Timeline for Are all ideals $I$ in the ring of smooth functions on a compact manifold M equal to a set of smooth functions that vanish in $Z \subset M$?
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| May 23, 2016 at 7:56 | comment | added | Mikhail Katz | @M10687 I am not sure what happened here: either I did not notice that the functions need to be continuous or I overlooked that the manifold is compact, I don't remember. | |
| May 22, 2016 at 17:10 | comment | added | M10687 | @MikhailKatz this is really old, but the claim that any maximal ideal is the ideal of functions vanishing at a point is true for compact $M$. See here: mathoverflow.net/questions/3871/… | |
| Jan 13, 2016 at 15:28 | comment | added | Mikhail Katz | @abx at any rate it is not true that every maximal ideal is the ideal of functions vanishing at a point. This usage of the term principal ideal can be found for instance in Hewitt, Edwin Rings of real-valued continuous functions. I. Trans. Amer. Math. Soc. 64, (1948), 45–99 where NONprincipal ideals called hyper-real ideals were first exploited. | |
| Jan 13, 2016 at 15:20 | comment | added | abx | You have a strange definition of a principal ideal! Look at "principal ideal" in Wikipedia. | |
| Jan 13, 2016 at 14:53 | comment | added | Mikhail Katz | @abx, the set of functions vanishing at a point $x\in M$ is by definition a principal ideal in the ring of functions. However, there are other maximal ones. | |
| Jan 13, 2016 at 14:51 | comment | added | abx | There is no mention of principal ideals in the OP question. | |
| Jan 13, 2016 at 14:29 | comment | added | Mikhail Katz | @abx, the relation is that the OP's question is based on an incorrect premise as I pointed out. | |
| Jan 13, 2016 at 14:09 | comment | added | abx | How is this related to the question? | |
| Jan 13, 2016 at 12:46 | history | answered | Mikhail Katz | CC BY-SA 3.0 |