# Tagged Questions

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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. ...
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### 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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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Only $f^\sharp_x$ makes the diagram commutative

By Algebraic Geometry I from GĂ¶rtz, 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $f^{-1} \mathcal{F} : \mathrm{Ouv} (X) \to \mathrm{Set}$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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### 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 ...
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### 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 $X$, ...
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### 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 ...
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