Why is $mathcal{O}_{V,W}$ a coordinate ring?
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I saw this claim: Let $Wsubset V$ be a codimension-$1$ irreducible subvariety of an $n$-dimensional normal, irreducible, affine variety $V$. Then $mathcal{O}_{V,W}$ — the coordinate ring at $W$— is a discrete valuation ring.
But why is $mathcal{O}_{V,W}$ a coordinate ring? A coordinate ring should be in the form of $k[x_1,...,x_n]/M$, but I can't see why is $mathcal{O}_{V,W}$ in that form? Also I feel confused with the definition of $mathcal{O}_{V,W}$.
algebraic-geometry
asked Nov 21 at 1:48
6666
1,219620
add a comment |
up vote
1
down vote
favorite
I saw this claim: Let $Wsubset V$ be a codimension-$1$ irreducible subvariety of an $n$-dimensional normal, irreducible, affine variety $V$. Then $mathcal{O}_{V,W}$ — the coordinate ring at $W$— is a discrete valuation ring.
But why is $mathcal{O}_{V,W}$ a coordinate ring? A coordinate ring should be in the form of $k[x_1,...,x_n]/M$, but I can't see why is $mathcal{O}_{V,W}$ in that form? Also I feel confused with the definition of $mathcal{O}_{V,W}$.
algebraic-geometry
asked Nov 21 at 1:48
6666
1,219620
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I saw this claim: Let $Wsubset V$ be a codimension-$1$ irreducible subvariety of an $n$-dimensional normal, irreducible, affine variety $V$. Then $mathcal{O}_{V,W}$ — the coordinate ring at $W$— is a discrete valuation ring.
But why is $mathcal{O}_{V,W}$ a coordinate ring? A coordinate ring should be in the form of $k[x_1,...,x_n]/M$, but I can't see why is $mathcal{O}_{V,W}$ in that form? Also I feel confused with the definition of $mathcal{O}_{V,W}$.
algebraic-geometry
asked Nov 21 at 1:48
6666
1,219620
I saw this claim: Let $Wsubset V$ be a codimension-$1$ irreducible subvariety of an $n$-dimensional normal, irreducible, affine variety $V$. Then $mathcal{O}_{V,W}$ — the coordinate ring at $W$— is a discrete valuation ring.
But why is $mathcal{O}_{V,W}$ a coordinate ring? A coordinate ring should be in the form of $k[x_1,...,x_n]/M$, but I can't see why is $mathcal{O}_{V,W}$ in that form? Also I feel confused with the definition of $mathcal{O}_{V,W}$.
algebraic-geometry
algebraic-geometry
asked Nov 21 at 1:48
6666
1,219620
asked Nov 21 at 1:48
6666
1,219620
edited Nov 21 at 2:09
asked Nov 21 at 1:48
6666
1,219620
asked Nov 21 at 1:48
6666
1,219620
asked Nov 21 at 1:48
6666
1,219620
1,219620
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
up vote
1
down vote
accepted
I suspect the confusion is a matter of terminology. If $W$ is a subvariety of $V$, then the coordinate ring of $V$ at $W$ isn't actually the coordinate ring of any variety, but instead it's defined to be the localization of the coordinate ring of $V$ at the prime ideal corresponding to $W$. I.e., if let $A$ be the coordinate ring of $V$, and $mathfrak{p}$ be the prime ideal of $A$ corresponding to the irreducible subvariety $W$, then $mathcal{O}_{V,W}:=A_{mathfrak{p}}$.
This is usually called the local ring of $V$ at $W$ though, so this terminology does strike me as a bit odd.
answered Nov 21 at 2:14
jgon
11.1k11839
So this is not related to the sheaf of regular functions, i.e. $mathcal{O}(V)$?
– 6666
Nov 21 at 2:29
1
Well, it's related, if you're familiar with sheaves, $mathcal{O}_{V,W}$ is the stalk of $mathcal{O}_V$ at the generic point of $W$.
– jgon
Nov 21 at 2:43
Just to complement the answer: if $i:Whookrightarrow V$ is the embedding of $W$ into $V$, the sheaf of regular functions of $W$ seen as a sheaf over $V$ is $i^*mathcal{O}_W$, or, abusing notation, $mathcal{O}_W$. This further shows that the notation is consistent. In this case the ring of regular functions of $W$ would again be $mathcal{O}_W(W)$.
– user347489
Nov 21 at 3:35
Is there a proof for that? can you tell any reference for a proof?
– 6666
Nov 21 at 6:58
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
I suspect the confusion is a matter of terminology. If $W$ is a subvariety of $V$, then the coordinate ring of $V$ at $W$ isn't actually the coordinate ring of any variety, but instead it's defined to be the localization of the coordinate ring of $V$ at the prime ideal corresponding to $W$. I.e., if let $A$ be the coordinate ring of $V$, and $mathfrak{p}$ be the prime ideal of $A$ corresponding to the irreducible subvariety $W$, then $mathcal{O}_{V,W}:=A_{mathfrak{p}}$.
This is usually called the local ring of $V$ at $W$ though, so this terminology does strike me as a bit odd.
answered Nov 21 at 2:14
jgon
11.1k11839
So this is not related to the sheaf of regular functions, i.e. $mathcal{O}(V)$?
– 6666
Nov 21 at 2:29
1
Well, it's related, if you're familiar with sheaves, $mathcal{O}_{V,W}$ is the stalk of $mathcal{O}_V$ at the generic point of $W$.
– jgon
Nov 21 at 2:43
Just to complement the answer: if $i:Whookrightarrow V$ is the embedding of $W$ into $V$, the sheaf of regular functions of $W$ seen as a sheaf over $V$ is $i^*mathcal{O}_W$, or, abusing notation, $mathcal{O}_W$. This further shows that the notation is consistent. In this case the ring of regular functions of $W$ would again be $mathcal{O}_W(W)$.
– user347489
Nov 21 at 3:35
Is there a proof for that? can you tell any reference for a proof?
– 6666
Nov 21 at 6:58
add a comment |
up vote
1
down vote
accepted
I suspect the confusion is a matter of terminology. If $W$ is a subvariety of $V$, then the coordinate ring of $V$ at $W$ isn't actually the coordinate ring of any variety, but instead it's defined to be the localization of the coordinate ring of $V$ at the prime ideal corresponding to $W$. I.e., if let $A$ be the coordinate ring of $V$, and $mathfrak{p}$ be the prime ideal of $A$ corresponding to the irreducible subvariety $W$, then $mathcal{O}_{V,W}:=A_{mathfrak{p}}$.
This is usually called the local ring of $V$ at $W$ though, so this terminology does strike me as a bit odd.
answered Nov 21 at 2:14
jgon
11.1k11839
So this is not related to the sheaf of regular functions, i.e. $mathcal{O}(V)$?
– 6666
Nov 21 at 2:29
1
Well, it's related, if you're familiar with sheaves, $mathcal{O}_{V,W}$ is the stalk of $mathcal{O}_V$ at the generic point of $W$.
– jgon
Nov 21 at 2:43
Just to complement the answer: if $i:Whookrightarrow V$ is the embedding of $W$ into $V$, the sheaf of regular functions of $W$ seen as a sheaf over $V$ is $i^*mathcal{O}_W$, or, abusing notation, $mathcal{O}_W$. This further shows that the notation is consistent. In this case the ring of regular functions of $W$ would again be $mathcal{O}_W(W)$.
– user347489
Nov 21 at 3:35
Is there a proof for that? can you tell any reference for a proof?
– 6666
Nov 21 at 6:58
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
I suspect the confusion is a matter of terminology. If $W$ is a subvariety of $V$, then the coordinate ring of $V$ at $W$ isn't actually the coordinate ring of any variety, but instead it's defined to be the localization of the coordinate ring of $V$ at the prime ideal corresponding to $W$. I.e., if let $A$ be the coordinate ring of $V$, and $mathfrak{p}$ be the prime ideal of $A$ corresponding to the irreducible subvariety $W$, then $mathcal{O}_{V,W}:=A_{mathfrak{p}}$.
This is usually called the local ring of $V$ at $W$ though, so this terminology does strike me as a bit odd.
answered Nov 21 at 2:14
jgon
11.1k11839
I suspect the confusion is a matter of terminology. If $W$ is a subvariety of $V$, then the coordinate ring of $V$ at $W$ isn't actually the coordinate ring of any variety, but instead it's defined to be the localization of the coordinate ring of $V$ at the prime ideal corresponding to $W$. I.e., if let $A$ be the coordinate ring of $V$, and $mathfrak{p}$ be the prime ideal of $A$ corresponding to the irreducible subvariety $W$, then $mathcal{O}_{V,W}:=A_{mathfrak{p}}$.
This is usually called the local ring of $V$ at $W$ though, so this terminology does strike me as a bit odd.
answered Nov 21 at 2:14
jgon
11.1k11839
answered Nov 21 at 2:14
jgon
11.1k11839
answered Nov 21 at 2:14
jgon
11.1k11839
answered Nov 21 at 2:14
jgon
11.1k11839
11.1k11839
So this is not related to the sheaf of regular functions, i.e. $mathcal{O}(V)$?
– 6666
Nov 21 at 2:29
1
Well, it's related, if you're familiar with sheaves, $mathcal{O}_{V,W}$ is the stalk of $mathcal{O}_V$ at the generic point of $W$.
– jgon
Nov 21 at 2:43
Just to complement the answer: if $i:Whookrightarrow V$ is the embedding of $W$ into $V$, the sheaf of regular functions of $W$ seen as a sheaf over $V$ is $i^*mathcal{O}_W$, or, abusing notation, $mathcal{O}_W$. This further shows that the notation is consistent. In this case the ring of regular functions of $W$ would again be $mathcal{O}_W(W)$.
– user347489
Nov 21 at 3:35
Is there a proof for that? can you tell any reference for a proof?
– 6666
Nov 21 at 6:58
add a comment |
So this is not related to the sheaf of regular functions, i.e. $mathcal{O}(V)$?
– 6666
Nov 21 at 2:29
1
Well, it's related, if you're familiar with sheaves, $mathcal{O}_{V,W}$ is the stalk of $mathcal{O}_V$ at the generic point of $W$.
– jgon
Nov 21 at 2:43
Just to complement the answer: if $i:Whookrightarrow V$ is the embedding of $W$ into $V$, the sheaf of regular functions of $W$ seen as a sheaf over $V$ is $i^*mathcal{O}_W$, or, abusing notation, $mathcal{O}_W$. This further shows that the notation is consistent. In this case the ring of regular functions of $W$ would again be $mathcal{O}_W(W)$.
– user347489
Nov 21 at 3:35
Is there a proof for that? can you tell any reference for a proof?
– 6666
Nov 21 at 6:58
So this is not related to the sheaf of regular functions, i.e. $mathcal{O}(V)$?
– 6666
Nov 21 at 2:29
So this is not related to the sheaf of regular functions, i.e. $mathcal{O}(V)$?
– 6666
Nov 21 at 2:29
1
1
Well, it's related, if you're familiar with sheaves, $mathcal{O}_{V,W}$ is the stalk of $mathcal{O}_V$ at the generic point of $W$.
– jgon
Nov 21 at 2:43
Well, it's related, if you're familiar with sheaves, $mathcal{O}_{V,W}$ is the stalk of $mathcal{O}_V$ at the generic point of $W$.
– jgon
Nov 21 at 2:43
Just to complement the answer: if $i:Whookrightarrow V$ is the embedding of $W$ into $V$, the sheaf of regular functions of $W$ seen as a sheaf over $V$ is $i^*mathcal{O}_W$, or, abusing notation, $mathcal{O}_W$. This further shows that the notation is consistent. In this case the ring of regular functions of $W$ would again be $mathcal{O}_W(W)$.
– user347489
Nov 21 at 3:35
Just to complement the answer: if $i:Whookrightarrow V$ is the embedding of $W$ into $V$, the sheaf of regular functions of $W$ seen as a sheaf over $V$ is $i^*mathcal{O}_W$, or, abusing notation, $mathcal{O}_W$. This further shows that the notation is consistent. In this case the ring of regular functions of $W$ would again be $mathcal{O}_W(W)$.
– user347489
Nov 21 at 3:35
Is there a proof for that? can you tell any reference for a proof?
– 6666
Nov 21 at 6:58
Is there a proof for that? can you tell any reference for a proof?
– 6666
Nov 21 at 6:58
add a comment |
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