Timeline for Is every flat unramified cover of quasi-projective curves profinite?
Current License: CC BY-SA 2.5
16 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 29, 2010 at 4:35 | vote | accept | H. Hasson | ||
| Jan 29, 2010 at 3:54 | answer | added | Kirsten Wickelgren | timeline score: 10 | |
| Jan 28, 2010 at 6:30 | comment | added | Pete L. Clark | Yes, exactly. Unramifiedness is not preserved by inverse limits. And by "not preserved" I don't just mean "not always preserved" but "hardly ever preserved except in rather trivial cases". I believe the following is the basic idea (I hope I'm not being misled by the analogy to the topological case): the fiber of an inverse limit of finite maps is going to be quasi-compact, but the fiberwise criterion for unramifiedness shows that unramified + quasi-compact fibers implies finite fibers. | |
| Jan 28, 2010 at 6:24 | answer | added | S. Carnahan♦ | timeline score: 9 | |
| Jan 28, 2010 at 5:59 | comment | added | H. Hasson | Oh, I see now what you're saying. But that seems odd to me. If we allow non-locally-notherian schemes, then the inverse limit of an inverse system of finite covers will exist (because finite maps are affine). Are you saying one of flat or unramified doesn't go through? | |
| Jan 28, 2010 at 5:45 | history | edited | H. Hasson | CC BY-SA 2.5 |
added 10 characters in body
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| Jan 28, 2010 at 5:43 | comment | added | H. Hasson | Yes. In defining the etale fundamental group we only consider the finite covers, and there was a rumor at the time that one motivation for doing this is that all etale covers are inverse limits of their finite subcovers. You claim this is not true? Do you have a counterexample? | |
| Jan 28, 2010 at 5:32 | comment | added | Pete L. Clark | @HH: Please modify the question accordingly. But anyway, you don't seem to be taking my point: the category of etale covers of a curve is far from being closed under passage to inverse limits. I am asking for a nontrivial example of an etale covering of a curve which is not just a finite covering. Is your example an inverse limit of finite coverings? | |
| Jan 28, 2010 at 5:17 | comment | added | H. Hasson | Well, that's really the point - is there such a connected example such that it's not an inverse limit of finite covers? (the connectedness condition should really have been in the stated in the question.) | |
| Jan 28, 2010 at 4:54 | comment | added | Pete L. Clark | My point is that this concept is not really an analogue of infinite degree covering maps in the topological case. It would help if you could give a nontrivial example of the sort of morphism you have in mind, where by "trivial" I mean something like a map where the "covering scheme" has infinitely many connected components. In fact, it seems to me that even a map from an infinite disjoint union of copies of a complex curve down to the curve is a counterexample to your claim. | |
| Jan 28, 2010 at 3:05 | comment | added | H. Hasson | "an infinite degree algebraic cover" was an intuitive way to describe what I later specified as: a scheme mapping onto the given quasi-projective variety in a way that is flat and unramified but not finite. | |
| Jan 28, 2010 at 2:02 | comment | added | Pete L. Clark | What do you mean by an "infinite degree algebraic cover"? | |
| Jan 28, 2010 at 2:02 | comment | added | Pete L. Clark | @CB: Yes, I sure did -- thanks. I will delete the above comment and reproduce the first sentence below. | |
| Jan 28, 2010 at 1:59 | comment | added | Clark Barwick | I think you meant to say: a morphism is étale if and only if it is flat and unramified. | |
| Jan 27, 2010 at 20:23 | answer | added | user19475 | timeline score: 1 | |
| Jan 27, 2010 at 20:15 | history | asked | H. Hasson | CC BY-SA 2.5 |