Timeline for preservation of localness among certain Krull domains
Current License: CC BY-SA 3.0
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| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
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| Sep 18, 2014 at 14:08 | comment | added | Neil Epstein | @Matthias The relevance of torsion in the class group is as follows. It is well known that $\mathfrak p$ is contained in the union of the other height one primes if and only if its class is non-torsion. If $\mathfrak p$ is not contained in said union (i.e. the class is torsion), then $\mathfrak p S$ is not a proper ideal of $S$. I was wondering (and presumably so was the OP at math.SE) whether, in the local case, the converse were true. | |
| Sep 17, 2014 at 19:30 | comment | added | Laurent Moret-Bailly | I agree with Mathias, but I can't prove anything yet. I suggest you look at the 2-dimensional case. In this case, if you denote by $U$ the complement of the closed point in $\mathrm{Spec}(R)$, then $S$ is the ring of global functions on $V:=U\smallsetminus\{\mathfrak{p}\}$, and I would rephrase Mathias's intuition by saying that $V$ might be affine (in which case, of course, $S$ is not local since $\mathrm{Spec}(S)=V$). | |
| Sep 17, 2014 at 16:54 | comment | added | Matthias Wendt | The ring $S$ does not look local to me. Using the characterization of normality, $S$ should contain the functions on $\operatorname{Spec}R$ which are defined away from the divisor corresponding to $\mathfrak{p}$. This looks much like the localization of a local ring, which in general has no reason for being local. Similarly, it seems to me that $\mathfrak{p}S$ would have a tendency of being everything. I have to admit that my intuition comes from the case where $\mathfrak{p}$ is principal, but I also do not see how class group statements would change much here... | |
| S Sep 17, 2014 at 16:43 | history | suggested | user26857 | CC BY-SA 3.0 |
fixed some Latex errors
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| Sep 17, 2014 at 16:22 | review | Suggested edits | |||
| S Sep 17, 2014 at 16:43 | |||||
| Sep 16, 2014 at 17:23 | history | asked | Neil Epstein | CC BY-SA 3.0 |