Sections of a locally free sheaf The Next CEO of Stack OverflowSections of locally free sheavesDescribing a locally free sheaf sitting between two locally free sheaves which are given as extensionsQuotient of locally free sheaf is locally free?Sheaf of sections vanishing at a point is $Gamma(E) otimes I_p$Sheaf of sections of vector bundle over a manifold is an $mathcal O_M$-moduleRegular sections of an invertible sheafTotal space of a finite rank locally free sheaf, Vakil's 17.1.4 & 17.1.GVector Bundle Locally Free SheafSection of pullback bundle isomorphic to the sheaf pullback of sectionsSheaf Hom between locally free sheaves
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Sections of a locally free sheaf
The Next CEO of Stack OverflowSections of locally free sheavesDescribing a locally free sheaf sitting between two locally free sheaves which are given as extensionsQuotient of locally free sheaf is locally free?Sheaf of sections vanishing at a point is $Gamma(E) otimes I_p$Sheaf of sections of vector bundle over a manifold is an $mathcal O_M$-moduleRegular sections of an invertible sheafTotal space of a finite rank locally free sheaf, Vakil's 17.1.4 & 17.1.GVector Bundle Locally Free SheafSection of pullback bundle isomorphic to the sheaf pullback of sectionsSheaf Hom between locally free sheaves
$begingroup$
Let $mathcalF$ be a locally free sheaf of rank $n$ on a scheme $X$. We know that we can associate to it a vector bundle $F$ on $X$ such that $F_x simeq mathcalF(x)$, where with $mathcalF(x)$ I mean the fibre of the sheaf over the point $x$. We also know that the sheaf of sections of the vector bundle $F$ is isomorphic to the locally free sheaf we started with.
My question is the following: a section of $F$ is by definition a regular map $sigma : X rightarrow F$ such that $sigma(x) in F_x$ for any $x in X$, whereas a section of $mathcalF$ can be interpreted as a regular map $psi : X rightarrow sqcup_x in X mathcalF_x$. I can see a natural map from “sections of $mathcalF$” to “sections of $F$”, but how can one claim this is an isomorphism? Am I mixing something up?
sheaf-theory vector-bundles schemes
asked yesterday
FedericoFederico
918313
$endgroup$
add a comment |
$begingroup$
Let $mathcalF$ be a locally free sheaf of rank $n$ on a scheme $X$. We know that we can associate to it a vector bundle $F$ on $X$ such that $F_x simeq mathcalF(x)$, where with $mathcalF(x)$ I mean the fibre of the sheaf over the point $x$. We also know that the sheaf of sections of the vector bundle $F$ is isomorphic to the locally free sheaf we started with.
My question is the following: a section of $F$ is by definition a regular map $sigma : X rightarrow F$ such that $sigma(x) in F_x$ for any $x in X$, whereas a section of $mathcalF$ can be interpreted as a regular map $psi : X rightarrow sqcup_x in X mathcalF_x$. I can see a natural map from “sections of $mathcalF$” to “sections of $F$”, but how can one claim this is an isomorphism? Am I mixing something up?
sheaf-theory vector-bundles schemes
asked yesterday
FedericoFederico
918313
$endgroup$
add a comment |
$begingroup$
Let $mathcalF$ be a locally free sheaf of rank $n$ on a scheme $X$. We know that we can associate to it a vector bundle $F$ on $X$ such that $F_x simeq mathcalF(x)$, where with $mathcalF(x)$ I mean the fibre of the sheaf over the point $x$. We also know that the sheaf of sections of the vector bundle $F$ is isomorphic to the locally free sheaf we started with.
My question is the following: a section of $F$ is by definition a regular map $sigma : X rightarrow F$ such that $sigma(x) in F_x$ for any $x in X$, whereas a section of $mathcalF$ can be interpreted as a regular map $psi : X rightarrow sqcup_x in X mathcalF_x$. I can see a natural map from “sections of $mathcalF$” to “sections of $F$”, but how can one claim this is an isomorphism? Am I mixing something up?
sheaf-theory vector-bundles schemes
asked yesterday
FedericoFederico
918313
$endgroup$
Let $mathcalF$ be a locally free sheaf of rank $n$ on a scheme $X$. We know that we can associate to it a vector bundle $F$ on $X$ such that $F_x simeq mathcalF(x)$, where with $mathcalF(x)$ I mean the fibre of the sheaf over the point $x$. We also know that the sheaf of sections of the vector bundle $F$ is isomorphic to the locally free sheaf we started with.
My question is the following: a section of $F$ is by definition a regular map $sigma : X rightarrow F$ such that $sigma(x) in F_x$ for any $x in X$, whereas a section of $mathcalF$ can be interpreted as a regular map $psi : X rightarrow sqcup_x in X mathcalF_x$. I can see a natural map from “sections of $mathcalF$” to “sections of $F$”, but how can one claim this is an isomorphism? Am I mixing something up?
sheaf-theory vector-bundles schemes
sheaf-theory vector-bundles schemes
asked yesterday
FedericoFederico
918313
asked yesterday
FedericoFederico
918313
asked yesterday
FedericoFederico
918313
asked yesterday
FedericoFederico
918313
asked yesterday
FedericoFederico
918313
918313
add a comment |
add a comment |
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