Mixing
Is $mathbb{A}^3 - {0}$ a separated variety?
$begingroup$
To check if $X=mathbb{A}^n -{0}$ is separated is equivalent to check that the diagonal is closed. In this case I think that: $X times X= {(x,y)| x,y in X }$ then $Delta(X)=Z(y-x)- {0}subset mathbb{A}^n $ but i'm not sure if this is closed or not. I think that it is open can you confirm that?
algebraic-geometry
asked Feb 3 at 0:13
andresandres
2439
$endgroup$
add a comment |
$begingroup$
To check if $X=mathbb{A}^n -{0}$ is separated is equivalent to check that the diagonal is closed. In this case I think that: $X times X= {(x,y)| x,y in X }$ then $Delta(X)=Z(y-x)- {0}subset mathbb{A}^n $ but i'm not sure if this is closed or not. I think that it is open can you confirm that?
algebraic-geometry
asked Feb 3 at 0:13
andresandres
2439
$endgroup$
add a comment |
$begingroup$
To check if $X=mathbb{A}^n -{0}$ is separated is equivalent to check that the diagonal is closed. In this case I think that: $X times X= {(x,y)| x,y in X }$ then $Delta(X)=Z(y-x)- {0}subset mathbb{A}^n $ but i'm not sure if this is closed or not. I think that it is open can you confirm that?
algebraic-geometry
asked Feb 3 at 0:13
andresandres
2439
$endgroup$
To check if $X=mathbb{A}^n -{0}$ is separated is equivalent to check that the diagonal is closed. In this case I think that: $X times X= {(x,y)| x,y in X }$ then $Delta(X)=Z(y-x)- {0}subset mathbb{A}^n $ but i'm not sure if this is closed or not. I think that it is open can you confirm that?
algebraic-geometry
algebraic-geometry
asked Feb 3 at 0:13
andresandres
2439
asked Feb 3 at 0:13
andresandres
2439
asked Feb 3 at 0:13
andresandres
2439
asked Feb 3 at 0:13
andresandres
2439
asked Feb 3 at 0:13
andresandres
2439
2439
add a comment |
add a comment |
2 Answers
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$begingroup$
There are several errors in your description of $Delta(X)$. First, it lives inside $Xtimes X$, not $Bbb A^n$. Secondly, you're using $x$ and $y$ for points of $X$ and then also as coordinates (when you subtract them), which doesn't really make sense. The second error is maybe not so bad, but the first is serious: it gives you the wrong answer. This variety is separated, as the diagonal copy of $X$ is in fact closed inside $Xtimes X$, exactly as the zero locus of $x_i-y_i$ where the $x_i$ and $y_i$ are coordinates on the first and second $X$, respectively.
answered Feb 3 at 0:34
KReiserKReiser
10.1k21435
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$begingroup$
The answer to the title question is yes: a locally closed subvariety of a separated variety is separated. Cf. these notes
answered Feb 3 at 0:39
BenBen
4,373617
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2 Answers
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2 Answers
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$begingroup$
There are several errors in your description of $Delta(X)$. First, it lives inside $Xtimes X$, not $Bbb A^n$. Secondly, you're using $x$ and $y$ for points of $X$ and then also as coordinates (when you subtract them), which doesn't really make sense. The second error is maybe not so bad, but the first is serious: it gives you the wrong answer. This variety is separated, as the diagonal copy of $X$ is in fact closed inside $Xtimes X$, exactly as the zero locus of $x_i-y_i$ where the $x_i$ and $y_i$ are coordinates on the first and second $X$, respectively.
answered Feb 3 at 0:34
KReiserKReiser
10.1k21435
$endgroup$
add a comment |
$begingroup$
There are several errors in your description of $Delta(X)$. First, it lives inside $Xtimes X$, not $Bbb A^n$. Secondly, you're using $x$ and $y$ for points of $X$ and then also as coordinates (when you subtract them), which doesn't really make sense. The second error is maybe not so bad, but the first is serious: it gives you the wrong answer. This variety is separated, as the diagonal copy of $X$ is in fact closed inside $Xtimes X$, exactly as the zero locus of $x_i-y_i$ where the $x_i$ and $y_i$ are coordinates on the first and second $X$, respectively.
answered Feb 3 at 0:34
KReiserKReiser
10.1k21435
$endgroup$
add a comment |
$begingroup$
There are several errors in your description of $Delta(X)$. First, it lives inside $Xtimes X$, not $Bbb A^n$. Secondly, you're using $x$ and $y$ for points of $X$ and then also as coordinates (when you subtract them), which doesn't really make sense. The second error is maybe not so bad, but the first is serious: it gives you the wrong answer. This variety is separated, as the diagonal copy of $X$ is in fact closed inside $Xtimes X$, exactly as the zero locus of $x_i-y_i$ where the $x_i$ and $y_i$ are coordinates on the first and second $X$, respectively.
answered Feb 3 at 0:34
KReiserKReiser
10.1k21435
$endgroup$
There are several errors in your description of $Delta(X)$. First, it lives inside $Xtimes X$, not $Bbb A^n$. Secondly, you're using $x$ and $y$ for points of $X$ and then also as coordinates (when you subtract them), which doesn't really make sense. The second error is maybe not so bad, but the first is serious: it gives you the wrong answer. This variety is separated, as the diagonal copy of $X$ is in fact closed inside $Xtimes X$, exactly as the zero locus of $x_i-y_i$ where the $x_i$ and $y_i$ are coordinates on the first and second $X$, respectively.
answered Feb 3 at 0:34
KReiserKReiser
10.1k21435
answered Feb 3 at 0:34
KReiserKReiser
10.1k21435
answered Feb 3 at 0:34
KReiserKReiser
10.1k21435
answered Feb 3 at 0:34
KReiserKReiser
10.1k21435
10.1k21435
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$begingroup$
The answer to the title question is yes: a locally closed subvariety of a separated variety is separated. Cf. these notes
answered Feb 3 at 0:39
BenBen
4,373617
$endgroup$
add a comment |
$begingroup$
The answer to the title question is yes: a locally closed subvariety of a separated variety is separated. Cf. these notes
answered Feb 3 at 0:39
BenBen
4,373617
$endgroup$
add a comment |
$begingroup$
The answer to the title question is yes: a locally closed subvariety of a separated variety is separated. Cf. these notes
answered Feb 3 at 0:39
BenBen
4,373617
$endgroup$
The answer to the title question is yes: a locally closed subvariety of a separated variety is separated. Cf. these notes
answered Feb 3 at 0:39
BenBen
4,373617
answered Feb 3 at 0:39
BenBen
4,373617
answered Feb 3 at 0:39
BenBen
4,373617
answered Feb 3 at 0:39
BenBen
4,373617
4,373617
add a comment |
add a comment |
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