# Tagged Questions

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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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### 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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### 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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### 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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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 $0\rightarrow \mathcal{O}_X(n)\rightarrow ... 1answer 39 views ### Why is$H^0(C,\mathcal O(D))$a vector space? Given a divisor$D$on a smooth curve, one can define the sheaf$\mathcal O(D)$by the prescription$\Gamma(U,\mathcal O(D) :=\{$meromorphic functions on$U$that satisfy$(f) + D \ge 0\}$. Then, ... 1answer 263 views ### Global sections of a tensor product of vector bundles on a smooth manifold This question is similar to Conditions such that taking global sections of line bundles commutes with tensor product? and Tensor product of invertible sheaves except that I am concerned with ... 0answers 62 views ### Cohomology of$\mathcal O(k)$I am reading a paper in which it is claimed that$H^1(\mathcal O(-k),\mathcal O)=0$, where$k\geqslant 1$. Moreover, the argument also requires that$H^2(\mathcal O(-k),\mathcal O)=0$. Here ... 1answer 105 views ### Identifying a line bundle on$\mathbb{P}^1$I have a geometric line bundle$L$on$\mathbb{P}^1 = \{[x_0:x_1]\}$. With respect to the standard affine cover$U_0 = \{x_0 \neq 0\}$and$U_1 = \{x_1 \neq 0\}$, I have the transition function ... 0answers 109 views ### A question of extension of vector bundles. Fix$p \in \mathbb{P}^1$. Let$X=\mathbb{P}^1\times \mathbb{P}^1$,$C_1=\mathbb{P}^1\times \{p\}$and$C_2=\{p\}\times \mathbb{P}^1$. Since$\mathrm{Ext}^1(\mathcal{O}_{C_2},\mathcal{O}_{C_1})\cong ...
Given 2 vector bundles $E$ and $F$ of ranks $r_1, r_2$, we can define $k$'th exterior power $\wedge^k (E \otimes F)$. Is there some simple way to decompose this into tensor products of various ...