Topological analogue of Zariski's Main Theorem?











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Zariski's main theorem says that over a nice base, a quasi finite separated morphism admits an open immersion into a finite morphism.



The rough topological translation I have in mind is that over a compact base, a universally cloesd bundle (continuous map) whose fibers are finite and discrete in the total space can be openly immersed in some sort of ramified cover. In other words, we can nicely arrange the sporadically spread fibers.



However:




  1. I don't see how to do something like this topologically.

  2. My topological translation is probably way off.


I remember there are topological translations of milder versions in Mumford's red book, but I would like a topological analogue of the modern (linked) version.



What is the topological picture here?






algebraic-geometry commutative-algebra schemes affine-geometry







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    up vote
    1
    down vote

    favorite












    Zariski's main theorem says that over a nice base, a quasi finite separated morphism admits an open immersion into a finite morphism.



    The rough topological translation I have in mind is that over a compact base, a universally cloesd bundle (continuous map) whose fibers are finite and discrete in the total space can be openly immersed in some sort of ramified cover. In other words, we can nicely arrange the sporadically spread fibers.



    However:




    1. I don't see how to do something like this topologically.

    2. My topological translation is probably way off.


    I remember there are topological translations of milder versions in Mumford's red book, but I would like a topological analogue of the modern (linked) version.



    What is the topological picture here?






    algebraic-geometry commutative-algebra schemes affine-geometry







    share|cite|improve this question
























      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      Zariski's main theorem says that over a nice base, a quasi finite separated morphism admits an open immersion into a finite morphism.



      The rough topological translation I have in mind is that over a compact base, a universally cloesd bundle (continuous map) whose fibers are finite and discrete in the total space can be openly immersed in some sort of ramified cover. In other words, we can nicely arrange the sporadically spread fibers.



      However:




      1. I don't see how to do something like this topologically.

      2. My topological translation is probably way off.


      I remember there are topological translations of milder versions in Mumford's red book, but I would like a topological analogue of the modern (linked) version.



      What is the topological picture here?






      algebraic-geometry commutative-algebra schemes affine-geometry







      share|cite|improve this question













      Zariski's main theorem says that over a nice base, a quasi finite separated morphism admits an open immersion into a finite morphism.



      The rough topological translation I have in mind is that over a compact base, a universally cloesd bundle (continuous map) whose fibers are finite and discrete in the total space can be openly immersed in some sort of ramified cover. In other words, we can nicely arrange the sporadically spread fibers.



      However:




      1. I don't see how to do something like this topologically.

      2. My topological translation is probably way off.


      I remember there are topological translations of milder versions in Mumford's red book, but I would like a topological analogue of the modern (linked) version.



      What is the topological picture here?






      algebraic-geometry commutative-algebra schemes affine-geometry




      algebraic-geometry commutative-algebra schemes affine-geometry






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Nov 21 at 22:16









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