0
votes
0answers
16 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 ...
4
votes
0answers
42 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 ...
2
votes
1answer
35 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 $E$ ...
1
vote
0answers
38 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 ...
0
votes
1answer
59 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 $H$ ...
3
votes
0answers
44 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 ...
2
votes
0answers
31 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 ...
0
votes
0answers
59 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
votes
1answer
97 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 ...
2
votes
0answers
35 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 ...
0
votes
0answers
26 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
votes
1answer
79 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
votes
1answer
38 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 ...
3
votes
1answer
54 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 ...
0
votes
0answers
61 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 ...
0
votes
1answer
72 views

Only $f^\sharp_x$ makes the diagram commutative

By Algebraic Geometry I from Görtz, Wedhorn page 60 $f^\sharp_x$ is the unique ring homomorphism which makes the diagram $A\to B \to B_{p_x}$, $A\to A_{p_{f(x)}}\to B_{p_x}$ commutative. The first ...
1
vote
1answer
63 views

Asterisk Notation

I am trying to read through the Cech Cohomology section in these notes: http://pub.math.leidenuniv.nl/~edixhovensj/teaching/2011-2012/AAG/lecture_14.pdf I would be really grateful if someone could ...
1
vote
1answer
55 views

Group of Automorphism of the Fiber of Fiber Bundle

Let us consider the Mobius Bundle. Can Someone Explain the part "There is no natural unique homeomorphism of $Y_x$with $Y$. However there are two such which differ by the map $g$ of $Y$ on itself ...
1
vote
1answer
28 views

Well-definedness of the action of the structure group of a principal bundle on the total space.

Find the definition of a fiber bundle here- Definition of Fiber Bundle I am having difficulty in proving that the natural action of $K$ on $X$ is well-defined: Let us recall how does K acts on X ...
1
vote
1answer
51 views

Principal Bundle- Definition cum Exercise from “Geometry and Topology” by Bredon

The definition of fiber bundle can be found from here: Definition of Fiber Bundle Then Bredon defines Principal bundle in the exercise as follows: I am not able to show how K acts naturally on ...
3
votes
1answer
84 views

Learning Fibre Bundle from “Topology and Geometry” by Bredon

Bredon defines bundle projection in the following way: $\bf13.1.$ Definition. Let $X,B$ and $F$ be Hausdorff spaces and $p:X\to B$ a map. Then $p$ is called a bundle projection with fiber $F$, if ...
5
votes
0answers
67 views

Leray's theorem for cech and derived sheaf cohomology.

My question is about the hypothesis of Leray's theorem. This theorem says that if $\mathcal{U}$ is an open cover of a topological space $X$, and $\mathcal{F}$ is a sheaf over $X$ and if ...
0
votes
0answers
38 views

Direct limit of $ \ \ \mathcal{D} = ((\mathcal{F} (U))_{U \in \mathcal{V}} \, \ (r \ : \ \mathcal{F} (U) \to \mathcal{F} (V))_{V \subset U}) $. [duplicate]

Let $ X $ be a topological space. Let $ \mathcal{F} $ be a sheaf on $ X $. Let $ U $ be an open subset of $ X $. Let $ \mathcal{V} $ the set of open neighborhoods of $ U $, which is the filter for ...
2
votes
1answer
80 views

The inverse image of a sheaf

By definition, the inverse image of the sheaf $ \mathcal{F} : \mathrm{Ouv} (Y) \to \mathrm {Set} $ is the sheaf associated to the presheaf $ f^{-1} \mathcal{F} : \mathrm{Ouv} (X) \to \mathrm{Set} $ ...
1
vote
0answers
31 views

Sheafification part 2: Uniqueness of $\tilde{\varphi}$ and a formal consequence

I'd like to go on discussing the proof which I started to discuss here. The book says sending $(s_x)_x\in\tilde{\mathscr{F}}(U)$ to $(\varphi_x(s_x))_x\in\tilde{\mathscr{G}}(U)$ defines a morphism ...
3
votes
1answer
77 views

Sheafification: Show that $\tilde{\mathscr{F}_x}=\mathscr{F}_x$.

My today's question is about a proof of this book. More precisely we are talking about the proof of Prop. 2.24 on page 52. The book says that we have $\tilde{\mathscr{F}_x}=\mathscr{F}_x$ for all ...
5
votes
1answer
72 views

A question about Hartshorne III 12.2

In Hartshorne III 12.2, $X\to \text{Spec}\ A$ is a morphism, $\mathcal{F}$ is a coherent sheaf on $X$, flat over $\text{Spec}\ A$, $M$ any $A$ module, then we can construct the sheaf associated to the ...
1
vote
0answers
89 views

Colimit and push forward of quasi-coherent sheaves in the analytic setting

I'm currently working on my master thesis and have some unresolved questions about quasi-coherent sheaves. Since I'm new to algebraic geometry, they might be rather trivial. I'm working in the ...
0
votes
1answer
70 views

Hartshorne Exersice 1.17 Skyscraper sheaf Chapter II Schemes

I am able to verify the statements about the stalk. I want to see how the direct image of the the skyscraper sheaf can be thought of as the constant sheaf. Observation- If $P\notin U$, then $U\cap ...
5
votes
1answer
79 views

Is there a coherent sheaf which is not a quotient of locally free sheaf?

Suppose $X$ is an algebraic variety, is there a coherent sheaf $\mathcal{F}$ on $X$ which is not a quotient of locally free sheaf? (Hartshorne II Cor 5.18 showed that on every projective variety, ...
9
votes
1answer
153 views

Hartshorne Theorem 8.17

I can't understand the proof of theorem 8.17 from Hartshorne's "Algebraic Geometry". Namely, he says that we have an exact sequence $$ 0 \to \mathcal J'/\mathcal J'^2 \to \Omega_{X/k} \otimes ...
1
vote
2answers
96 views

Soft sheaves adapted to $f_!$

I'm reading Gelfand-Manin, Homological Algebra. I understand that the class of soft sheaves is sufficiently large, because every injective sheaf is soft. Now to see that this class is adapted to ...
0
votes
1answer
56 views

What is a Presheaf (intuitively) and help with the technical machinery.

I have come across things such as that a Presheaf $\mathcal{F}$ associates data (such as rings, groups, other sets etc.) to open sets $U$ of $X$. That the Presheaf $\mathcal{F}$ becomes a Sheaf if ...
0
votes
1answer
64 views

Prove: $U \mapsto \mathrm{Hom}(U, Y)$

Rewording this problem via what Zhen Lin's notion of the original question is. For $X$ and $Y$ ringed spaces Prove: For each open $U \subset X$ the Presheaf $U \mapsto \mathrm{Hom}(U, Y)$ is a ...
4
votes
1answer
83 views

Exact sequence of sheaves of holomorphic functions

This is from Exercise 2.4.P. June 2013 version of Ravi Vakil's Math 216 notes. The idea is to show $\mathscr{O}_X \xrightarrow{\text{exp}} \mathscr{O}^*_X$ is an epimorphism. It seems ...
1
vote
1answer
52 views

Connected scheme but not quasicompact

I am searching for a connected scheme which is not quasi compact. My try: Suppose $A_i$=Spec$k[x]$, where $k$ is algebraically closed field are affine schmes . I am planning to glue $0$ of the ...
0
votes
0answers
46 views

Sheaf of ideals [duplicate]

Let $(X, \mathcal{O}_X)$ be a scheme and and $f \in \mathcal{O}_X(X)$. We define the sheaf of ideals by the assignment $$ U \longmapsto f|_U \mathcal{O}_X(U) $$ and denote this sheaf by ...
1
vote
0answers
53 views

Hom sheaf over a scheme in the case of quasi-coherent sheaf at first argument

Let $X$ be a scheme and $\mathcal{F},\mathcal{G}$ be two sheaves of $\mathcal{O}_X$-modules. I showed that the presheaf which assigns each open subset $U$ of $X$, $$ U \longmapsto ...
4
votes
0answers
31 views

Are acyclic coverings cofinal in the set of coverings?

I am interested by the following question in algebraic geometry. Recall that a covering $\mathfrak{U}$ of a topological space $X$ is acyclic for a sheaf $\mathscr{F}$ if we have $H^q(U_{i_0,\cdots, ...
1
vote
1answer
83 views

How to prove directly that, if $A$ is Noetherian and $X=\operatorname{Spec} A,$ then $\mathscr O_X$ is a coherent $\mathscr O_X$-module?

I use the following definition: Definition Let $(X,\mathscr O_X)$ be a locally ringed space. An $\mathscr O_X$-module $\mathscr F$ is coherent if (i) it is locally finitely generated. (ii) for every ...
1
vote
1answer
156 views

Is the cotangent sheaf quasi-coherent?

Allow me to reconstruct what is written here, in order for me to present the question. Let $(\mathscr{M},\mathscr{O}_{\mathscr{M}})$ be a smooth manifold, and let $\mathscr{M\times M}$ be the ...
6
votes
1answer
64 views

exact sequence induced by restriction to closed subscheme

I'm somewhat embarrassed to be confused about this issue after a while studying algebraic geometry, but here goes. Let $\iota: Y \hookrightarrow X$ be the inclusion of a closed subscheme, $U$ is the ...
2
votes
1answer
40 views

What is a sheaf of rings? (question regarding the definition)

I was reading some notes on Algebraic Geometry and it says, "Suppose $O_X$ is a sheaf on rings on a topological space $X$ (i.e., a sheaf on $X$ with values in the category of rings)." What does this ...
0
votes
1answer
40 views

Kernel,image of quasi-coherent sheaf is quasi-coherent for ringed spaces?

If $X$ is a ringed or locally ring space (not necessarily scheme), do we still have the kernel, image of quasi-coherent sheaves quasi-coherent?
1
vote
1answer
43 views

Pushforward of pullback of an etale sheaf

I hope this question is not too elementary, but I'm a bit lost : Let $S$ be a finite set of closed points of $\mathbb{P^1_{C}}$ and $j : U \longrightarrow \mathbb{P^1_{C}}$ be the inclusion of the ...
4
votes
3answers
88 views

What is the inverse image of a sheaf

Let $f : X \rightarrow Y$ be a continuous map of topological spaces and $\mathcal{G}$ a sheaf on $Y$. What exactly is $f^{-1}\mathcal{G}$? It seems like we should be able to describe the sections ...
0
votes
2answers
190 views

Non-cohomological proof that a quasi-coherent sheaf over an affine scheme is quasi-flasque

Let $\mathcal F$ be a quasi-coherent sheaf over an affine scheme $X$. Let $0 \rightarrow \mathcal F \rightarrow \mathcal G \rightarrow \mathcal H \rightarrow 0$ be an exact sequence of sheaves on $X$, ...
4
votes
1answer
51 views

Do I correctly understand the constructions involved in definition of Cartier divisor?

Let $(X,\mathcal O)$ be a ringed space, where $\mathcal O$ is a sheaf of unital commutative integrity domains. Let $\widetilde{\mathcal M}_U$ be the field of fractions of ring $\mathcal O_U$ for any ...
1
vote
0answers
41 views

Describing a locally free sheaf sitting between two locally free sheaves which are given as extensions

Assume $X$ is a two dimensional scheme with $Pic(X)=\mathbb{Z}$ such that every rank two locally free sheaf $\mathcal{E}$ is given by an exact sequence $0\rightarrow \mathcal{O}_X(n)\rightarrow ...
4
votes
2answers
129 views

Geometric interpretation and computation of the Normal bundle

My question concern the definition, geometric meaning, and usage of the normal bundle in algebraic geometry. Let $X$ be a nonsingular variety over an algebraically closed field $k$, and $Y\subseteq ...