Recall a path from a point $x$ to a point $y$ in a topological space $X$ is a continuous function $f$ from the unit interval $[0,1]$ to $X$ with $f(0) = x$ and $f(1) = y$. Now let $R$ be a commutative ring with $1$ and $Spec(R)$ be the set of all prime ideals of $R$ with the Zariski topology. It is well-known that if $P$ and $Q$ are two prime ideals of $R$ with $P\subseteq Q$, then there exists a path from $P$ to $Q$. Now let $\frak{p}_1$ and $\frak{p}_2$ be two minimal prime ideals of $R$. I am looking for equivalent conditions under which there exists a path between $\frak{p}_1$ and $\frak{p}_2$.
2
-
$\begingroup$ Shouldn't this just be equivalent to the existence of a zig-zag of specialisations from one to the other? $\endgroup$– R. van Dobben de BruynCommented Aug 17, 2020 at 17:52
-
$\begingroup$ @R. van Dobben de Bruyn: as you stated, if there exists a finite zig-zag of specializations from one to the other we are done. But does the existence of a path imply a finite zig-zag of specializations from one to the other? Also, infinite zig-zag of specializations (especially uncountable) is also a problem. $\endgroup$– AntonioCommented Aug 17, 2020 at 18:25
Add a comment
|