8
$\begingroup$

More precisely, my question is as follows: Let $X$ be a qcqs scheme, $Z \subset X$ a closed subscheme and assume that there exists an open affine subscheme $U \subset X$ containing $Z$ such that $Z$ is a connected component of $U$. Does this imply that $Z$ is a connected component of $X$ ?

I do not know whether we need the algebro-geometric setup since this question only regards the underlying topological space of the scheme. Note that since $X$ is qcqs, every connected component is the intersection of its clopen supersets in $X$.

Improve this question
$\endgroup$
21
  • 3
    $\begingroup$ qcqs = quasi-compact quasi-separated. "quasi-compact" is just "compact" in the mainstream English terminology. quasi-separated is not just a condition on the underlying topological space, it involves the topology on $X\times X$, which is not determined by the topology on $X$. $\endgroup$
    – YCor
    Commented Feb 7, 2018 at 15:17
  • 2
    $\begingroup$ If $Z$ is a connected component of $U$, then $Z$ is both open and closed in $U$. Since $Z$ is open in $U$, and $U$ is open in $X$, also $Z$ is open in $X$. Since you also assume that $Z$ is closed, the scheme $Z$ is both open and closed in $X$. Since $Z$ is connected, the scheme $Z$ is a connected component of $X$. $\endgroup$
    – Jason Starr
    Commented Feb 7, 2018 at 16:26
  • 1
    $\begingroup$ @JasonStarr, so you used scheme-ness only to say that connected components are open, right? $\endgroup$
    – LSpice
    Commented Feb 7, 2018 at 16:51
  • 3
    $\begingroup$ @LSpice If the scheme is Noetherian, that is true. However, the OP does not specify that the scheme is Noetherian. $\endgroup$
    – Jason Starr
    Commented Feb 7, 2018 at 17:59
  • 2
    $\begingroup$ @AntonGeraschenko, I'm afraid that I don't understand some definition. I don't know what the deleted comb space is, but if a large space is connected, then surely no proper subset of it can be maximal connected? $\endgroup$
    – LSpice
    Commented Feb 7, 2018 at 18:16

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .