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.

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 $\mathcal F(V)$, $V\subseteq U$ is obtained by restricting the domain and not all elements of $\mathcal F(U)$ can be obtained by restricting a global section $\in\mathcal F(X)$.

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Given a sheaf $\mathcal{N}$ of $\mathcal{O}_Y$-modules defined by a finitely generated $\mathbb{C}[x]$-module, what is $f^*\mathcal{N}$?

Say $f : \mathbb{A}^2 \rightarrow \mathbb{A}$ is projection on the first coordinate. Given a sheaf $\mathcal{N}$ of $\mathcal{O}_Y$-modules defined by a finitely generated $\mathbb{C}[x]$-module, what is $f^*\mathcal{N}$? Does it come from a…
user99716
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Equivalent definitions of the sheafification of a presheaf

According to Hartshorne's book a sheafification is: I saw a different definition elsewhere Definition 2 Given a presheaf $\mathcal F$ in a topological space $X$ we can associate a $\mathcal F$ a sheaf called $\mathcal F^+$: $\mathcal F^+(U):=\{t\in…
user42912
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$K$-Category $M(0, 0) = M(A, 0) = M(0, A)$ using definition from Swan's 'Sheaf Theory'

I'm using the following definition: A category $\mathcal C$ is given by the following: A collection of objects $A$ A set $M(A, B)$ for any two objects $A, B \in \mathcal C$. A function $M(B, C) \times M(A, B) \to M(A, C)$ for each triple of objects…
Robert Cardona
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If $\varphi$ is an isomorphism, so is $\varphi_p$.

I'm trying to prove this statement in Hartshorne's book: MY ATTEMPT Injectivity part If $\varphi$ is an isomorphism, then $\varphi_U$ is an isomorphism of abelian groups for every open subset of $U$ So for $s\in F(U)$ and $t\in F(V)$, with $p\in…
user42912
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Ringed space: interpretation of quotient.

Let $(X,O_X)$ a locally ringed space, I'd like to define the tangent space $T_x$ for $x \in X$. We can consider the local ring (stalk) $R_x$ in $x$ with maximal ideal $m_x$ and from invertibility we have that $k_x:= R_x/m_x$ is a field. How can I…
ArthurStuart
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Questions about a natural map: $f^{-1}f_{*}\mathcal{F} \to \mathcal{F}$.

By definition $$f^{-1}f_{*}\mathcal{F}(U)=\lim_{\substack{\rightarrow\\V \supseteq f(U)}} \mathcal{F}(f^{-1}(V)).$$ If $V \supseteq f(U)$, then $U \subseteq f^{-1}(V)$. Since $\mathcal{F}$ is a sheaf, we have the restriction map:…
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supple, non flabby sheaf

Can anyone give an example of a sheaf that is supple, but not flabby? Consider sheafs $\mathcal{F}$ of Abelian groups over $X$. it is flabby if for any $U$ open subset of $X$, the restriction $\rho_{V,U} : \mathcal{F}(U)\to\mathcal{F}(V)$ is…
Yul Otani
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A question of isomorphism of sheaves

How do I verify the following: If $F$ is a sheaf of abelian groups on $X$, then show that $\operatorname{Hom}_{AbX}(Z,F)$ is isomorphic to $F$ (an isomorphism of sheaves of abelian groups).
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Exactness of direct image functor

Let $f:X\to Y$ be a morphism of schemes. Then there exists a functor $f_*:{Sh}X\to {Sh}Y$ with $f_*\mathcal{F}(U)=\mathcal{f^{-1}(U)}$ whenever $\mathcal{F}$ is asheaf on $X$. It is proved that the direct image functor is a left adjoint functor.…
user38585
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A question of sheaf

Let $\mathscr F$ and $\mathscr G$ be sheaves. $\mathscr{H}\hspace{-3pt}om$ does not commute with taking stalks. More precisely: It is not true in general that $\mathscr{H}\hspace{-3pt}om(\mathscr F, \mathscr G)_p$ is isomorphic to…
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Study of category and functor

I am going to start study of Category and Functors and sheaf theory. What are good text/ lecture notes to start with? I have not done any prior course on category theory and homological algebra or Algebraic topology. But I had done courses in…
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what is the difference between sections and germs in a sheaf?

As abovementioned, what is the difference between sections and germs in a sheaf? By definition, it seems that section is same as germ.
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What is the difference between $\mathbb C_X$-modules and sheaves of $\mathbb C$-vector spaces?

In the literature I have encountered the notions "category $\mathbb C_X-Mod$ of $\mathbb C_X$-modules" and "category $Sh_{\mathbb C}(X)$ of sheaves of $\mathbb C$-vector spaces". Does anyone know if both names and designations describe the same…
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Factorization of a morphism of LRS

The questions, in particular, are the two written in italics, but any other correction is welcome. Thank you in advance. Let $(f,f^\#):(Y,\mathcal O_{Y})\to (X,\mathcal O_{X})$ be a morphism of LRS, so $f^\#:\mathcal O_X\to f_*(\mathcal O_{Y})$ is…
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Closed immersions of schemes vs closed immersions of LRS

In my course (that vaguely follows Liu's Algebraic Geometry) the definition of closed (resp. open) immersion $f:Y\to X$, is that $f$ induces a homeomorphism onto $f(Y)$, $f(Y)$ is closed (resp. open) in $X$, and ($*$) $i^\#_y:\mathcal…
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