isomorphism between an ideal and its double dual
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Let $R=mathbb{Z}[sqrt{-14}]$, $I_1 = (3, 1 + sqrt{-14})$ and $I_2 = (3, 1 - sqrt{-14})$.
I need to check that the natural maps $I_1 longrightarrow I_1^{veevee}$ and $I_2 longrightarrow I_2^{veevee}$ are isomorphisms.
By natural map I mean the map that sends every element $a longrightarrow ev_a$. Where $ev_a$ is the evaluation map $ev_a: phi longrightarrow phi(a)$.
I have already proved that $I_1^{vee} cong I_2$ and $I_2^{vee} cong I_1$.
But I do not know how to check that these maps are isomorphisms.
abstract-algebra commutative-algebra modules ideals
asked Nov 16 at 16:16
idrisk
214
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up vote
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favorite
Let $R=mathbb{Z}[sqrt{-14}]$, $I_1 = (3, 1 + sqrt{-14})$ and $I_2 = (3, 1 - sqrt{-14})$.
I need to check that the natural maps $I_1 longrightarrow I_1^{veevee}$ and $I_2 longrightarrow I_2^{veevee}$ are isomorphisms.
By natural map I mean the map that sends every element $a longrightarrow ev_a$. Where $ev_a$ is the evaluation map $ev_a: phi longrightarrow phi(a)$.
I have already proved that $I_1^{vee} cong I_2$ and $I_2^{vee} cong I_1$.
But I do not know how to check that these maps are isomorphisms.
abstract-algebra commutative-algebra modules ideals
asked Nov 16 at 16:16
idrisk
214
To check isomorphism, it suffices to localize and this will become free module as those ideals are rank 1 projective line bundle over spec(R). This reduces to checking the localized morphism being canonical map ismorphism $Vto V^{starstar}$ of 1 generator. The result is trivial for PID which will be the case after localization.
– user45765
Nov 17 at 0:51
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $R=mathbb{Z}[sqrt{-14}]$, $I_1 = (3, 1 + sqrt{-14})$ and $I_2 = (3, 1 - sqrt{-14})$.
I need to check that the natural maps $I_1 longrightarrow I_1^{veevee}$ and $I_2 longrightarrow I_2^{veevee}$ are isomorphisms.
By natural map I mean the map that sends every element $a longrightarrow ev_a$. Where $ev_a$ is the evaluation map $ev_a: phi longrightarrow phi(a)$.
I have already proved that $I_1^{vee} cong I_2$ and $I_2^{vee} cong I_1$.
But I do not know how to check that these maps are isomorphisms.
abstract-algebra commutative-algebra modules ideals
asked Nov 16 at 16:16
idrisk
214
Let $R=mathbb{Z}[sqrt{-14}]$, $I_1 = (3, 1 + sqrt{-14})$ and $I_2 = (3, 1 - sqrt{-14})$.
I need to check that the natural maps $I_1 longrightarrow I_1^{veevee}$ and $I_2 longrightarrow I_2^{veevee}$ are isomorphisms.
By natural map I mean the map that sends every element $a longrightarrow ev_a$. Where $ev_a$ is the evaluation map $ev_a: phi longrightarrow phi(a)$.
I have already proved that $I_1^{vee} cong I_2$ and $I_2^{vee} cong I_1$.
But I do not know how to check that these maps are isomorphisms.
abstract-algebra commutative-algebra modules ideals
abstract-algebra commutative-algebra modules ideals
asked Nov 16 at 16:16
idrisk
214
asked Nov 16 at 16:16
idrisk
214
asked Nov 16 at 16:16
idrisk
214
asked Nov 16 at 16:16
idrisk
214
asked Nov 16 at 16:16
idrisk
214
214
To check isomorphism, it suffices to localize and this will become free module as those ideals are rank 1 projective line bundle over spec(R). This reduces to checking the localized morphism being canonical map ismorphism $Vto V^{starstar}$ of 1 generator. The result is trivial for PID which will be the case after localization.
– user45765
Nov 17 at 0:51
add a comment |
To check isomorphism, it suffices to localize and this will become free module as those ideals are rank 1 projective line bundle over spec(R). This reduces to checking the localized morphism being canonical map ismorphism $Vto V^{starstar}$ of 1 generator. The result is trivial for PID which will be the case after localization.
– user45765
Nov 17 at 0:51
To check isomorphism, it suffices to localize and this will become free module as those ideals are rank 1 projective line bundle over spec(R). This reduces to checking the localized morphism being canonical map ismorphism $Vto V^{starstar}$ of 1 generator. The result is trivial for PID which will be the case after localization.
– user45765
Nov 17 at 0:51
To check isomorphism, it suffices to localize and this will become free module as those ideals are rank 1 projective line bundle over spec(R). This reduces to checking the localized morphism being canonical map ismorphism $Vto V^{starstar}$ of 1 generator. The result is trivial for PID which will be the case after localization.
– user45765
Nov 17 at 0:51
add a comment |
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To check isomorphism, it suffices to localize and this will become free module as those ideals are rank 1 projective line bundle over spec(R). This reduces to checking the localized morphism being canonical map ismorphism $Vto V^{starstar}$ of 1 generator. The result is trivial for PID which will be the case after localization.
– user45765
Nov 17 at 0:51