# Questions tagged [sheaf-theory]

For questions about sheaves on a topological space. Usually you think of a sheaf on a space as the data of functions defined on that space, although there is a more general interpretation in terms of category theory. Use this tag with the broader (algebraic-geometry) tag.

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### Map of global sections surjective under a local condition

Let $M$ be a compact complex manifold and $p \in M$. Let $L \to M$ be a line bundle on $M$ and $\mathcal{F}_{\{p\}}$ be the sheaf of holomorphic sections of $L$ that vanish at $p$. Locally around a ...
• 63
64 views

### Qing Liu's Algebraic Geometry, Chapter 2, Exercise 2.13.

Part (a) of Exercise 2.13: Let $f:X\mapsto Y$ be a continuous map of topological spaces. Let $\mathcal{F}$ be a sheaf on $X$, and $\mathcal{G}$ be a sheaf on $Y$. (a) Show that there exist canonical ...
• 63
1 vote
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• 125
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### definition of closed immersion of schemes

The definition that I found on books for definition of closed immersion of schemes is the following: A closed immersion $i:Z\hookrightarrow X$ is a morphism which satifies: (1) The underlying ...
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### Proper direct image of sheaves along the inclusion of a locally closed subset $W$ is f.f. with essential image equal to sheaves with supp. $\subset W$

$\def\F{\mathcal{F}} \def\Supp{\operatorname{Supp}} \def\E{\mathcal{E}}$In B. Iversen, Cohomology of Sheaves, Section II.6, we find the definition of the proper direct image functor $h_!$ (aka direct ...
1 vote
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### Need Help Proving That These Functions are Natural Transformations

In MacLane's Sheaves in Geometry and Logic, it is stated the following. Now, consider a given presheaf $P$ and the sheaf $\Gamma \Lambda_P$ of sections of the bundle $\Lambda_P \rightarrow X$. For ...
• 315
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### $f^{-1}(\varphi)_{x} = \varphi_{f(x)}$ ( Stalk of the inverse image )?

It seems easy question but don't find rigorous argument about it until now. Let $f : X\to Y$ be a continuous map. Then we obtain a functor $f^{-1}$ from the category of presheaves on $Y$ to the ...
• 2,644
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### How to show that a sheaf is itself a sheaf of modules?

I am currently writing the proof for the following proposition: **Show that the sheaf of sections on a vector bundle $V$ over $X$ is a sheaf of modules over a sheaf of continuous function on $X$. ** I ...
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### For a scheme $Z$, let $z\in Z, t\in \mathscr O_Z(Z)$, what does $t(z)$ mean?

For a scheme $Z$, let $z\in Z, t\in \mathscr O_Z(Z)$, let $Z_t:=\{z\in Z:t(z)\not= 0\}$. The author claims $Z_t$ is open subset of $Z$. But what does $t(z)$ mean?
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1 vote
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• 61
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### Sections of an algebraic tensor product of sheaves

Let $\mathcal{O}_X$ be the sheaf of functions on a smooth affine variety $X$. Then, for every Zariski open set $U$ in $X$, there is an isomorphism: \mathcal{O}_X(U)\otimes_{\mathbb C}\...
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### Equivalence of the Definition of closed immersion and closed subscheme

In wedhorn and Görtz‘s algebraic geometry book Algebraic Geometry I:Schemes page 86 I found the following definition. (1)A closed subscheme of $X$ is given by a closed subset $Z\subset X$ and an ideal ...
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### Uniqueness of Harder-Narasimhan filtration for coherent sheafs

I am reading the following lecture notes https://people.math.harvard.edu/%7Elurie/205notes/Lecture20-HarderNarasimhan.pdf on the Harder-Narasimhan filtration. I am having trouble understanding the ...
1 vote
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