Quasicoherent sheaves on the groupoid of vector bundles on a surface
$begingroup$
Consider the groupoid $Vect_n(S)$ of rank n-vector bundles over a projective surface $S$. What does it mean to have a sheaf $$mathcal Lin QCoh(Vect_n(S))?$$ A notion of quasicoherent sheaf on a groupoid should be involved, but I guess that the groupoid should have "some topology".
Sorry for the vagueness of the question, but I have been spoken about this and I realise I don't quite understand the notion.
Thank you in advance.
algebraic-geometry vector-bundles quasicoherent-sheaves
asked Dec 9 '18 at 15:42
W. RetherW. Rether
738417
$endgroup$
add a comment |
$begingroup$
Consider the groupoid $Vect_n(S)$ of rank n-vector bundles over a projective surface $S$. What does it mean to have a sheaf $$mathcal Lin QCoh(Vect_n(S))?$$ A notion of quasicoherent sheaf on a groupoid should be involved, but I guess that the groupoid should have "some topology".
Sorry for the vagueness of the question, but I have been spoken about this and I realise I don't quite understand the notion.
Thank you in advance.
algebraic-geometry vector-bundles quasicoherent-sheaves
asked Dec 9 '18 at 15:42
W. RetherW. Rether
738417
$endgroup$
$begingroup$
Do you know what is a stack ?
$endgroup$
– Nicolas Hemelsoet
Dec 9 '18 at 21:22
$begingroup$
Yes. Does this follow from the notion of sheaf on a stack? And if so, how?
$endgroup$
– W. Rether
Dec 9 '18 at 21:33
1
$begingroup$
Qiaochu answered for me : $Vect_n(S)$ is a stack and there is a notion of sheaf on a stack which is what is meant here.
$endgroup$
– Nicolas Hemelsoet
Dec 9 '18 at 21:50
add a comment |
$begingroup$
Consider the groupoid $Vect_n(S)$ of rank n-vector bundles over a projective surface $S$. What does it mean to have a sheaf $$mathcal Lin QCoh(Vect_n(S))?$$ A notion of quasicoherent sheaf on a groupoid should be involved, but I guess that the groupoid should have "some topology".
Sorry for the vagueness of the question, but I have been spoken about this and I realise I don't quite understand the notion.
Thank you in advance.
algebraic-geometry vector-bundles quasicoherent-sheaves
asked Dec 9 '18 at 15:42
W. RetherW. Rether
738417
$endgroup$
Consider the groupoid $Vect_n(S)$ of rank n-vector bundles over a projective surface $S$. What does it mean to have a sheaf $$mathcal Lin QCoh(Vect_n(S))?$$ A notion of quasicoherent sheaf on a groupoid should be involved, but I guess that the groupoid should have "some topology".
Sorry for the vagueness of the question, but I have been spoken about this and I realise I don't quite understand the notion.
Thank you in advance.
algebraic-geometry vector-bundles quasicoherent-sheaves
algebraic-geometry vector-bundles quasicoherent-sheaves
asked Dec 9 '18 at 15:42
W. RetherW. Rether
738417
asked Dec 9 '18 at 15:42
W. RetherW. Rether
738417
asked Dec 9 '18 at 15:42
W. RetherW. Rether
738417
asked Dec 9 '18 at 15:42
W. RetherW. Rether
738417
asked Dec 9 '18 at 15:42
W. RetherW. Rether
738417
738417
$begingroup$
Do you know what is a stack ?
$endgroup$
– Nicolas Hemelsoet
Dec 9 '18 at 21:22
$begingroup$
Yes. Does this follow from the notion of sheaf on a stack? And if so, how?
$endgroup$
– W. Rether
Dec 9 '18 at 21:33
1
$begingroup$
Qiaochu answered for me : $Vect_n(S)$ is a stack and there is a notion of sheaf on a stack which is what is meant here.
$endgroup$
– Nicolas Hemelsoet
Dec 9 '18 at 21:50
add a comment |
$begingroup$
Do you know what is a stack ?
$endgroup$
– Nicolas Hemelsoet
Dec 9 '18 at 21:22
$begingroup$
Yes. Does this follow from the notion of sheaf on a stack? And if so, how?
$endgroup$
– W. Rether
Dec 9 '18 at 21:33
1
$begingroup$
Qiaochu answered for me : $Vect_n(S)$ is a stack and there is a notion of sheaf on a stack which is what is meant here.
$endgroup$
– Nicolas Hemelsoet
Dec 9 '18 at 21:50
$begingroup$
Do you know what is a stack ?
$endgroup$
– Nicolas Hemelsoet
Dec 9 '18 at 21:22
$begingroup$
Do you know what is a stack ?
$endgroup$
– Nicolas Hemelsoet
Dec 9 '18 at 21:22
$begingroup$
Yes. Does this follow from the notion of sheaf on a stack? And if so, how?
$endgroup$
– W. Rether
Dec 9 '18 at 21:33
$begingroup$
Yes. Does this follow from the notion of sheaf on a stack? And if so, how?
$endgroup$
– W. Rether
Dec 9 '18 at 21:33
1
1
$begingroup$
Qiaochu answered for me : $Vect_n(S)$ is a stack and there is a notion of sheaf on a stack which is what is meant here.
$endgroup$
– Nicolas Hemelsoet
Dec 9 '18 at 21:50
$begingroup$
Qiaochu answered for me : $Vect_n(S)$ is a stack and there is a notion of sheaf on a stack which is what is meant here.
$endgroup$
– Nicolas Hemelsoet
Dec 9 '18 at 21:50
add a comment |
1 Answer
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$begingroup$
$text{Vect}_n(S)$ is a stack, not a groupoid. Its functor of points sends a commutative ring $R$ to the groupoid of rank $n$ vector bundles on $S times text{Spec } R$. $L$ is a quasicoherent sheaf on this stack.
answered Dec 9 '18 at 21:36
Qiaochu YuanQiaochu Yuan
279k32588932
$endgroup$
add a comment |
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1 Answer
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1 Answer
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$begingroup$
$text{Vect}_n(S)$ is a stack, not a groupoid. Its functor of points sends a commutative ring $R$ to the groupoid of rank $n$ vector bundles on $S times text{Spec } R$. $L$ is a quasicoherent sheaf on this stack.
answered Dec 9 '18 at 21:36
Qiaochu YuanQiaochu Yuan
279k32588932
$endgroup$
add a comment |
$begingroup$
$text{Vect}_n(S)$ is a stack, not a groupoid. Its functor of points sends a commutative ring $R$ to the groupoid of rank $n$ vector bundles on $S times text{Spec } R$. $L$ is a quasicoherent sheaf on this stack.
answered Dec 9 '18 at 21:36
Qiaochu YuanQiaochu Yuan
279k32588932
$endgroup$
add a comment |
$begingroup$
$text{Vect}_n(S)$ is a stack, not a groupoid. Its functor of points sends a commutative ring $R$ to the groupoid of rank $n$ vector bundles on $S times text{Spec } R$. $L$ is a quasicoherent sheaf on this stack.
answered Dec 9 '18 at 21:36
Qiaochu YuanQiaochu Yuan
279k32588932
$endgroup$
$text{Vect}_n(S)$ is a stack, not a groupoid. Its functor of points sends a commutative ring $R$ to the groupoid of rank $n$ vector bundles on $S times text{Spec } R$. $L$ is a quasicoherent sheaf on this stack.
answered Dec 9 '18 at 21:36
Qiaochu YuanQiaochu Yuan
279k32588932
answered Dec 9 '18 at 21:36
Qiaochu YuanQiaochu Yuan
279k32588932
answered Dec 9 '18 at 21:36
Qiaochu YuanQiaochu Yuan
279k32588932
answered Dec 9 '18 at 21:36
Qiaochu YuanQiaochu Yuan
279k32588932
279k32588932
add a comment |
add a comment |
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$begingroup$
Do you know what is a stack ?
$endgroup$
– Nicolas Hemelsoet
Dec 9 '18 at 21:22
$begingroup$
Yes. Does this follow from the notion of sheaf on a stack? And if so, how?
$endgroup$
– W. Rether
Dec 9 '18 at 21:33
1
$begingroup$
Qiaochu answered for me : $Vect_n(S)$ is a stack and there is a notion of sheaf on a stack which is what is meant here.
$endgroup$
– Nicolas Hemelsoet
Dec 9 '18 at 21:50