Isomorphism between an algebraic set and a cartesian product [closed]
Let W, X and Y be algebraic sets and $gamma_1$: W $rightarrow$ X, $gamma_2$: W $rightarrow$ Y, two morphisms which verify that, given Z an algebraic set and two morphisms $alpha$: Z $rightarrow$ X and $beta$: Z $rightarrow$ Y, exists only one morphism $phi$: Z $rightarrow$ W verifying $alpha = gamma_1phi$ and $beta = gamma_2phi$. Prove that W is isomorphic to X x Y.
algebraic-geometry commutative-algebra
asked Nov 28 at 11:48
Julia Vazquez Escobar
31
closed as off-topic by amWhy, Alexander Gruber♦ Nov 30 at 3:37
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – amWhy, Alexander Gruber
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
Let W, X and Y be algebraic sets and $gamma_1$: W $rightarrow$ X, $gamma_2$: W $rightarrow$ Y, two morphisms which verify that, given Z an algebraic set and two morphisms $alpha$: Z $rightarrow$ X and $beta$: Z $rightarrow$ Y, exists only one morphism $phi$: Z $rightarrow$ W verifying $alpha = gamma_1phi$ and $beta = gamma_2phi$. Prove that W is isomorphic to X x Y.
algebraic-geometry commutative-algebra
asked Nov 28 at 11:48
Julia Vazquez Escobar
31
closed as off-topic by amWhy, Alexander Gruber♦ Nov 30 at 3:37
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – amWhy, Alexander Gruber
If this question can be reworded to fit the rules in the help center, please edit the question.
Hi Julia, please write what did you try and what do you think about this problem if you would like useful feedback from the MSE community.
– Nicolas Hemelsoet
Nov 28 at 11:59
I've tried choosing X x Y as Z, so $alpha$ and $beta$ would be the prejction morphisms. I've stated that exists $psi$: W $rightarrow$ X x Y expressed as $psi(t) = (gamma_1(t), gamma_2(t))$ that clearly is a morphism, and $phi$: X x Y $rightarrow$ W would be the inverse, we know that exists from the problem statements. But I'm having a hard time proving $phi circ psi$ is the identity.
– Julia Vazquez Escobar
Nov 28 at 12:08
Thank you for your responses
– Julia Vazquez Escobar
Nov 28 at 14:05
add a comment |
Let W, X and Y be algebraic sets and $gamma_1$: W $rightarrow$ X, $gamma_2$: W $rightarrow$ Y, two morphisms which verify that, given Z an algebraic set and two morphisms $alpha$: Z $rightarrow$ X and $beta$: Z $rightarrow$ Y, exists only one morphism $phi$: Z $rightarrow$ W verifying $alpha = gamma_1phi$ and $beta = gamma_2phi$. Prove that W is isomorphic to X x Y.
algebraic-geometry commutative-algebra
asked Nov 28 at 11:48
Julia Vazquez Escobar
31
Let W, X and Y be algebraic sets and $gamma_1$: W $rightarrow$ X, $gamma_2$: W $rightarrow$ Y, two morphisms which verify that, given Z an algebraic set and two morphisms $alpha$: Z $rightarrow$ X and $beta$: Z $rightarrow$ Y, exists only one morphism $phi$: Z $rightarrow$ W verifying $alpha = gamma_1phi$ and $beta = gamma_2phi$. Prove that W is isomorphic to X x Y.
algebraic-geometry commutative-algebra
algebraic-geometry commutative-algebra
asked Nov 28 at 11:48
Julia Vazquez Escobar
31
asked Nov 28 at 11:48
Julia Vazquez Escobar
31
asked Nov 28 at 11:48
Julia Vazquez Escobar
31
asked Nov 28 at 11:48
Julia Vazquez Escobar
31
asked Nov 28 at 11:48
Julia Vazquez Escobar
31
31
closed as off-topic by amWhy, Alexander Gruber♦ Nov 30 at 3:37
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – amWhy, Alexander Gruber
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by amWhy, Alexander Gruber♦ Nov 30 at 3:37
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – amWhy, Alexander Gruber
If this question can be reworded to fit the rules in the help center, please edit the question.
Hi Julia, please write what did you try and what do you think about this problem if you would like useful feedback from the MSE community.
– Nicolas Hemelsoet
Nov 28 at 11:59
I've tried choosing X x Y as Z, so $alpha$ and $beta$ would be the prejction morphisms. I've stated that exists $psi$: W $rightarrow$ X x Y expressed as $psi(t) = (gamma_1(t), gamma_2(t))$ that clearly is a morphism, and $phi$: X x Y $rightarrow$ W would be the inverse, we know that exists from the problem statements. But I'm having a hard time proving $phi circ psi$ is the identity.
– Julia Vazquez Escobar
Nov 28 at 12:08
Thank you for your responses
– Julia Vazquez Escobar
Nov 28 at 14:05
add a comment |
Hi Julia, please write what did you try and what do you think about this problem if you would like useful feedback from the MSE community.
– Nicolas Hemelsoet
Nov 28 at 11:59
I've tried choosing X x Y as Z, so $alpha$ and $beta$ would be the prejction morphisms. I've stated that exists $psi$: W $rightarrow$ X x Y expressed as $psi(t) = (gamma_1(t), gamma_2(t))$ that clearly is a morphism, and $phi$: X x Y $rightarrow$ W would be the inverse, we know that exists from the problem statements. But I'm having a hard time proving $phi circ psi$ is the identity.
– Julia Vazquez Escobar
Nov 28 at 12:08
Thank you for your responses
– Julia Vazquez Escobar
Nov 28 at 14:05
Hi Julia, please write what did you try and what do you think about this problem if you would like useful feedback from the MSE community.
– Nicolas Hemelsoet
Nov 28 at 11:59
Hi Julia, please write what did you try and what do you think about this problem if you would like useful feedback from the MSE community.
– Nicolas Hemelsoet
Nov 28 at 11:59
I've tried choosing X x Y as Z, so $alpha$ and $beta$ would be the prejction morphisms. I've stated that exists $psi$: W $rightarrow$ X x Y expressed as $psi(t) = (gamma_1(t), gamma_2(t))$ that clearly is a morphism, and $phi$: X x Y $rightarrow$ W would be the inverse, we know that exists from the problem statements. But I'm having a hard time proving $phi circ psi$ is the identity.
– Julia Vazquez Escobar
Nov 28 at 12:08
I've tried choosing X x Y as Z, so $alpha$ and $beta$ would be the prejction morphisms. I've stated that exists $psi$: W $rightarrow$ X x Y expressed as $psi(t) = (gamma_1(t), gamma_2(t))$ that clearly is a morphism, and $phi$: X x Y $rightarrow$ W would be the inverse, we know that exists from the problem statements. But I'm having a hard time proving $phi circ psi$ is the identity.
– Julia Vazquez Escobar
Nov 28 at 12:08
Thank you for your responses
– Julia Vazquez Escobar
Nov 28 at 14:05
Thank you for your responses
– Julia Vazquez Escobar
Nov 28 at 14:05
add a comment |
1 Answer
1
active
oldest
votes
Well, you have to show that $Xtimes Y$ satisfies the same condition as $Z$. Doing this, you will have that there is a unique morphism $Xtimes Yto Z$ and a unique morphism $Zto Xtimes Y$ (such that the respective diagrams commute), since there is a unique morphism such that makes the diagram commutes and such that $Zto Z$ is the identity (same argument for $Xtimes Yto Xtimes Y$). You will have that $Z$ is isomorphic (with unique isomorphism) to $Xtimes Y$.
Now, forget the algebraic structure for a second. $Z$ satisfies the product universal property for sets, so that $Z$ is isomorphic to $Xtimes Y$ (see the above argument) with a unique isomorphism. Now, if $phi:Xtimes Yto Z$ is the biyection (iso of sets) you have to prove that it is an isomorphism of algebraic varieties. This is, if $psi:Zto Xtimes Y$ if the (set) inverse of $phi$: Probe that $phi$ and $psi$ are morphisms. But this will follow from the diagrams and using the definition of algebraic set morphism.
answered Nov 28 at 14:16
José Alejandro Aburto Araneda
40619
Thank you very much :)
– Julia Vazquez Escobar
Nov 28 at 20:39
You are welcome. And good luck with algebraic geometry, it's a really beatiful (and hard) subject
– José Alejandro Aburto Araneda
Nov 29 at 0:01
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
Well, you have to show that $Xtimes Y$ satisfies the same condition as $Z$. Doing this, you will have that there is a unique morphism $Xtimes Yto Z$ and a unique morphism $Zto Xtimes Y$ (such that the respective diagrams commute), since there is a unique morphism such that makes the diagram commutes and such that $Zto Z$ is the identity (same argument for $Xtimes Yto Xtimes Y$). You will have that $Z$ is isomorphic (with unique isomorphism) to $Xtimes Y$.
Now, forget the algebraic structure for a second. $Z$ satisfies the product universal property for sets, so that $Z$ is isomorphic to $Xtimes Y$ (see the above argument) with a unique isomorphism. Now, if $phi:Xtimes Yto Z$ is the biyection (iso of sets) you have to prove that it is an isomorphism of algebraic varieties. This is, if $psi:Zto Xtimes Y$ if the (set) inverse of $phi$: Probe that $phi$ and $psi$ are morphisms. But this will follow from the diagrams and using the definition of algebraic set morphism.
answered Nov 28 at 14:16
José Alejandro Aburto Araneda
40619
Thank you very much :)
– Julia Vazquez Escobar
Nov 28 at 20:39
You are welcome. And good luck with algebraic geometry, it's a really beatiful (and hard) subject
– José Alejandro Aburto Araneda
Nov 29 at 0:01
add a comment |
Well, you have to show that $Xtimes Y$ satisfies the same condition as $Z$. Doing this, you will have that there is a unique morphism $Xtimes Yto Z$ and a unique morphism $Zto Xtimes Y$ (such that the respective diagrams commute), since there is a unique morphism such that makes the diagram commutes and such that $Zto Z$ is the identity (same argument for $Xtimes Yto Xtimes Y$). You will have that $Z$ is isomorphic (with unique isomorphism) to $Xtimes Y$.
Now, forget the algebraic structure for a second. $Z$ satisfies the product universal property for sets, so that $Z$ is isomorphic to $Xtimes Y$ (see the above argument) with a unique isomorphism. Now, if $phi:Xtimes Yto Z$ is the biyection (iso of sets) you have to prove that it is an isomorphism of algebraic varieties. This is, if $psi:Zto Xtimes Y$ if the (set) inverse of $phi$: Probe that $phi$ and $psi$ are morphisms. But this will follow from the diagrams and using the definition of algebraic set morphism.
answered Nov 28 at 14:16
José Alejandro Aburto Araneda
40619
Thank you very much :)
– Julia Vazquez Escobar
Nov 28 at 20:39
You are welcome. And good luck with algebraic geometry, it's a really beatiful (and hard) subject
– José Alejandro Aburto Araneda
Nov 29 at 0:01
add a comment |
Well, you have to show that $Xtimes Y$ satisfies the same condition as $Z$. Doing this, you will have that there is a unique morphism $Xtimes Yto Z$ and a unique morphism $Zto Xtimes Y$ (such that the respective diagrams commute), since there is a unique morphism such that makes the diagram commutes and such that $Zto Z$ is the identity (same argument for $Xtimes Yto Xtimes Y$). You will have that $Z$ is isomorphic (with unique isomorphism) to $Xtimes Y$.
Now, forget the algebraic structure for a second. $Z$ satisfies the product universal property for sets, so that $Z$ is isomorphic to $Xtimes Y$ (see the above argument) with a unique isomorphism. Now, if $phi:Xtimes Yto Z$ is the biyection (iso of sets) you have to prove that it is an isomorphism of algebraic varieties. This is, if $psi:Zto Xtimes Y$ if the (set) inverse of $phi$: Probe that $phi$ and $psi$ are morphisms. But this will follow from the diagrams and using the definition of algebraic set morphism.
answered Nov 28 at 14:16
José Alejandro Aburto Araneda
40619
Well, you have to show that $Xtimes Y$ satisfies the same condition as $Z$. Doing this, you will have that there is a unique morphism $Xtimes Yto Z$ and a unique morphism $Zto Xtimes Y$ (such that the respective diagrams commute), since there is a unique morphism such that makes the diagram commutes and such that $Zto Z$ is the identity (same argument for $Xtimes Yto Xtimes Y$). You will have that $Z$ is isomorphic (with unique isomorphism) to $Xtimes Y$.
Now, forget the algebraic structure for a second. $Z$ satisfies the product universal property for sets, so that $Z$ is isomorphic to $Xtimes Y$ (see the above argument) with a unique isomorphism. Now, if $phi:Xtimes Yto Z$ is the biyection (iso of sets) you have to prove that it is an isomorphism of algebraic varieties. This is, if $psi:Zto Xtimes Y$ if the (set) inverse of $phi$: Probe that $phi$ and $psi$ are morphisms. But this will follow from the diagrams and using the definition of algebraic set morphism.
answered Nov 28 at 14:16
José Alejandro Aburto Araneda
40619
edited Nov 28 at 23:58
answered Nov 28 at 14:16
José Alejandro Aburto Araneda
40619
answered Nov 28 at 14:16
José Alejandro Aburto Araneda
40619
answered Nov 28 at 14:16
José Alejandro Aburto Araneda
40619
40619
Thank you very much :)
– Julia Vazquez Escobar
Nov 28 at 20:39
You are welcome. And good luck with algebraic geometry, it's a really beatiful (and hard) subject
– José Alejandro Aburto Araneda
Nov 29 at 0:01
add a comment |
Thank you very much :)
– Julia Vazquez Escobar
Nov 28 at 20:39
You are welcome. And good luck with algebraic geometry, it's a really beatiful (and hard) subject
– José Alejandro Aburto Araneda
Nov 29 at 0:01
Thank you very much :)
– Julia Vazquez Escobar
Nov 28 at 20:39
Thank you very much :)
– Julia Vazquez Escobar
Nov 28 at 20:39
You are welcome. And good luck with algebraic geometry, it's a really beatiful (and hard) subject
– José Alejandro Aburto Araneda
Nov 29 at 0:01
You are welcome. And good luck with algebraic geometry, it's a really beatiful (and hard) subject
– José Alejandro Aburto Araneda
Nov 29 at 0:01
add a comment |
Hi Julia, please write what did you try and what do you think about this problem if you would like useful feedback from the MSE community.
– Nicolas Hemelsoet
Nov 28 at 11:59
I've tried choosing X x Y as Z, so $alpha$ and $beta$ would be the prejction morphisms. I've stated that exists $psi$: W $rightarrow$ X x Y expressed as $psi(t) = (gamma_1(t), gamma_2(t))$ that clearly is a morphism, and $phi$: X x Y $rightarrow$ W would be the inverse, we know that exists from the problem statements. But I'm having a hard time proving $phi circ psi$ is the identity.
– Julia Vazquez Escobar
Nov 28 at 12:08
Thank you for your responses
– Julia Vazquez Escobar
Nov 28 at 14:05