1
$\begingroup$

Assume $X$ to be a Notherian topological space such that any irreducible closed subset has a unique generic point. Consider $Y\subseteq X$ as a topological space with the induced topology from $X$. Is it true that any irreducible closed subset of $Y$ has a unique generic point?

What properties on $Y$ make above question to be true?

Any information even in the case that $X= \text{Spec R}$ with the Zariski topology (R is a Notherian ring) is also useful.

$\endgroup$
  • 2
    $\begingroup$ Open subspaces of sober spaces are sober, as are closed subspaces. In general, not every subspace is sober. See Lemma 1.2.5 in [Sketches of an elephant, Part C]. $\endgroup$ – Zhen Lin Jun 22 '14 at 17:08
4
$\begingroup$

Surely not: the subspace of $\text{Spec}(\mathbb{Z})$ consisting of nonzero primes, which is homeomorphic to $\mathbb{N}$ with the cofinite topology, is not sober. See also Wikipedia.

$\endgroup$
  • $\begingroup$ So if $M$ is not a finitely generated module, we still can not say any thing about being Sober of $Y=\text{Supp}_RM$ as a topological subspace of $X=\text{Spec} R$. $\endgroup$ – user49402 Jun 22 '14 at 18:21
  • $\begingroup$ Yes, for example the sum of all $\mathbb{Z}/(p)$ has support given by the nonzero primes $p$. $\endgroup$ – Todd Trimble Jun 22 '14 at 19:01

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.