Is $mathbb{A}^3 - {0}$ a separated variety?












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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?










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    $begingroup$


    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?










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    $endgroup$















      1












      1








      1





      $begingroup$


      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?










      share|cite|improve this question









      $endgroup$




      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






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      asked Feb 3 at 0:13









      andresandres

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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.






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            The answer to the title question is yes: a locally closed subvariety of a separated variety is separated. Cf. these notes






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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.






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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.






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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.






                  share|cite|improve this answer









                  $endgroup$



                  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.







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                  answered Feb 3 at 0:34









                  KReiserKReiser

                  10.1k21435




                  10.1k21435























                      0












                      $begingroup$

                      The answer to the title question is yes: a locally closed subvariety of a separated variety is separated. Cf. these notes






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                        0












                        $begingroup$

                        The answer to the title question is yes: a locally closed subvariety of a separated variety is separated. Cf. these notes






                        share|cite|improve this answer









                        $endgroup$
















                          0












                          0








                          0





                          $begingroup$

                          The answer to the title question is yes: a locally closed subvariety of a separated variety is separated. Cf. these notes






                          share|cite|improve this answer









                          $endgroup$



                          The answer to the title question is yes: a locally closed subvariety of a separated variety is separated. Cf. these notes







                          share|cite|improve this answer












                          share|cite|improve this answer



                          share|cite|improve this answer










                          answered Feb 3 at 0:39









                          BenBen

                          4,373617




                          4,373617






























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