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
50 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
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1answer
39 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?
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2answers
61 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 ...
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0answers
41 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
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2answers
250 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 $$ ...
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1answer
16 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
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2answers
61 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
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0answers
43 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
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0answers
35 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 $\mathcal{F}'$ ...
3
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0answers
65 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
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1answer
50 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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0answers
36 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 ...
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2answers
47 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
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1answer
40 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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0answers
15 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 ...
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1answer
40 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, ...
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1answer
37 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 $F\in PSh(\mathcal{C})$ ...
3
votes
2answers
58 views

Action on sheaf cohomology in Bott-Borel-Weil theorem

Let $G$ be simply connected complex semisimple Lie grou and $P \subseteq G$ parabolic subgroup. Suppose $V$ is finite dimensional irreducible representation of $P$ with highest weight $\lambda$, and ...
8
votes
1answer
167 views

Why are these two definitions of the left-adjoint to $u^p\colon PShv(D)\to PShv(C)$ equivalent?

Suppose $u\colon C\to D$ is a functor between categories. Then there is a functor $$ u^p\colon PShv(D)\to PShv(C) $$ between the associated presheaf categories by precomposition with $u$ as it is ...
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0answers
50 views

Splitting of short exact sequence of sheaves

Let $X$ be a smooth projective variety over a field, say $k$. Consider the short exact sequence of $k$-modules, $$0 \to A_1 \to A \to A_2 \to 0$$ where $A$ and $A_2$ are $k$-algebras. Since these can ...
6
votes
1answer
111 views

When does a smooth projective variety X have a free Grothendieck group

Let $X$ be a smooth projective variety (e.g. Grassmannians). Since $X$ is smooth, the groups $G_0(X):=K_0(CohX)$ and $K_0(X):=K_0(VectX)$, the Grothendieck groups of coherent sheaves of modules on $X$ ...
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0answers
45 views

Vector bundles on projective varieties

For instance, in algebraic category, given a vector bundle on a smooth projective variety (irreducible), then is it always true that such bundle can be embedded as a subbundle of a trivial vector ...
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0answers
24 views

Exact sequence of sheaves from the exactness of sections

Let $X$ be a topological space, and $\mathcal{F}_1$, $\mathcal{F}_2$ and $\mathcal{F}_3$ be sheaves on $X$. Suppose for all $U$ open in $X$ we have, $0\longrightarrow ...
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0answers
40 views

Reference request: Cartier divisors versus invertible sheaves by Kleiman

Please delete this question if it is deemed inappropriate. Could someone link me to the paper "Cartier divisors versus invertible sheaves" by Kleiman please? My library doesn't provide access to it. ...
2
votes
2answers
50 views

Why is this locally free sheaf free?

Let $f:X\longrightarrow Y$ be a finite, flat morphism of schemes. Then, we know that $f_*\mathcal{O}_X$ is flat over $Y$, and also that $f_*\mathcal{O}_X$ is a coherent $\mathcal{O}_Y$ module. We ...
2
votes
1answer
30 views

Construct natural transformation $u^* R^i f_* \rightarrow R^i g_* v^*$ without assumption of quasi-coherence

I am reading Hartshorne Algebraic geometry. Chapter 3 Proposition 9.3 (in particular remark 9.3.1). It states that if we have commutative diagram in category of schemes (namely morphisms $f, g, h, u$ ...
1
vote
1answer
31 views

Open and closed localization of sheaves

In this paper: http://www-math.mit.edu/~hrm/papers/ss.pdf the author claims that Leray originally developed sheaves over closed sets rather than open sets and that it was Cartan who later realized ...
1
vote
1answer
53 views

Why is the h-topology not subcanonical?

The h-topology introduced by Voevodsky on the category $Sch/K$ of separated schemes of finite type over a field $K$ is the Grothendieck topology associated with the pretopology whose coverings are of ...
1
vote
1answer
42 views

Isomorphisms of sheaves, are they equal in this situation?

Suppose I have a topological space $X$ and sheaves of rings $F$ and $G$ on $X$. Suppose I also have two isomorphisms $\phi, \psi : F \rightarrow G$ and that $\phi(X) = \psi(X)$ as maps from $F(X)$ to ...
0
votes
1answer
40 views

Gluing schemes together

Let $X_i$ be schemes $(i \in I)$, $X_{ij} \subseteq X_i$ open subschemes with $X_{ii} = X_i$ along with isomorphisms $f_{ij}:X_{ij} \rightarrow X_{ji}$ such that $f_{ik} |_{X_{ij} \cap X_{ik}} ...
1
vote
1answer
49 views

Maps between direct limits and functoriality of $f^{-1}:Shv(Y) \rightarrow Shv(X)$ and $f_{*}:Shv(X) \rightarrow Shv(Y)$

My question rises from exercise 1.18 in chapter 2 of Hartshorne. Given a continuous function $f:X \rightarrow Y$, one has to show there is a natural bijection between ...
1
vote
1answer
92 views

Definitions of a very ample invertible sheaf

At the moment, I'm struggling with the following definitions i) and ii). I'd like to know why they are equivalent: Let $\mathcal{L}$ be an invertible sheaf on a variety X. i) $\mathcal{L}$ is ...
2
votes
0answers
48 views

Gluing sheaves together

I am doing the following exercise 2.7D from Ravi Vakil's notes. Suppose $X = \cup U_i$ is an open cover of $X$ and we have sheave $F_i$ on $U_i$ with the following isomorphisms $\phi_{ij}:U_i ...
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votes
3answers
33 views

basic question regarding the definition of sheaf of rings

I was wondering and got confused about something. Say we have a sheaf of rings $F$ on a topological space $X$. Let $U$ be an open set of $X$, then by the definition $F(U)$ is a ring. I was wondering, ...
1
vote
1answer
59 views

Categories and the direct image functor

I've just started learning about sheaves and yesterday I had a look at MacLane/Moerdijk's book "Sheaves in Geometry and Logic". I've discovered something in this book on which I'd like some ...
1
vote
1answer
28 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} ...
2
votes
2answers
71 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 ...
1
vote
0answers
28 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 ...
2
votes
1answer
86 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 $U \cong V$ ...
0
votes
1answer
52 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
votes
1answer
49 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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votes
0answers
20 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} ...
3
votes
1answer
60 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 ...
1
vote
1answer
25 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 ...
0
votes
1answer
75 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 ...
1
vote
1answer
53 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 ...
0
votes
0answers
48 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
vote
1answer
37 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 ...
0
votes
1answer
44 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 ...
0
votes
1answer
44 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 ...