Closed morphism of products of k-schemes with a field extension of k implies closed
2
$begingroup$
This is from exercise 9.2.J of Ravi Vakil's notes on algebraic geometry here
We have a morphism of $k$ schemes $pi:X rightarrow Y$. $l/k$ is a field extension. We want to show if $pitimes_k l: Xtimes_k lrightarrow Ytimes_k l$ is a closed embedding then $pi$ also is one.
If you assume $pi$ is affine this is fairly simple. I am having a lot of trouble proving it without that hypothesis. Any help would be appreciated.
algebraic-geometry extension-field schemes
asked Dec 13 '18 at 22:18
NiarehNiareh
112
$endgroup$
add a comment |
2
$begingroup$
This is from exercise 9.2.J of Ravi Vakil's notes on algebraic geometry here
We have a morphism of $k$ schemes $pi:X rightarrow Y$. $l/k$ is a field extension. We want to show if $pitimes_k l: Xtimes_k lrightarrow Ytimes_k l$ is a closed embedding then $pi$ also is one.
If you assume $pi$ is affine this is fairly simple. I am having a lot of trouble proving it without that hypothesis. Any help would be appreciated.
algebraic-geometry extension-field schemes
asked Dec 13 '18 at 22:18
NiarehNiareh
112
$endgroup$
$begingroup$
Have you done the exercise establishing that "closed immersions are affine-local on the target"? If so, what difficulties do you encounter applying that to this exercise?
$endgroup$
– KReiser
Dec 13 '18 at 22:22
$begingroup$
So I can reduce it to the case where $Y$ can be affine but $X$ is still arbitrary. Now I am able to show $pitimes_k l$ will induce an isomorphism between $text{Spec}(Y/Jotimes_{k}l)$ and $Xtimes_k l$ where $J$ is some ideal of $Y$. I am unable to show X must be $text{Spec}(Y/J)$.
$endgroup$
– Niareh
Dec 13 '18 at 23:06
$begingroup$
In the earlier comment I had misunderstood some of the subtlety involved here, and for that I apologize. One potential answer is that the property of $pi$ being affine is stable under descent, so in fact $pi_l$ affine (as it's a closed immersion) implies $pi$ affine. A source for this is Stacksproject 02L5 or EGA IV2 2.7.1.
$endgroup$
– KReiser
Dec 14 '18 at 3:58
$begingroup$
Thanks this seems to be exactly what I need.
$endgroup$
– Niareh
Dec 16 '18 at 16:56
add a comment |
2
2
2
$begingroup$
This is from exercise 9.2.J of Ravi Vakil's notes on algebraic geometry here
We have a morphism of $k$ schemes $pi:X rightarrow Y$. $l/k$ is a field extension. We want to show if $pitimes_k l: Xtimes_k lrightarrow Ytimes_k l$ is a closed embedding then $pi$ also is one.
If you assume $pi$ is affine this is fairly simple. I am having a lot of trouble proving it without that hypothesis. Any help would be appreciated.
algebraic-geometry extension-field schemes
asked Dec 13 '18 at 22:18
NiarehNiareh
112
$endgroup$
This is from exercise 9.2.J of Ravi Vakil's notes on algebraic geometry here
We have a morphism of $k$ schemes $pi:X rightarrow Y$. $l/k$ is a field extension. We want to show if $pitimes_k l: Xtimes_k lrightarrow Ytimes_k l$ is a closed embedding then $pi$ also is one.
If you assume $pi$ is affine this is fairly simple. I am having a lot of trouble proving it without that hypothesis. Any help would be appreciated.
algebraic-geometry extension-field schemes
algebraic-geometry extension-field schemes
asked Dec 13 '18 at 22:18
NiarehNiareh
112
asked Dec 13 '18 at 22:18
NiarehNiareh
112
asked Dec 13 '18 at 22:18
NiarehNiareh
112
asked Dec 13 '18 at 22:18
NiarehNiareh
112
asked Dec 13 '18 at 22:18
NiarehNiareh
112
112
$begingroup$
Have you done the exercise establishing that "closed immersions are affine-local on the target"? If so, what difficulties do you encounter applying that to this exercise?
$endgroup$
– KReiser
Dec 13 '18 at 22:22
$begingroup$
So I can reduce it to the case where $Y$ can be affine but $X$ is still arbitrary. Now I am able to show $pitimes_k l$ will induce an isomorphism between $text{Spec}(Y/Jotimes_{k}l)$ and $Xtimes_k l$ where $J$ is some ideal of $Y$. I am unable to show X must be $text{Spec}(Y/J)$.
$endgroup$
– Niareh
Dec 13 '18 at 23:06
$begingroup$
In the earlier comment I had misunderstood some of the subtlety involved here, and for that I apologize. One potential answer is that the property of $pi$ being affine is stable under descent, so in fact $pi_l$ affine (as it's a closed immersion) implies $pi$ affine. A source for this is Stacksproject 02L5 or EGA IV2 2.7.1.
$endgroup$
– KReiser
Dec 14 '18 at 3:58
$begingroup$
Thanks this seems to be exactly what I need.
$endgroup$
– Niareh
Dec 16 '18 at 16:56
add a comment |
$begingroup$
Have you done the exercise establishing that "closed immersions are affine-local on the target"? If so, what difficulties do you encounter applying that to this exercise?
$endgroup$
– KReiser
Dec 13 '18 at 22:22
$begingroup$
So I can reduce it to the case where $Y$ can be affine but $X$ is still arbitrary. Now I am able to show $pitimes_k l$ will induce an isomorphism between $text{Spec}(Y/Jotimes_{k}l)$ and $Xtimes_k l$ where $J$ is some ideal of $Y$. I am unable to show X must be $text{Spec}(Y/J)$.
$endgroup$
– Niareh
Dec 13 '18 at 23:06
$begingroup$
In the earlier comment I had misunderstood some of the subtlety involved here, and for that I apologize. One potential answer is that the property of $pi$ being affine is stable under descent, so in fact $pi_l$ affine (as it's a closed immersion) implies $pi$ affine. A source for this is Stacksproject 02L5 or EGA IV2 2.7.1.
$endgroup$
– KReiser
Dec 14 '18 at 3:58
$begingroup$
Thanks this seems to be exactly what I need.
$endgroup$
– Niareh
Dec 16 '18 at 16:56
$begingroup$
Have you done the exercise establishing that "closed immersions are affine-local on the target"? If so, what difficulties do you encounter applying that to this exercise?
$endgroup$
– KReiser
Dec 13 '18 at 22:22
$begingroup$
Have you done the exercise establishing that "closed immersions are affine-local on the target"? If so, what difficulties do you encounter applying that to this exercise?
$endgroup$
– KReiser
Dec 13 '18 at 22:22
$begingroup$
So I can reduce it to the case where $Y$ can be affine but $X$ is still arbitrary. Now I am able to show $pitimes_k l$ will induce an isomorphism between $text{Spec}(Y/Jotimes_{k}l)$ and $Xtimes_k l$ where $J$ is some ideal of $Y$. I am unable to show X must be $text{Spec}(Y/J)$.
$endgroup$
– Niareh
Dec 13 '18 at 23:06
$begingroup$
So I can reduce it to the case where $Y$ can be affine but $X$ is still arbitrary. Now I am able to show $pitimes_k l$ will induce an isomorphism between $text{Spec}(Y/Jotimes_{k}l)$ and $Xtimes_k l$ where $J$ is some ideal of $Y$. I am unable to show X must be $text{Spec}(Y/J)$.
$endgroup$
– Niareh
Dec 13 '18 at 23:06
$begingroup$
In the earlier comment I had misunderstood some of the subtlety involved here, and for that I apologize. One potential answer is that the property of $pi$ being affine is stable under descent, so in fact $pi_l$ affine (as it's a closed immersion) implies $pi$ affine. A source for this is Stacksproject 02L5 or EGA IV2 2.7.1.
$endgroup$
– KReiser
Dec 14 '18 at 3:58
$begingroup$
In the earlier comment I had misunderstood some of the subtlety involved here, and for that I apologize. One potential answer is that the property of $pi$ being affine is stable under descent, so in fact $pi_l$ affine (as it's a closed immersion) implies $pi$ affine. A source for this is Stacksproject 02L5 or EGA IV2 2.7.1.
$endgroup$
– KReiser
Dec 14 '18 at 3:58
$begingroup$
Thanks this seems to be exactly what I need.
$endgroup$
– Niareh
Dec 16 '18 at 16:56
$begingroup$
Thanks this seems to be exactly what I need.
$endgroup$
– Niareh
Dec 16 '18 at 16:56
add a comment |
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$begingroup$
Have you done the exercise establishing that "closed immersions are affine-local on the target"? If so, what difficulties do you encounter applying that to this exercise?
$endgroup$
– KReiser
Dec 13 '18 at 22:22
$begingroup$
So I can reduce it to the case where $Y$ can be affine but $X$ is still arbitrary. Now I am able to show $pitimes_k l$ will induce an isomorphism between $text{Spec}(Y/Jotimes_{k}l)$ and $Xtimes_k l$ where $J$ is some ideal of $Y$. I am unable to show X must be $text{Spec}(Y/J)$.
$endgroup$
– Niareh
Dec 13 '18 at 23:06
$begingroup$
In the earlier comment I had misunderstood some of the subtlety involved here, and for that I apologize. One potential answer is that the property of $pi$ being affine is stable under descent, so in fact $pi_l$ affine (as it's a closed immersion) implies $pi$ affine. A source for this is Stacksproject 02L5 or EGA IV2 2.7.1.
$endgroup$
– KReiser
Dec 14 '18 at 3:58
$begingroup$
Thanks this seems to be exactly what I need.
$endgroup$
– Niareh
Dec 16 '18 at 16:56