Mixing
Question regarding flat localization of left modules
$begingroup$
This is taken from Rosenberg's non-commutative affine scheme https://sasharosenberg.com/?x-portfolio=noncommutative-affine-schemes page 3-4, where he talks about flat localizations of $R$-$Mod$.
Let $R$ be an associative ring with unity $M$ be a left-$R$ module. $F$ be a radical/gabriel filter of left ideals of $R$. Define $$H'_{F}M:=Colim(Hom_{R}(m,M)|min F)$$
where the arrows of the directed set taken w.r.t reverse inclusion. Now, clearly the colimit is an abelian group. But how is it possible to give it a left $R$-module structure? For example, if we consider $$Rtimes H'_{F}Mrightarrow H'_{F}M \ (r,[g,m])rightarrow [rg,m]$$
where $gin Hom_{R}(m,M)$. The problem is $rg$ need not be a left $R$-module homomorphism and I don't see any other way out. Please have a look in the page $3$-$4$ of the paper as i fear I'm misunderstanding it.
Thanks in advance!
noncommutative-algebra
asked Jan 27 at 4:47
solgaleosolgaleo
7611
$endgroup$
add a comment |
$begingroup$
This is taken from Rosenberg's non-commutative affine scheme https://sasharosenberg.com/?x-portfolio=noncommutative-affine-schemes page 3-4, where he talks about flat localizations of $R$-$Mod$.
Let $R$ be an associative ring with unity $M$ be a left-$R$ module. $F$ be a radical/gabriel filter of left ideals of $R$. Define $$H'_{F}M:=Colim(Hom_{R}(m,M)|min F)$$
where the arrows of the directed set taken w.r.t reverse inclusion. Now, clearly the colimit is an abelian group. But how is it possible to give it a left $R$-module structure? For example, if we consider $$Rtimes H'_{F}Mrightarrow H'_{F}M \ (r,[g,m])rightarrow [rg,m]$$
where $gin Hom_{R}(m,M)$. The problem is $rg$ need not be a left $R$-module homomorphism and I don't see any other way out. Please have a look in the page $3$-$4$ of the paper as i fear I'm misunderstanding it.
Thanks in advance!
noncommutative-algebra
asked Jan 27 at 4:47
solgaleosolgaleo
7611
$endgroup$
add a comment |
$begingroup$
This is taken from Rosenberg's non-commutative affine scheme https://sasharosenberg.com/?x-portfolio=noncommutative-affine-schemes page 3-4, where he talks about flat localizations of $R$-$Mod$.
Let $R$ be an associative ring with unity $M$ be a left-$R$ module. $F$ be a radical/gabriel filter of left ideals of $R$. Define $$H'_{F}M:=Colim(Hom_{R}(m,M)|min F)$$
where the arrows of the directed set taken w.r.t reverse inclusion. Now, clearly the colimit is an abelian group. But how is it possible to give it a left $R$-module structure? For example, if we consider $$Rtimes H'_{F}Mrightarrow H'_{F}M \ (r,[g,m])rightarrow [rg,m]$$
where $gin Hom_{R}(m,M)$. The problem is $rg$ need not be a left $R$-module homomorphism and I don't see any other way out. Please have a look in the page $3$-$4$ of the paper as i fear I'm misunderstanding it.
Thanks in advance!
noncommutative-algebra
asked Jan 27 at 4:47
solgaleosolgaleo
7611
$endgroup$
This is taken from Rosenberg's non-commutative affine scheme https://sasharosenberg.com/?x-portfolio=noncommutative-affine-schemes page 3-4, where he talks about flat localizations of $R$-$Mod$.
Let $R$ be an associative ring with unity $M$ be a left-$R$ module. $F$ be a radical/gabriel filter of left ideals of $R$. Define $$H'_{F}M:=Colim(Hom_{R}(m,M)|min F)$$
where the arrows of the directed set taken w.r.t reverse inclusion. Now, clearly the colimit is an abelian group. But how is it possible to give it a left $R$-module structure? For example, if we consider $$Rtimes H'_{F}Mrightarrow H'_{F}M \ (r,[g,m])rightarrow [rg,m]$$
where $gin Hom_{R}(m,M)$. The problem is $rg$ need not be a left $R$-module homomorphism and I don't see any other way out. Please have a look in the page $3$-$4$ of the paper as i fear I'm misunderstanding it.
Thanks in advance!
noncommutative-algebra
noncommutative-algebra
asked Jan 27 at 4:47
solgaleosolgaleo
7611
asked Jan 27 at 4:47
solgaleosolgaleo
7611
edited Jan 29 at 10:10
solgaleo
asked Jan 27 at 4:47
solgaleosolgaleo
7611
asked Jan 27 at 4:47
solgaleosolgaleo
7611
asked Jan 27 at 4:47
solgaleosolgaleo
7611
7611
add a comment |
add a comment |
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