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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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 $0\rightarrow \mathcal{O}_X(n)\rightarrow \mathcal{E}\rightarrow\mathcal{I}_Z(m)\rightarrow 0$,…
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Sheaves: pretopology versus comma pretopology

I'm reading about sheaves on sites and I have a question about a particular example in these notes: http://www.math.harvard.edu/~nasko/documents/stacks.pdf http://homepage.sns.it/vistoli/descent.pdf Let $\mathcal{C}$ be a small category with a…
John M
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Section of pre-sheaf.

The notation is from O. Forster, Lectures on Riemann Surfaces, Springer-Verlag, 1981. Let $\mathcal{F}$ be a presheaf on the topological space $X$. Let $U\subset X$. Let $s:U\rightarrow |\mathcal{F}|$ be a section of $p$ over $U$. Show that $s$ is…
user44322
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Linear systems and rational maps

I'm following Beauville's book on Complex Algebraic Surfaces. If $D$ is a divisor on a surface $S$, we write $|D|$ for the set of all effective divisors linear equivalent to $D$ and we call it a complete linear system on a surface $S$. At page 14,…
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About the sheafification

Look at the following definition of the shefification (the source is the Stacks Project): I don't understand what is the projection $\prod_{u\in U}\mathcal F_u\longrightarrow\prod_{v\in V}\mathcal F_v$. In particular, if $u\in U\setminus V$ what…
Dubious
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In a quasicompact scheme every nonreduced point has a closed nonreduced point in its closure

Let $x\in X$ be a non reduced point. Then I can find a nilpotent $f\in \Gamma(U, \mathcal O_X)$, where $U$ is some open neighborhood of $x$. I also know that when $X$ is assumed to be quasicompact, every point has a closed point in its closure. Now,…
Rodrigo
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Associated sheaf to a $\mathbb{C}$-module

In section II.5 Hartshorne describes the notion of a sheaf $\tilde{M}$ associated to a module M over a ring A. Now in the case $A:= \mathbb{C}$, we have $\operatorname{Spec}(\mathbb{C}) = {0}$. Unless I'm mistaken (very probable) since there is…
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$\operatorname{Hom}(f^*F, G) = \operatorname{Hom}(F, f_*G)$

For topological spaces $X, Y$ and the continuous morphism $f \colon Y \to X$, consider the sheaf $F$ on $X$ and $G$ on $Y$. A very famous formula says that ${\mathrm{Hom}_{\mathrm{sh}}}(f^*F,G) = {\mathrm{Hom}_{\mathrm{sh}}}(F,f_*G) \cdots (*)$,…
Pierre
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What is $\mathcal{O}_{X,x}$?

I read from Liu's "Algebraic Geometry and Arithmetic Curves the following definition: A ringed topological space consists of a topological space $X$ endowed with a sheaf of rings $\mathcal{O}_X$ on $X$ such that $\mathcal{O}_{X,x}$ is a local ring…
studying
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Orientability of Ringed Space

Differential manifold can be defined in two ways. One definition is a topological space equipped with an atlas and transition maps. Another definition is a topological space equipped with a sheaf of functions. Using first definition orientability of…
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Characterization of locally free sheaf of rank 1 (invertible sheaf)

Is it true that Any locally free sheaf of rank 1 over X is isomorphic to $\mathcal O_X (n)$ for some $n\in \mathbb N$
Babai
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Isomorphism in derived category of coherent sheaves

this question might seem a bit special, but it came up at a crucial point of a proof I read and so I would be very obliged if someone could explain this to me: given is a smooth projective variety $X$ with canonical sheaf $\omega_X$ and a closed…
Descartes
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Counterexample to $\operatorname{Supp} \mathcal G \subset Z \hookrightarrow X$ but $\mathcal G\cong i_*i^{-1}\mathcal G$ when $Z$ is not closed

Let $\mathcal G$ be a sheaf on a topological space and $X$ (say a sheaf of sets) and suppose its support is contained in a subset $Z$ of $X$, i.e. $\operatorname{Supp} \mathcal G \subset Z \hookrightarrow X$. Let $i$ denote this inclusion, $i_*$ the…
Rodrigo
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Explicitly showing cokernel of exponential sequence is not a sheaf

In the classical example of short exact sequence of presheaves $$0\rightarrow \mathbb Z \rightarrow \mathcal O_X \rightarrow \mathcal F \rightarrow 1,$$ it is well known that $\mathcal F$ is not a sheaf because there are functions that admit a…
Rodrigo
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Example of sheaf hom not commuting with stalk

I came up with the following example and I wanted to know if it is correct. Let $\mathcal F$ and $\mathcal G$ be sheaves of smooth functions on $\mathbb R$. Consider the smooth function $f$ constructed in the following manner (I think the heart of…
Rodrigo
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