Every closed immersion is a finite morphism. Can you give an example of quasi-projective varieties $X\subset Y$ such that inclusion $X\hookrightarrow Y$ is not finite? Same with Y projective?


Edit: Sorry this question is very simple, I made a mistake asking the question. For a corrected version, check out this one.


closed as no longer relevant by Harry Gindi, Gjergji Zaimi, Yemon Choi, Scott Morrison Mar 10 '10 at 2:06

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  • 2
    $\begingroup$ Yes I surely can. Is this homework? What did you try? $\endgroup$ – Kevin Buzzard Mar 9 '10 at 21:51
  • $\begingroup$ Oops... sorry. $\endgroup$ – Qing Liu Mar 9 '10 at 22:13
  • $\begingroup$ You really can't change the question like that. $\endgroup$ – Harry Gindi Mar 10 '10 at 0:38
  • $\begingroup$ Well, at least after you've already received an answer. $\endgroup$ – Harry Gindi Mar 10 '10 at 0:39
  • $\begingroup$ Should I delete and resubmit? $\endgroup$ – Paul Yuryev Mar 10 '10 at 0:39

An open immersion is never finite unless it is also a closed immersion (for finite morphisms are proper). So you just need to take a non-empty open subset $X$ which is not a connected component in $Y$.


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