Timeline for Does regularity of a prime ideal in the fibre imply regularity of the prime?
Current License: CC BY-SA 2.5
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| Oct 16, 2010 at 3:17 | comment | added | BCnrd | Harry, the general principle underlying Matt's answer is that if $f:X \rightarrow S$ is an flat map of finite type between noetherian schemes then for many interesting "homological" properties $P$ of local noetherian rings the validity of $P$ for $O_{X,x}$ is equivalent to the same for $O_{S,s}$ and $O_{X_s,x}$ together, where $s = f(x)$. This is just a principle rather than a "meta-theorem" (whatever that may mean), but it (or a mild variant) is true for many $P$. Look at the section "flatness and fibers" in Matsumura's "Commutative Ring Theory" for examples (including regularity). | |
| Oct 16, 2010 at 3:05 | history | edited | Emerton | CC BY-SA 2.5 |
added 750 characters in body; edited body
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| Oct 16, 2010 at 3:02 | comment | added | Hailong Dao | Matt, beautiful answer, as usual! | |
| Oct 16, 2010 at 2:54 | vote | accept | Harry Gindi | ||
| Oct 16, 2010 at 2:52 | history | answered | Emerton | CC BY-SA 2.5 |