working on part c) which is to prove that for a Noetherian scheme $X$, a coherent sheaf $\mathscr{F}$ is invertible (locally free of rank 1) iff there exists a coherent sheaf $\mathscr{G}$ such ...
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0answers
28 views

Two different definitions of sheaf of $K$-modules and tensor products.

I am confused by two different approaches to defining sheafs of modules. In Hartshorne there is the concept of a sheaf $F$ of modules $O_X$-modules, where $F(U)$ is a module over $O_X(U)$ with ...
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0answers
44 views

Endomorphisms of constant sheaves on connected spaces

In a paper by Deligne and Lusztig it says An endomorphism of a constant sheaf over a connected base is constant My interpretation of this statement is that given a (non-empty) connected ...
1
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1answer
27 views

Extension of coherent sheaf from open subspace.

I'm trying to solve following problem: Let $X$ be a noetherian scheme, $U$ an open subspace of $X$, $\mathcal F \in Qcoh(X), \mathcal G\in Coh(U), \mathcal G \subset \mathcal F|_{U},$ then there ...
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1answer
40 views

etale sheafification of a presheaf

Consider the following presheaf on the big etale site of smooth schemes over a field $k$: to every smooth $k$-scheme $U$, associate $$F(U):= \{f: U \to \mathbb A^1_k ~|~ \text{$f$ factors through the ...
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1answer
41 views

Understanding the mechanics of gluing sections of presheaves to obtain sheaves?

Could anyone give me a couple specific examples of how sections of a Presheaf on discrete topology would or could glue together? If I am correct, it depends on the mapping one defines. Now I have ...
3
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1answer
59 views

Why is $\phi:\mathbb{P}^n\rightarrow \mathbb{P}^m$ constant if dim $\phi(\mathbb{P}^n)<n$?

Let $\phi:\mathbb{P}^n\rightarrow \mathbb{P}^m$, $n\leq m$. I want to demonstrate that if dim $\phi(\mathbb{P}^n)<n$ then $\phi(\mathbb{P}^n)=pt$ (ex. 7.3(a), ch.II from Hartshorne). It's well ...
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1answer
31 views

Acyclic but not flasque sheaf of abelian group?

Is there a sheaf of abelian groups which is acyclic but not flasque? Maybe we can try $0\to \mathcal{F'}\to \mathcal{F}\to \mathcal{F''}\to 0$ where $\mathcal{F',F''}$ are flasque but $\mathcal{F}$ ...
2
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1answer
55 views

Restriction of sheaf via inclusion induces isomorphism on stalks

Let $i: Z\rightarrow X$ be the inclusion of $Z $ as a subspace of $X $. Let $\mathscr{F}$ be a sheaf on $X$. The restriction of $\mathscr{F}$ to $ Z $ is defined as the sheafification of $U\mapsto ...
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0answers
33 views

On structure sheaf of an affine scheme

I am reading the algebraic geometry notes by Ravi Vakil. When he proves that the structure sheaf on affine scheme is indeed a sheaf (Thm 4.1.2. in his notes), he first proves that it gives a sheaf ...
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1answer
55 views

When do functions turn a space into a locally ringed space?

Let $X$ be a topological space, and consider for each open set $U \subseteq X$ a set $F_U$ of functions $U \to k$ into some fixed field $k$. Let $\mathcal{O}$ be the sheaf of $k$-algebras induced by ...
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0answers
51 views

Possible mistake in Gortz-Wedhorn's algebraic geometry book

I'm trying to solve exercise 2.14c in Gortz-Wedhorn's book on algebraic geometry, and it looks to me like it's wrong. Here's the statement. Let $X$ be a topological space and $i:Z \rightarrow X$ the ...
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0answers
20 views

Is there an explicite description for injective sheaves?

I want to find a criterion for sheaves of modules to be injective. It would be great if one can such a criterion for sheaves of modules over a ringed space. But an answer for sheaves of abelian groups ...
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0answers
37 views

Can regular functions be specified simply as the sections of a bundle?

(following on from this question of mine). In Hartshorne Algebraic Geometry the definition of the sheaf of rings attached to the spectrum of a ring $A$ to define an affine scheme has the following: ...
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1answer
29 views

Co-domain of a sheaf

We know that sheaves take open sets in some topology to values in some category. I want to know if it's possible for a sheaf $\mathcal{F}$ to take values in some category $\mathcal{C}$ which has no ...
2
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1answer
59 views

Abstract nonsense proof that stalks of $\mathcal{O}_X$ modules are modules over $\mathcal{O}_X$-stalks

Let $(X,\mathcal{O}_X)$ be a ringed space and $\mathcal{F}$ be an $O_X$-module. Then for $x \in X$, $\mathcal{F}_x$ has a natural structure of an $O_{X, x}$-module. Question: Is there some ...
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0answers
22 views

Finding a sheaf lying above an object satisfying some property

I have a sheaf $\mathcal F$ over a topological space $X$ (valued in a category $\mathcal C$ over which I don't think I should assume anything except existence of kernels... feel free to assume ...
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1answer
45 views

Étalé space for sheaf of sections of a fiber bundle

Let $X$ be a topological space, $\pi:E\to X$ a fiber bundle over $X$ with fiber $F$ and structure group $G$. Let $\mathcal{F}$ denote the sheaf of continuous sections of the bundle. I probably want to ...
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1answer
47 views

Sheaf associated to sheaf on basis

Statement: If we have a basis $B$ for a topological space $X$, then a sheaf defined on $B$ defines uniquely a sheaf on $X$. I was wondering if the following proof is correct: Let $\mathcal{F}$ be a ...
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1answer
54 views

Is the constant sheaf $\mathbb{Q}$ injective?

Let $X$ be a topological space, and let $\mathbb{Q}$ be the constant sheaf of abelian groups on $X$ associated to the group of rational numbers under addition. Is $\mathbb{Q}$ an injective object in ...
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71 views

Showing Grothendieck's Vanishing Theorem provides a strict bound

The following result is due to Grothendieck: If $X$ is a noetherian topological space of dimension $n$, then for all $i>n$ and all sheaves of abelian groups $\mathscr{F}$ on $X$, we have ...
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1answer
81 views

How does a section of a stack give a sheaf?

At nLab in the article constant stack and a few other related articles, a pattern is mentioned where a section of a constant sheaf is a locally constant function, a section of a constant stack is a ...
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1answer
45 views

Showing the image of $H^j(X;\mathbb C^\times)$ lies in the torsion subgroup of $H^{j+1}(X;\mathbb Z)$

Let $X$ be a (compact, if necessary) topological space. Then from the short exact sequence of constant sheaves $$ 0 \to \mathbb Z \to \mathbb C \to \mathbb C^\times \to 0 $$ we have a connecting ...
2
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1answer
41 views

Sheaf of sections vanishing at a point is $\Gamma(E) \otimes I_p$

Notation: $E$ is a vector bundle with sheaf of sections $\Gamma(E)$. $I_p$ is the sheaf of regular functions on the base space vanishing at a point $p$. $\Gamma_p(E)$ is the sheaf of sections of $E$ ...
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0answers
42 views

differential of $f:X\to\Sigma$ as an elliptic surface,

Let $X$ be an algebraic surface surface and $\sum$ an algebraic curve, and assume, $f:X\to\Sigma$ be an elliptic surface, my question is Why the differential $df$ can be viewed as an injection of ...
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1answer
68 views

Quotient of locally free sheaf is locally free?

If $0\rightarrow F\rightarrow G\rightarrow H \rightarrow 0$ is an extension of $\mathcal{O}$-modules with $F$ and $G$ locally free (each of constant finite rank, i.e. vector bundles), then is $H$ ...
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0answers
51 views

Hartshorne Exercise III 6.2 (a)

Let $X=\mathbb{P}^1_k$, with $k$ an infinite field. Show there does not exist a projective object $\mathcal{P}\to\mathcal{O}_X\to 0$. The author suggests to consider surjections of the form ...
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1answer
34 views

Stacks versus sheaves with values in categories

A (small) category is a perfectly valid algebraic structure like Groups, Rings, vector spaces, groupoids etc. So on a topological space or more generally on a site, it makes perfectly sense to ...
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0answers
41 views

Rank, degree and slope of a general coherent sheaf

Let $(X,\mathcal O_X)$ be a ringed space and $\mathcal F$ be a coherent sheaf of $\mathcal O_X$-modules on $(X,\mathcal O_X)$. Are there the definitions of rank, degree and slope of $\mathcal F$ in ...
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0answers
60 views

Finding the ring of regular functions on $X-S$

I am studying for an exam of algebraic geometry, and I would like to know if the following is correct. Let $X$ be an affine variety, and let $\mathcal{O}_{X}$ denote its sheaf of regular functions. ...
3
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1answer
108 views

sheafification definition?

I came across two different definitions of sheafification and I'm not sure how they are equivalent. One of them is here: About the sheafification Another one is from Tennison's sheaf theory: Given a ...
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0answers
36 views

Flat families of semistable sheaves parametrized by $\mathbb{A}^1$.

Suppose we have a non trivial short exact sequence, $$0\longrightarrow F'\longrightarrow F\longrightarrow F''\longrightarrow0,$$ where $F$, $F'$ and $F''$ are semistable sheaves with the same reduced ...
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0answers
28 views

Invariant differentials on group schemes

I'm studying group schemes from http://www.math.ru.nl/~bmoonen/BookAV/BasGrSch.pdf and I have some trouble with the following proposition. (3.15)Proposition Let $\pi:G\to S$ be a group scheme. Then ...
5
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1answer
89 views

Hypercohomology: finding a resolution for the de Rham complex of $\mathbb{CP}^1 $

Let $\mathbb{P}^1 $ be the complex projective line. Using the standard affine cover, $\mathcal{U} = \lbrace U,U' \rbrace, \ \ $ we can define some quasi-coherent sheaves on $\mathbb{P}^1 $. We can ...
3
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1answer
41 views

Adjunction counit for sheaves is isomorphism

Let $f\colon X \to S$ be a proper morphism of varieties over $\mathbb{C}$ with $f_* \mathcal{O}_X = \mathcal{O}_S$ and $\mathcal{G}$ be a coherent sheaf on $S$. Then we have a natural morphism ...
2
votes
1answer
59 views

Prove that $H^1(\mathcal{M}^*)=0$.

Let $X$ be a compact Riemann surface. For an open set $U$, let $\mathcal{M}^*(U)$ be the multiplicative group of nonzero meromorphic functions on $U$ ("nonzero" meaning "not identically zero"). This ...
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1answer
67 views

Cohomology Calculation

A couple of days ago I asked this Question on calculating hypercohomology I tried a similar example for $(\mathbb{C}^*)^2$, and I have a couple of questions. Here is my calculation: We have a ...
2
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1answer
46 views

Is this sequence of presheaves exact?

On p.298 of his Homological Algebra text, Rotman considers the sequence of presheaves on $X=\mathbb{C}-\{0\}$ : $0 \to \mathbb{Z}\to \mathcal{O}\to \mathcal{O}^\times \to 0$ where $\mathbb{Z}$ is the ...
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0answers
65 views

Connection between differential form of a manifold and Sheaf of relative differential of a map of schemes.

I was wondering whether there is a connection between the differential form of a manifold and Sheaf of Relative differential of a Scheme map. Definitions: Differential form on a manifold M is a map ...