When is it the case that all closed immersions of all irreducible components are flat?
$begingroup$
Let $X$ be a projective, reduced scheme of pure dimension one (irreducible components have dimension one) over some field $k$. Let $X_1,ldots,X_r$ be the irreducible components of $X$ and let $f_i: X_i to X$ denote the corresponding closed immersions.
I was wondering when (for what kind of schemes resp. for what kind of intersection behaviour of the components) the following happens: The closed immersions $f_i$ are all flat.
My background for this is: I have $mathcal{O}_X$-modules $mathcal{F}$ that are embedded into $mathcal{K}_X$, the sheaf of meromorphic functions on $X$, and want to restrict them to the $X_i$. I am trying to find reasonable assumptions on $mathcal{F}$ such that its restriction to the components is again embedded in $mathcal{K}_{X_i}$. Doing the local analysis I find that even ideal sheaves do not restrict to something embedded.
I am grateful for any kind of input or help.
algebraic-geometry sheaf-theory algebraic-curves
asked Dec 7 '18 at 8:38
windsheafwindsheaf
607312
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add a comment |
$begingroup$
Let $X$ be a projective, reduced scheme of pure dimension one (irreducible components have dimension one) over some field $k$. Let $X_1,ldots,X_r$ be the irreducible components of $X$ and let $f_i: X_i to X$ denote the corresponding closed immersions.
I was wondering when (for what kind of schemes resp. for what kind of intersection behaviour of the components) the following happens: The closed immersions $f_i$ are all flat.
My background for this is: I have $mathcal{O}_X$-modules $mathcal{F}$ that are embedded into $mathcal{K}_X$, the sheaf of meromorphic functions on $X$, and want to restrict them to the $X_i$. I am trying to find reasonable assumptions on $mathcal{F}$ such that its restriction to the components is again embedded in $mathcal{K}_{X_i}$. Doing the local analysis I find that even ideal sheaves do not restrict to something embedded.
I am grateful for any kind of input or help.
algebraic-geometry sheaf-theory algebraic-curves
asked Dec 7 '18 at 8:38
windsheafwindsheaf
607312
$endgroup$
add a comment |
$begingroup$
Let $X$ be a projective, reduced scheme of pure dimension one (irreducible components have dimension one) over some field $k$. Let $X_1,ldots,X_r$ be the irreducible components of $X$ and let $f_i: X_i to X$ denote the corresponding closed immersions.
I was wondering when (for what kind of schemes resp. for what kind of intersection behaviour of the components) the following happens: The closed immersions $f_i$ are all flat.
My background for this is: I have $mathcal{O}_X$-modules $mathcal{F}$ that are embedded into $mathcal{K}_X$, the sheaf of meromorphic functions on $X$, and want to restrict them to the $X_i$. I am trying to find reasonable assumptions on $mathcal{F}$ such that its restriction to the components is again embedded in $mathcal{K}_{X_i}$. Doing the local analysis I find that even ideal sheaves do not restrict to something embedded.
I am grateful for any kind of input or help.
algebraic-geometry sheaf-theory algebraic-curves
asked Dec 7 '18 at 8:38
windsheafwindsheaf
607312
$endgroup$
Let $X$ be a projective, reduced scheme of pure dimension one (irreducible components have dimension one) over some field $k$. Let $X_1,ldots,X_r$ be the irreducible components of $X$ and let $f_i: X_i to X$ denote the corresponding closed immersions.
I was wondering when (for what kind of schemes resp. for what kind of intersection behaviour of the components) the following happens: The closed immersions $f_i$ are all flat.
My background for this is: I have $mathcal{O}_X$-modules $mathcal{F}$ that are embedded into $mathcal{K}_X$, the sheaf of meromorphic functions on $X$, and want to restrict them to the $X_i$. I am trying to find reasonable assumptions on $mathcal{F}$ such that its restriction to the components is again embedded in $mathcal{K}_{X_i}$. Doing the local analysis I find that even ideal sheaves do not restrict to something embedded.
I am grateful for any kind of input or help.
algebraic-geometry sheaf-theory algebraic-curves
algebraic-geometry sheaf-theory algebraic-curves
asked Dec 7 '18 at 8:38
windsheafwindsheaf
607312
asked Dec 7 '18 at 8:38
windsheafwindsheaf
607312
asked Dec 7 '18 at 8:38
windsheafwindsheaf
607312
asked Dec 7 '18 at 8:38
windsheafwindsheaf
607312
asked Dec 7 '18 at 8:38
windsheafwindsheaf
607312
607312
add a comment |
add a comment |
1 Answer
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A flat map locally of finite presentation is open, so asking for a closed immersion (which is certainly locally of finite presentation) to also be flat means that it should be both a closed and open map. This only occurs when the irreducible component is also a connected component. So your scheme must be a disjoint union of it's irreducible components.
answered Dec 7 '18 at 9:06
KReiserKReiser
9,64221435
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$begingroup$
I already suspected that. Thank you for your answer.
$endgroup$
– windsheaf
Dec 7 '18 at 9:14
add a comment |
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1 Answer
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1 Answer
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active
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$begingroup$
A flat map locally of finite presentation is open, so asking for a closed immersion (which is certainly locally of finite presentation) to also be flat means that it should be both a closed and open map. This only occurs when the irreducible component is also a connected component. So your scheme must be a disjoint union of it's irreducible components.
answered Dec 7 '18 at 9:06
KReiserKReiser
9,64221435
$endgroup$
$begingroup$
I already suspected that. Thank you for your answer.
$endgroup$
– windsheaf
Dec 7 '18 at 9:14
add a comment |
$begingroup$
A flat map locally of finite presentation is open, so asking for a closed immersion (which is certainly locally of finite presentation) to also be flat means that it should be both a closed and open map. This only occurs when the irreducible component is also a connected component. So your scheme must be a disjoint union of it's irreducible components.
answered Dec 7 '18 at 9:06
KReiserKReiser
9,64221435
$endgroup$
$begingroup$
I already suspected that. Thank you for your answer.
$endgroup$
– windsheaf
Dec 7 '18 at 9:14
add a comment |
$begingroup$
A flat map locally of finite presentation is open, so asking for a closed immersion (which is certainly locally of finite presentation) to also be flat means that it should be both a closed and open map. This only occurs when the irreducible component is also a connected component. So your scheme must be a disjoint union of it's irreducible components.
answered Dec 7 '18 at 9:06
KReiserKReiser
9,64221435
$endgroup$
A flat map locally of finite presentation is open, so asking for a closed immersion (which is certainly locally of finite presentation) to also be flat means that it should be both a closed and open map. This only occurs when the irreducible component is also a connected component. So your scheme must be a disjoint union of it's irreducible components.
answered Dec 7 '18 at 9:06
KReiserKReiser
9,64221435
answered Dec 7 '18 at 9:06
KReiserKReiser
9,64221435
answered Dec 7 '18 at 9:06
KReiserKReiser
9,64221435
answered Dec 7 '18 at 9:06
KReiserKReiser
9,64221435
9,64221435
$begingroup$
I already suspected that. Thank you for your answer.
$endgroup$
– windsheaf
Dec 7 '18 at 9:14
add a comment |
$begingroup$
I already suspected that. Thank you for your answer.
$endgroup$
– windsheaf
Dec 7 '18 at 9:14
$begingroup$
I already suspected that. Thank you for your answer.
$endgroup$
– windsheaf
Dec 7 '18 at 9:14
$begingroup$
I already suspected that. Thank you for your answer.
$endgroup$
– windsheaf
Dec 7 '18 at 9:14
add a comment |
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