What structure do patch-topology closed subsets of a scheme have?

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It is well-known that when we have a spectral space $Y$ and a subset $E subseteq Y$ that is closed with respect to the patch topology on $Y$, then $E$, with the subspace topology induced by $Y$, is itself a spectral space.



Now let $X$ be a scheme. Since $X$ carries the structure of a topological space, we can still form the patch topology on $X$. Suppose that $E subseteq X$ is closed in the patch topology.




What can we say about the restriction of $X$ to $E$?




It is clear that whenever $E$ is an affine open subset (hence closed in the patch topology), we have an affine scheme. Also, trivially, whenever $E$ is contained in some open affine, the underlying set is a spectral space. But I fail to see much more.




schemes




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    It is well-known that when we have a spectral space $Y$ and a subset $E subseteq Y$ that is closed with respect to the patch topology on $Y$, then $E$, with the subspace topology induced by $Y$, is itself a spectral space.



    Now let $X$ be a scheme. Since $X$ carries the structure of a topological space, we can still form the patch topology on $X$. Suppose that $E subseteq X$ is closed in the patch topology.




    What can we say about the restriction of $X$ to $E$?




    It is clear that whenever $E$ is an affine open subset (hence closed in the patch topology), we have an affine scheme. Also, trivially, whenever $E$ is contained in some open affine, the underlying set is a spectral space. But I fail to see much more.




    schemes




    share|cite|improve this question





















      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      It is well-known that when we have a spectral space $Y$ and a subset $E subseteq Y$ that is closed with respect to the patch topology on $Y$, then $E$, with the subspace topology induced by $Y$, is itself a spectral space.



      Now let $X$ be a scheme. Since $X$ carries the structure of a topological space, we can still form the patch topology on $X$. Suppose that $E subseteq X$ is closed in the patch topology.




      What can we say about the restriction of $X$ to $E$?




      It is clear that whenever $E$ is an affine open subset (hence closed in the patch topology), we have an affine scheme. Also, trivially, whenever $E$ is contained in some open affine, the underlying set is a spectral space. But I fail to see much more.




      schemes




      share|cite|improve this question











      It is well-known that when we have a spectral space $Y$ and a subset $E subseteq Y$ that is closed with respect to the patch topology on $Y$, then $E$, with the subspace topology induced by $Y$, is itself a spectral space.



      Now let $X$ be a scheme. Since $X$ carries the structure of a topological space, we can still form the patch topology on $X$. Suppose that $E subseteq X$ is closed in the patch topology.




      What can we say about the restriction of $X$ to $E$?




      It is clear that whenever $E$ is an affine open subset (hence closed in the patch topology), we have an affine scheme. Also, trivially, whenever $E$ is contained in some open affine, the underlying set is a spectral space. But I fail to see much more.





      schemes





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      share|cite|improve this question




      share|cite|improve this question









      asked Jul 30 at 21:25









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