Mixing
Primality in Analytic Ideal Sheaves
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Let $I subset mathscrO$ be an ideal sheaf, $mathscrO$ the algebraic structure sheaf of $X$. If I want to show $I(U) subset mathscrO(U)$ is a prime ideal one strategy would be:
1) Verify $Spec(mathscrO(U)) = U$ is connected;
2) Verify $I_x subset mathscrO_x$ is a prime ideal for all $x in U$.
2) checks that my ideal is prime at all its maximal ideals; 1) verifies the spectrum of my ring is connected and so primality is a local condition.
All of the above depended (to my mind) on taking the algebraic structure sheaf $mathscrO$. My question is: Is it possible to argue in the same/similar way if I take my structure sheaf to be analytic instead? In principal it seems that even if I could reduce primality to a question of what's happening at maximal ideals just checking at maximal ideals $m_x$ would cause me to miss some of the more exotic maximal ideals in $mathscrO(U)$?
Any guidance or references would be appreciated.
algebraic-geometry analytic-geometry
asked Jul 16 at 3:05
do_math
365
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up vote
0
down vote
favorite
Let $I subset mathscrO$ be an ideal sheaf, $mathscrO$ the algebraic structure sheaf of $X$. If I want to show $I(U) subset mathscrO(U)$ is a prime ideal one strategy would be:
1) Verify $Spec(mathscrO(U)) = U$ is connected;
2) Verify $I_x subset mathscrO_x$ is a prime ideal for all $x in U$.
2) checks that my ideal is prime at all its maximal ideals; 1) verifies the spectrum of my ring is connected and so primality is a local condition.
All of the above depended (to my mind) on taking the algebraic structure sheaf $mathscrO$. My question is: Is it possible to argue in the same/similar way if I take my structure sheaf to be analytic instead? In principal it seems that even if I could reduce primality to a question of what's happening at maximal ideals just checking at maximal ideals $m_x$ would cause me to miss some of the more exotic maximal ideals in $mathscrO(U)$?
Any guidance or references would be appreciated.
algebraic-geometry analytic-geometry
asked Jul 16 at 3:05
do_math
365
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $I subset mathscrO$ be an ideal sheaf, $mathscrO$ the algebraic structure sheaf of $X$. If I want to show $I(U) subset mathscrO(U)$ is a prime ideal one strategy would be:
1) Verify $Spec(mathscrO(U)) = U$ is connected;
2) Verify $I_x subset mathscrO_x$ is a prime ideal for all $x in U$.
2) checks that my ideal is prime at all its maximal ideals; 1) verifies the spectrum of my ring is connected and so primality is a local condition.
All of the above depended (to my mind) on taking the algebraic structure sheaf $mathscrO$. My question is: Is it possible to argue in the same/similar way if I take my structure sheaf to be analytic instead? In principal it seems that even if I could reduce primality to a question of what's happening at maximal ideals just checking at maximal ideals $m_x$ would cause me to miss some of the more exotic maximal ideals in $mathscrO(U)$?
Any guidance or references would be appreciated.
algebraic-geometry analytic-geometry
asked Jul 16 at 3:05
do_math
365
Let $I subset mathscrO$ be an ideal sheaf, $mathscrO$ the algebraic structure sheaf of $X$. If I want to show $I(U) subset mathscrO(U)$ is a prime ideal one strategy would be:
1) Verify $Spec(mathscrO(U)) = U$ is connected;
2) Verify $I_x subset mathscrO_x$ is a prime ideal for all $x in U$.
2) checks that my ideal is prime at all its maximal ideals; 1) verifies the spectrum of my ring is connected and so primality is a local condition.
All of the above depended (to my mind) on taking the algebraic structure sheaf $mathscrO$. My question is: Is it possible to argue in the same/similar way if I take my structure sheaf to be analytic instead? In principal it seems that even if I could reduce primality to a question of what's happening at maximal ideals just checking at maximal ideals $m_x$ would cause me to miss some of the more exotic maximal ideals in $mathscrO(U)$?
Any guidance or references would be appreciated.
algebraic-geometry analytic-geometry
asked Jul 16 at 3:05
do_math
365
asked Jul 16 at 3:05
do_math
365
asked Jul 16 at 3:05
do_math
365
asked Jul 16 at 3:05
do_math
365
365
add a comment |Â
add a comment |Â
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