Timeline for Adjunction for underlying reduced subschemes
Current License: CC BY-SA 2.5
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 3, 2010 at 23:51 | vote | accept | B. Cais | ||
| Feb 3, 2010 at 23:51 | comment | added | B. Cais | Sure, and in fact there are lots of examples that arise "in nature". For example, the regular proper model of the modular curve $X_1(N)$ over $\mathbf{Z}_p$ for $p$ dividing $N$ to high order (and $N$ divisible by some other sufficiently large prime, say, to avoid stacky issues) has closed fiber that is not reduced. However, CM is stable under base change, and regular implies CM, so... | |
| Feb 3, 2010 at 23:20 | comment | added | David E Speyer | On the other hand, every zero dimensional noetherian ring is CM, and almost none of them are reduced. So there are examples. | |
| Feb 3, 2010 at 23:19 | comment | added | David E Speyer | A basic comment: it is somewhat difficult to be CM and not reduced. This is because CM implies S_1, basically by definition. A result of Serre is that (R_0 and S_1) implies reduced. So any example must fail to be R_0, that is to say, it must not be generically reduced. | |
| Feb 3, 2010 at 19:49 | answer | added | Emerton | timeline score: 3 | |
| Feb 3, 2010 at 18:03 | history | asked | B. Cais | CC BY-SA 2.5 |