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If $p$ and $q $ are two prime ideals of a commutative ring $R $ such that $p \subseteq q$, then we can easily define a continuous function (a path) $f$ from the unit interval $ [0,1]$ to the prime spectrum $ Spec (R)$, with the Zariski topology, such that $f (0)=p$ and $f (1)=q$. (Also, in general, if there exists a zigzag of specializations of prime ideals between $p $ and $q $, then we can find a path between $p $ and $q $). Now if there exists no zigzag of specializations of prime ideals between $p $ and $q $, I am looking for some conditions (on $p $ and $q$) under which there exists a path between $ p$ and $q $.

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    $\begingroup$ The components (in the partial-order sense, i.e., equivalence classes of the "exists a zigzag" relation) of the partial order of prime ideals are open and closed in the Zariski topology, so the image of a continuous map from $[0,1]$ (or from any connected space) cannot meet more than one of them. $\endgroup$
    – Andreas Blass
    Commented Mar 15, 2021 at 18:22
  • $\begingroup$ @AndreasBlass That's not quite right. If $V$ is a valuation ring of infinite rank, you can have the situation $\mathrm{Spec}(V)=\{0,1,\ldots,\infty\}$ with the topology for which the open sets are all $\{n,n+1,\ldots,\infty\}$. In that case there is a map $[0,1]\to \mathrm{Spec}(V)$ connecting $0$ to $\infty$: On $[0,1/2]$, map to $0$, on $(1/2,3/4]$ map to $1$, on $(3/4,7/8]$ map to $2$ etc., finally mapping $1$ to $\infty$. $\endgroup$
    – Peter Scholze
    Commented Mar 17, 2021 at 14:38
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    $\begingroup$ There is a rich theory of homotopy types of finite non-Hausdorff spaces: see ncatlab.org/nlab/show/finite+topological+space for some references. Similar phenomena will appear if you study homotopy types of Zariski spectra of rings. $\endgroup$
    – Neil Strickland
    Commented Mar 17, 2021 at 15:55
  • $\begingroup$ @PeterScholze I don't (yet) understand your comment. In your example, isn't the partial order of prime ideals just a linear order, so every two ideals are joined by a zigzag (a single zig or a single zag)? $\endgroup$
    – Andreas Blass
    Commented Mar 17, 2021 at 17:51
  • $\begingroup$ @AndreasBlass Oops, I'm sorry, you are of course right. But can you not build a version of this example that includes specializations back and forth? I think this ought to be a spectral space, still. $\endgroup$
    – Peter Scholze
    Commented Mar 17, 2021 at 19:42

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