Mixing
Abelian $operatornameHom(-,X)$ in $mathcalAlg_R/A$
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How do I find all objects $X$ such that $operatornameHom(-,X)$ is an abelian group in the category $mathcalAlg_R/A$ of algebras over a fixed $R-$algebra $A$?
I have been considering what being over $A$ induces, but am making no progress right now. The claim is that they are all of the form $Altimes M$ for $M$ an $A-$module.
abstract-algebra category-theory modules
asked Aug 6 at 10:50
user492530
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up vote
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down vote
favorite
How do I find all objects $X$ such that $operatornameHom(-,X)$ is an abelian group in the category $mathcalAlg_R/A$ of algebras over a fixed $R-$algebra $A$?
I have been considering what being over $A$ induces, but am making no progress right now. The claim is that they are all of the form $Altimes M$ for $M$ an $A-$module.
abstract-algebra category-theory modules
asked Aug 6 at 10:50
user492530
142
Do you want the group structure to be natural?
– Arnaud D.
Aug 6 at 12:02
When $Altimes Mto A$ is considered as an abelian group object, how do you retrieve $M$? Once you work that out, you should be able to extract a module $M_X$ from any abelian group object $Xto A$ such that $A ltimes M_X to A$ is the object $Xto A$ you started from.
– Pece
Aug 6 at 14:15
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
How do I find all objects $X$ such that $operatornameHom(-,X)$ is an abelian group in the category $mathcalAlg_R/A$ of algebras over a fixed $R-$algebra $A$?
I have been considering what being over $A$ induces, but am making no progress right now. The claim is that they are all of the form $Altimes M$ for $M$ an $A-$module.
abstract-algebra category-theory modules
asked Aug 6 at 10:50
user492530
142
How do I find all objects $X$ such that $operatornameHom(-,X)$ is an abelian group in the category $mathcalAlg_R/A$ of algebras over a fixed $R-$algebra $A$?
I have been considering what being over $A$ induces, but am making no progress right now. The claim is that they are all of the form $Altimes M$ for $M$ an $A-$module.
abstract-algebra category-theory modules
asked Aug 6 at 10:50
user492530
142
edited Aug 6 at 11:11
asked Aug 6 at 10:50
user492530
142
asked Aug 6 at 10:50
user492530
142
asked Aug 6 at 10:50
user492530
142
142
Do you want the group structure to be natural?
– Arnaud D.
Aug 6 at 12:02
When $Altimes Mto A$ is considered as an abelian group object, how do you retrieve $M$? Once you work that out, you should be able to extract a module $M_X$ from any abelian group object $Xto A$ such that $A ltimes M_X to A$ is the object $Xto A$ you started from.
– Pece
Aug 6 at 14:15
add a comment |Â
Do you want the group structure to be natural?
– Arnaud D.
Aug 6 at 12:02
When $Altimes Mto A$ is considered as an abelian group object, how do you retrieve $M$? Once you work that out, you should be able to extract a module $M_X$ from any abelian group object $Xto A$ such that $A ltimes M_X to A$ is the object $Xto A$ you started from.
– Pece
Aug 6 at 14:15
Do you want the group structure to be natural?
– Arnaud D.
Aug 6 at 12:02
Do you want the group structure to be natural?
– Arnaud D.
Aug 6 at 12:02
When $Altimes Mto A$ is considered as an abelian group object, how do you retrieve $M$? Once you work that out, you should be able to extract a module $M_X$ from any abelian group object $Xto A$ such that $A ltimes M_X to A$ is the object $Xto A$ you started from.
– Pece
Aug 6 at 14:15
When $Altimes Mto A$ is considered as an abelian group object, how do you retrieve $M$? Once you work that out, you should be able to extract a module $M_X$ from any abelian group object $Xto A$ such that $A ltimes M_X to A$ is the object $Xto A$ you started from.
– Pece
Aug 6 at 14:15
add a comment |Â
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Do you want the group structure to be natural?
– Arnaud D.
Aug 6 at 12:02
When $Altimes Mto A$ is considered as an abelian group object, how do you retrieve $M$? Once you work that out, you should be able to extract a module $M_X$ from any abelian group object $Xto A$ such that $A ltimes M_X to A$ is the object $Xto A$ you started from.
– Pece
Aug 6 at 14:15