Does there exist an ideal sheaf $mathcal F$ on some affine scheme $X$ such that $mathcal F$ is not quasi-coherent?

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Please give an example such that $X$ is an affine scheme, and there exists an ideal sheaf $mathcal F$ on $X$ such that $mathcal F$ is not quasi-coherent.

algebraic-geometry sheaf-theory affine-schemes

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edited Jul 31 at 8:42

asked Jul 31 at 8:37

Born to be proud

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Please give an example such that $X$ is an affine scheme, and there exists an ideal sheaf $mathcal F$ on $X$ such that $mathcal F$ is not quasi-coherent.

algebraic-geometry sheaf-theory affine-schemes

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edited Jul 31 at 8:42

asked Jul 31 at 8:37

Born to be proud

41429

add a comment | 

up vote
1
down vote

favorite

up vote
1
down vote

favorite

Please give an example such that $X$ is an affine scheme, and there exists an ideal sheaf $mathcal F$ on $X$ such that $mathcal F$ is not quasi-coherent.

algebraic-geometry sheaf-theory affine-schemes

share|cite|improve this question

edited Jul 31 at 8:42

asked Jul 31 at 8:37

Born to be proud

41429

Please give an example such that $X$ is an affine scheme, and there exists an ideal sheaf $mathcal F$ on $X$ such that $mathcal F$ is not quasi-coherent.

algebraic-geometry sheaf-theory affine-schemes

share|cite|improve this question

edited Jul 31 at 8:42

asked Jul 31 at 8:37

Born to be proud

41429

share|cite|improve this question

share|cite|improve this question

edited Jul 31 at 8:42

edited Jul 31 at 8:42

edited Jul 31 at 8:42

asked Jul 31 at 8:37

Born to be proud

41429

asked Jul 31 at 8:37

Born to be proud

41429

asked Jul 31 at 8:37

Born to be proud

41429

41429

add a comment | 

add a comment | 

2 Answers
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1
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It often helps to translate the problem into commutative algebra.
I will define ideal sheaves on a scheme $X$ to be an $mathcalO_X$ module $mathcalI$ such that for all open sets $Usubset X$, $mathcalI(U)$ is an ideal of $mathcalO_X(U)$. So let $mathcalI$ be an ideal sheaf on $X=operatornameSpecA$. Recall that quasicoherence can be checked by showing that the natural map $Gamma(operatornameSpecA,mathcalI)_frightarrow Gamma(operatornameSpecA_f,mathcalI)$ is an isomorphism for all $f$. We clearly have $Gamma(operatornameSpecA,mathcalI)_f=I_f$ where $I$ is some ideal of $A$. Also $Gamma(operatornameSpecA_f,mathcalI)=J$ where $J$ is some ideal of $A_f$. Can you find an ideal $J$ of $A_f$ that doesn't come from localization of an ideal $I$ of $A$?

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answered Jul 31 at 17:09

Samir Canning

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Now that I understood the problem with my (deleted) answer, I try to write an easy example of an ideal sheaf that it is not quasicoherent:

Let $R$ be a discrete valuation ring, with field of fractions $K$, and let $X=operatorname Spec(R)=x,eta$ be the associated affine scheme with generic point $eta$ and closed point $x$. Consider now the ideal sheaf $mathcal Isubset mathcal O_X$ defined by $mathcal I(X)=0$ and $mathcal I(eta)=K$.

This ideal sheaf is not quasicoherent.

For example, you can see that the "associated closed subscheme", in case it were quasicoherent, would correspond to the set $eta$, which it is not closed.

Or you can follow Samir Canning suggestion and localize at the uniformizer $pi$ of $R$. Then $R_pi=K$, $Gamma(operatornameSpec(R_pi),mathcal I)=K$, while $Gamma(operatornameSpec(R),mathcal I)_pi=0$.

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answered Jul 31 at 20:36

xarles

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2 Answers
2

active

oldest

votes

2 Answers
2

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
1
down vote

It often helps to translate the problem into commutative algebra.
I will define ideal sheaves on a scheme $X$ to be an $mathcalO_X$ module $mathcalI$ such that for all open sets $Usubset X$, $mathcalI(U)$ is an ideal of $mathcalO_X(U)$. So let $mathcalI$ be an ideal sheaf on $X=operatornameSpecA$. Recall that quasicoherence can be checked by showing that the natural map $Gamma(operatornameSpecA,mathcalI)_frightarrow Gamma(operatornameSpecA_f,mathcalI)$ is an isomorphism for all $f$. We clearly have $Gamma(operatornameSpecA,mathcalI)_f=I_f$ where $I$ is some ideal of $A$. Also $Gamma(operatornameSpecA_f,mathcalI)=J$ where $J$ is some ideal of $A_f$. Can you find an ideal $J$ of $A_f$ that doesn't come from localization of an ideal $I$ of $A$?

share|cite|improve this answer

answered Jul 31 at 17:09

Samir Canning

38629

add a comment | 

up vote
1
down vote

It often helps to translate the problem into commutative algebra.
I will define ideal sheaves on a scheme $X$ to be an $mathcalO_X$ module $mathcalI$ such that for all open sets $Usubset X$, $mathcalI(U)$ is an ideal of $mathcalO_X(U)$. So let $mathcalI$ be an ideal sheaf on $X=operatornameSpecA$. Recall that quasicoherence can be checked by showing that the natural map $Gamma(operatornameSpecA,mathcalI)_frightarrow Gamma(operatornameSpecA_f,mathcalI)$ is an isomorphism for all $f$. We clearly have $Gamma(operatornameSpecA,mathcalI)_f=I_f$ where $I$ is some ideal of $A$. Also $Gamma(operatornameSpecA_f,mathcalI)=J$ where $J$ is some ideal of $A_f$. Can you find an ideal $J$ of $A_f$ that doesn't come from localization of an ideal $I$ of $A$?

share|cite|improve this answer

answered Jul 31 at 17:09

Samir Canning

38629

add a comment | 

up vote
1
down vote

up vote
1
down vote

It often helps to translate the problem into commutative algebra.
I will define ideal sheaves on a scheme $X$ to be an $mathcalO_X$ module $mathcalI$ such that for all open sets $Usubset X$, $mathcalI(U)$ is an ideal of $mathcalO_X(U)$. So let $mathcalI$ be an ideal sheaf on $X=operatornameSpecA$. Recall that quasicoherence can be checked by showing that the natural map $Gamma(operatornameSpecA,mathcalI)_frightarrow Gamma(operatornameSpecA_f,mathcalI)$ is an isomorphism for all $f$. We clearly have $Gamma(operatornameSpecA,mathcalI)_f=I_f$ where $I$ is some ideal of $A$. Also $Gamma(operatornameSpecA_f,mathcalI)=J$ where $J$ is some ideal of $A_f$. Can you find an ideal $J$ of $A_f$ that doesn't come from localization of an ideal $I$ of $A$?

share|cite|improve this answer

answered Jul 31 at 17:09

Samir Canning

38629

It often helps to translate the problem into commutative algebra.
I will define ideal sheaves on a scheme $X$ to be an $mathcalO_X$ module $mathcalI$ such that for all open sets $Usubset X$, $mathcalI(U)$ is an ideal of $mathcalO_X(U)$. So let $mathcalI$ be an ideal sheaf on $X=operatornameSpecA$. Recall that quasicoherence can be checked by showing that the natural map $Gamma(operatornameSpecA,mathcalI)_frightarrow Gamma(operatornameSpecA_f,mathcalI)$ is an isomorphism for all $f$. We clearly have $Gamma(operatornameSpecA,mathcalI)_f=I_f$ where $I$ is some ideal of $A$. Also $Gamma(operatornameSpecA_f,mathcalI)=J$ where $J$ is some ideal of $A_f$. Can you find an ideal $J$ of $A_f$ that doesn't come from localization of an ideal $I$ of $A$?

share|cite|improve this answer

answered Jul 31 at 17:09

Samir Canning

38629

share|cite|improve this answer

share|cite|improve this answer

answered Jul 31 at 17:09

Samir Canning

38629

answered Jul 31 at 17:09

Samir Canning

38629

answered Jul 31 at 17:09

Samir Canning

38629

38629

add a comment | 

add a comment | 

up vote
1
down vote

Now that I understood the problem with my (deleted) answer, I try to write an easy example of an ideal sheaf that it is not quasicoherent:

Let $R$ be a discrete valuation ring, with field of fractions $K$, and let $X=operatorname Spec(R)=x,eta$ be the associated affine scheme with generic point $eta$ and closed point $x$. Consider now the ideal sheaf $mathcal Isubset mathcal O_X$ defined by $mathcal I(X)=0$ and $mathcal I(eta)=K$.

This ideal sheaf is not quasicoherent.

For example, you can see that the "associated closed subscheme", in case it were quasicoherent, would correspond to the set $eta$, which it is not closed.

Or you can follow Samir Canning suggestion and localize at the uniformizer $pi$ of $R$. Then $R_pi=K$, $Gamma(operatornameSpec(R_pi),mathcal I)=K$, while $Gamma(operatornameSpec(R),mathcal I)_pi=0$.

share|cite|improve this answer

answered Jul 31 at 20:36

xarles

83558

add a comment | 

up vote
1
down vote

Now that I understood the problem with my (deleted) answer, I try to write an easy example of an ideal sheaf that it is not quasicoherent:

Let $R$ be a discrete valuation ring, with field of fractions $K$, and let $X=operatorname Spec(R)=x,eta$ be the associated affine scheme with generic point $eta$ and closed point $x$. Consider now the ideal sheaf $mathcal Isubset mathcal O_X$ defined by $mathcal I(X)=0$ and $mathcal I(eta)=K$.

This ideal sheaf is not quasicoherent.

For example, you can see that the "associated closed subscheme", in case it were quasicoherent, would correspond to the set $eta$, which it is not closed.

Or you can follow Samir Canning suggestion and localize at the uniformizer $pi$ of $R$. Then $R_pi=K$, $Gamma(operatornameSpec(R_pi),mathcal I)=K$, while $Gamma(operatornameSpec(R),mathcal I)_pi=0$.

share|cite|improve this answer

answered Jul 31 at 20:36

xarles

83558

add a comment | 

up vote
1
down vote

up vote
1
down vote

Now that I understood the problem with my (deleted) answer, I try to write an easy example of an ideal sheaf that it is not quasicoherent:

Let $R$ be a discrete valuation ring, with field of fractions $K$, and let $X=operatorname Spec(R)=x,eta$ be the associated affine scheme with generic point $eta$ and closed point $x$. Consider now the ideal sheaf $mathcal Isubset mathcal O_X$ defined by $mathcal I(X)=0$ and $mathcal I(eta)=K$.

This ideal sheaf is not quasicoherent.

For example, you can see that the "associated closed subscheme", in case it were quasicoherent, would correspond to the set $eta$, which it is not closed.

Or you can follow Samir Canning suggestion and localize at the uniformizer $pi$ of $R$. Then $R_pi=K$, $Gamma(operatornameSpec(R_pi),mathcal I)=K$, while $Gamma(operatornameSpec(R),mathcal I)_pi=0$.

share|cite|improve this answer

answered Jul 31 at 20:36

xarles

83558

Now that I understood the problem with my (deleted) answer, I try to write an easy example of an ideal sheaf that it is not quasicoherent:

Let $R$ be a discrete valuation ring, with field of fractions $K$, and let $X=operatorname Spec(R)=x,eta$ be the associated affine scheme with generic point $eta$ and closed point $x$. Consider now the ideal sheaf $mathcal Isubset mathcal O_X$ defined by $mathcal I(X)=0$ and $mathcal I(eta)=K$.

This ideal sheaf is not quasicoherent.

For example, you can see that the "associated closed subscheme", in case it were quasicoherent, would correspond to the set $eta$, which it is not closed.

Or you can follow Samir Canning suggestion and localize at the uniformizer $pi$ of $R$. Then $R_pi=K$, $Gamma(operatornameSpec(R_pi),mathcal I)=K$, while $Gamma(operatornameSpec(R),mathcal I)_pi=0$.

share|cite|improve this answer

answered Jul 31 at 20:36

xarles

83558

share|cite|improve this answer

share|cite|improve this answer

answered Jul 31 at 20:36

xarles

83558

answered Jul 31 at 20:36

xarles

83558

answered Jul 31 at 20:36

xarles

83558

83558

add a comment | 

add a comment | 

 

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