Timeline for Do regular noetherian schemes of dimension one only have finitely many etale covers of bounded degree
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7 events
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| Jan 9, 2013 at 19:19 | comment | added | Masse | @pranavk. It seems that I underestimated the difficulty of this question. When you said "a zoo", it became clear to me that this question is more difficult than I had expected. As you can read in the question, I was motivated by the arithmetic and the geometric case. I was too optimistic and hoped for an easy "classification" of Dedekind schemes which would lead to a comprehensible answer. But it seems that this is too difficult. | |
| Jan 9, 2013 at 3:18 | comment | added | user30379 | @Masse: What is the goal? It seems misguided to do a "case-by-case analysis" when no goal has been articulated; one is just wandering around in the wilderness of general connected Dedekind schemes, which is a zoo. Why not guide yourself via something that actually uses modern algebraic geometry (such as the later parts of Qing Liu's excellent textbook about arithmetic curves, or the classification of smooth projective surfaces over an algebraically closed field, or Mumford's book on abelian varieties, or Neron models, ...)? You'll develop good taste for useful generalizations that way. | |
| Jan 8, 2013 at 15:07 | comment | added | Qing Liu | @Masse: pranavk's counterexample is dimension one and not affine. | |
| Jan 7, 2013 at 18:21 | comment | added | Masse | Your counterexample (an open affine subscheme of Spec $R$) is two-dimensional, no? Also, I'm not sure how to exclude such examples. It might be possible to do a case-by-case analysis. | |
| Jan 7, 2013 at 14:58 | comment | added | user30379 | Are you looking for "interesting" examples of connected Dedekind $\mathbf{Q}$-schemes admitting infinitely many non-isomorphic connected finite etale covers of some bounded degree? If $R$ is a 2-dimensional normal local noetherian domain (not necessarily henselian) whose residue field $k$ has characteristic 0 and admits infinitely many finite Galois extensions of degree $\le d$ then the complement of the closed point in ${\rm{Spec}}(R)$ is such a non-affine example. Such an $R$ need not admit a $k$-algebra structure (since no henselian hypothesis), so how would you rule it out? | |
| Jan 7, 2013 at 10:16 | comment | added | Piotr Achinger | As far as I know, the only known proof of this result for curves goes through the comparison with the topological fundamental group over $\mathbb{C}$. | |
| Jan 7, 2013 at 10:00 | history | asked | Masse | CC BY-SA 3.0 |