Timeline for What is the completion at a family of ideals?
Current License: CC BY-SA 2.5
10 events
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| Sep 16, 2010 at 22:17 | answer | added | LK K | timeline score: 0 | |
| Aug 5, 2010 at 18:12 | comment | added | Ulrik Buchholtz | Just for reference: In other contexts, completion with respect to a family of ideals does occur, where one takes the directed system of finite (possibly repeated) products of ideals in the family, and then takes the limit of the resulting diagram. This nicely generalizes the completion wrt a single ideal. | |
| Aug 5, 2010 at 16:13 | answer | added | Carl Weisman | timeline score: 1 | |
| Aug 5, 2010 at 14:58 | comment | added | BCnrd | @Ricky: Kevin's interpretation is definitely the correct one. The construction in your question seems not interesting for its own sake. (Of course, sometimes in proofs one may pass to a completion, and then later to another, and so on, such as mixtures of formal and restricted power series, but I've never seen this general concept arise in a form for which the iterated process is intrinsically a topic of interest. The order of completions makes a huge difference; just try $x$-adic and $y$-adic completion of $k[x,y]$ for a field $k$.) | |
| Aug 5, 2010 at 14:48 | comment | added | Ricky | @Kevin That makes sense, I'll check whether the proof work with yours interpretation. In any case, the construction of taking repeatedly the completion is not interesting? | |
| Aug 5, 2010 at 14:40 | comment | added | Kevin Buzzard | Let me hypothesise that he does not mean to complete R at these prime ideals all at once, but rather one at a time (i.e. reduce to applying the case of a discrete valuating ring but possibly infinitely many times). | |
| Aug 5, 2010 at 14:37 | history | edited | Ben Webster♦ | CC BY-SA 2.5 |
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| Aug 5, 2010 at 14:30 | comment | added | Ricky | Let $R$ be a $p$-adically complete $\mathbb Z_p$-algebra. The sentence is "taking the completion of $R$ at the prime ideals of the special fiber, we can assume that it is a discrete valutation ring". | |
| Aug 5, 2010 at 14:23 | comment | added | BCnrd | Can you please fill in the rest of the sentence you cite so it is clear that the author doesn't simply mean "completion at each prime ideal such that..."? | |
| Aug 5, 2010 at 13:46 | history | asked | Ricky | CC BY-SA 2.5 |