Localization of finitely generated algebra
$begingroup$
Let $R$ be a reduced finitely generated algebra over $Bbb Z$. Let $T$ be a finite set of prime ideals of $R$.
Let $S = bigcap_{p notin T} R setminus p$.
1) Is it true that $A := S^{-1}R$ is finitely generated as an algebra over $Bbb Z$?
2) Is it true that $B := bigcap_{p not in T} ((R setminus p)^{-1}R)$ is finitely generated as an algebra over $Bbb Z$?
It seems to work if $R$ is the ring of integers of a number field, and we have $A cong B$. I am wondering about the general case. Maybe I should add some assumptions to make this true (e.g. $R$ integral domain...).
commutative-algebra localization finitely-generated
asked Nov 30 '18 at 9:45
AlphonseAlphonse
2,178623
$endgroup$
add a comment |
$begingroup$
Let $R$ be a reduced finitely generated algebra over $Bbb Z$. Let $T$ be a finite set of prime ideals of $R$.
Let $S = bigcap_{p notin T} R setminus p$.
1) Is it true that $A := S^{-1}R$ is finitely generated as an algebra over $Bbb Z$?
2) Is it true that $B := bigcap_{p not in T} ((R setminus p)^{-1}R)$ is finitely generated as an algebra over $Bbb Z$?
It seems to work if $R$ is the ring of integers of a number field, and we have $A cong B$. I am wondering about the general case. Maybe I should add some assumptions to make this true (e.g. $R$ integral domain...).
commutative-algebra localization finitely-generated
asked Nov 30 '18 at 9:45
AlphonseAlphonse
2,178623
$endgroup$
add a comment |
$begingroup$
Let $R$ be a reduced finitely generated algebra over $Bbb Z$. Let $T$ be a finite set of prime ideals of $R$.
Let $S = bigcap_{p notin T} R setminus p$.
1) Is it true that $A := S^{-1}R$ is finitely generated as an algebra over $Bbb Z$?
2) Is it true that $B := bigcap_{p not in T} ((R setminus p)^{-1}R)$ is finitely generated as an algebra over $Bbb Z$?
It seems to work if $R$ is the ring of integers of a number field, and we have $A cong B$. I am wondering about the general case. Maybe I should add some assumptions to make this true (e.g. $R$ integral domain...).
commutative-algebra localization finitely-generated
asked Nov 30 '18 at 9:45
AlphonseAlphonse
2,178623
$endgroup$
Let $R$ be a reduced finitely generated algebra over $Bbb Z$. Let $T$ be a finite set of prime ideals of $R$.
Let $S = bigcap_{p notin T} R setminus p$.
1) Is it true that $A := S^{-1}R$ is finitely generated as an algebra over $Bbb Z$?
2) Is it true that $B := bigcap_{p not in T} ((R setminus p)^{-1}R)$ is finitely generated as an algebra over $Bbb Z$?
It seems to work if $R$ is the ring of integers of a number field, and we have $A cong B$. I am wondering about the general case. Maybe I should add some assumptions to make this true (e.g. $R$ integral domain...).
commutative-algebra localization finitely-generated
commutative-algebra localization finitely-generated
asked Nov 30 '18 at 9:45
AlphonseAlphonse
2,178623
asked Nov 30 '18 at 9:45
AlphonseAlphonse
2,178623
edited Nov 30 '18 at 9:57
Alphonse
asked Nov 30 '18 at 9:45
AlphonseAlphonse
2,178623
asked Nov 30 '18 at 9:45
AlphonseAlphonse
2,178623
asked Nov 30 '18 at 9:45
AlphonseAlphonse
2,178623
2,178623
add a comment |
add a comment |
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