$G$-Action on a Ring extends to a Module
$begingroup$
Let $A$ be ring endowed with $G$-action by a group. Take any arbitrary $A$-module $M$.
Is there a canonical way to extend the $G$-action to $M$? I'm not sure how to avoid the problem with well definess if I try to settle the action in naive way via $am mapsto a^g m$.
The background of my question is Automorphism Group of a Variety acts on Local Sections
commutative-algebra group-actions
asked Jan 13 at 3:18
KarlPeterKarlPeter
6271315
$endgroup$
add a comment |
$begingroup$
Let $A$ be ring endowed with $G$-action by a group. Take any arbitrary $A$-module $M$.
Is there a canonical way to extend the $G$-action to $M$? I'm not sure how to avoid the problem with well definess if I try to settle the action in naive way via $am mapsto a^g m$.
The background of my question is Automorphism Group of a Variety acts on Local Sections
commutative-algebra group-actions
asked Jan 13 at 3:18
KarlPeterKarlPeter
6271315
$endgroup$
add a comment |
$begingroup$
Let $A$ be ring endowed with $G$-action by a group. Take any arbitrary $A$-module $M$.
Is there a canonical way to extend the $G$-action to $M$? I'm not sure how to avoid the problem with well definess if I try to settle the action in naive way via $am mapsto a^g m$.
The background of my question is Automorphism Group of a Variety acts on Local Sections
commutative-algebra group-actions
asked Jan 13 at 3:18
KarlPeterKarlPeter
6271315
$endgroup$
Let $A$ be ring endowed with $G$-action by a group. Take any arbitrary $A$-module $M$.
Is there a canonical way to extend the $G$-action to $M$? I'm not sure how to avoid the problem with well definess if I try to settle the action in naive way via $am mapsto a^g m$.
The background of my question is Automorphism Group of a Variety acts on Local Sections
commutative-algebra group-actions
commutative-algebra group-actions
asked Jan 13 at 3:18
KarlPeterKarlPeter
6271315
asked Jan 13 at 3:18
KarlPeterKarlPeter
6271315
asked Jan 13 at 3:18
KarlPeterKarlPeter
6271315
asked Jan 13 at 3:18
KarlPeterKarlPeter
6271315
asked Jan 13 at 3:18
KarlPeterKarlPeter
6271315
6271315
add a comment |
add a comment |
1 Answer
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No, there is not. For example, in the case where $M$ is $A$ itself, in order for the action of $G$ on $M$ to deserve the name "canonical", it would need to commute with any automorphism of $M$ as a module. Concretely, that means that every unit of $A$ would have to be fixed by the action of $G$, which certainly is not true in general.
answered Jan 13 at 3:53
Eric WofseyEric Wofsey
184k13211338
$endgroup$
$begingroup$
This is a simple and intuitive answer.
$endgroup$
– Youngsu
Jan 13 at 19:14
$begingroup$
Yes, this is the point. Do you maybe have an idea which $G$-action on an $A$-module @Mohan mean in math.stackexchange.com/questions/3071478/… ?
$endgroup$
– KarlPeter
Jan 13 at 22:32
$begingroup$
I think he was confused.
$endgroup$
– Eric Wofsey
Jan 13 at 22:36
add a comment |
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1 Answer
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1 Answer
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$begingroup$
No, there is not. For example, in the case where $M$ is $A$ itself, in order for the action of $G$ on $M$ to deserve the name "canonical", it would need to commute with any automorphism of $M$ as a module. Concretely, that means that every unit of $A$ would have to be fixed by the action of $G$, which certainly is not true in general.
answered Jan 13 at 3:53
Eric WofseyEric Wofsey
184k13211338
$endgroup$
$begingroup$
This is a simple and intuitive answer.
$endgroup$
– Youngsu
Jan 13 at 19:14
$begingroup$
Yes, this is the point. Do you maybe have an idea which $G$-action on an $A$-module @Mohan mean in math.stackexchange.com/questions/3071478/… ?
$endgroup$
– KarlPeter
Jan 13 at 22:32
$begingroup$
I think he was confused.
$endgroup$
– Eric Wofsey
Jan 13 at 22:36
add a comment |
$begingroup$
No, there is not. For example, in the case where $M$ is $A$ itself, in order for the action of $G$ on $M$ to deserve the name "canonical", it would need to commute with any automorphism of $M$ as a module. Concretely, that means that every unit of $A$ would have to be fixed by the action of $G$, which certainly is not true in general.
answered Jan 13 at 3:53
Eric WofseyEric Wofsey
184k13211338
$endgroup$
$begingroup$
This is a simple and intuitive answer.
$endgroup$
– Youngsu
Jan 13 at 19:14
$begingroup$
Yes, this is the point. Do you maybe have an idea which $G$-action on an $A$-module @Mohan mean in math.stackexchange.com/questions/3071478/… ?
$endgroup$
– KarlPeter
Jan 13 at 22:32
$begingroup$
I think he was confused.
$endgroup$
– Eric Wofsey
Jan 13 at 22:36
add a comment |
$begingroup$
No, there is not. For example, in the case where $M$ is $A$ itself, in order for the action of $G$ on $M$ to deserve the name "canonical", it would need to commute with any automorphism of $M$ as a module. Concretely, that means that every unit of $A$ would have to be fixed by the action of $G$, which certainly is not true in general.
answered Jan 13 at 3:53
Eric WofseyEric Wofsey
184k13211338
$endgroup$
No, there is not. For example, in the case where $M$ is $A$ itself, in order for the action of $G$ on $M$ to deserve the name "canonical", it would need to commute with any automorphism of $M$ as a module. Concretely, that means that every unit of $A$ would have to be fixed by the action of $G$, which certainly is not true in general.
answered Jan 13 at 3:53
Eric WofseyEric Wofsey
184k13211338
answered Jan 13 at 3:53
Eric WofseyEric Wofsey
184k13211338
answered Jan 13 at 3:53
Eric WofseyEric Wofsey
184k13211338
answered Jan 13 at 3:53
Eric WofseyEric Wofsey
184k13211338
184k13211338
$begingroup$
This is a simple and intuitive answer.
$endgroup$
– Youngsu
Jan 13 at 19:14
$begingroup$
Yes, this is the point. Do you maybe have an idea which $G$-action on an $A$-module @Mohan mean in math.stackexchange.com/questions/3071478/… ?
$endgroup$
– KarlPeter
Jan 13 at 22:32
$begingroup$
I think he was confused.
$endgroup$
– Eric Wofsey
Jan 13 at 22:36
add a comment |
$begingroup$
This is a simple and intuitive answer.
$endgroup$
– Youngsu
Jan 13 at 19:14
$begingroup$
Yes, this is the point. Do you maybe have an idea which $G$-action on an $A$-module @Mohan mean in math.stackexchange.com/questions/3071478/… ?
$endgroup$
– KarlPeter
Jan 13 at 22:32
$begingroup$
I think he was confused.
$endgroup$
– Eric Wofsey
Jan 13 at 22:36
$begingroup$
This is a simple and intuitive answer.
$endgroup$
– Youngsu
Jan 13 at 19:14
$begingroup$
This is a simple and intuitive answer.
$endgroup$
– Youngsu
Jan 13 at 19:14
$begingroup$
Yes, this is the point. Do you maybe have an idea which $G$-action on an $A$-module @Mohan mean in math.stackexchange.com/questions/3071478/… ?
$endgroup$
– KarlPeter
Jan 13 at 22:32
$begingroup$
Yes, this is the point. Do you maybe have an idea which $G$-action on an $A$-module @Mohan mean in math.stackexchange.com/questions/3071478/… ?
$endgroup$
– KarlPeter
Jan 13 at 22:32
$begingroup$
I think he was confused.
$endgroup$
– Eric Wofsey
Jan 13 at 22:36
$begingroup$
I think he was confused.
$endgroup$
– Eric Wofsey
Jan 13 at 22:36
add a comment |
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