Timeline for Local characterization of semistability
Current License: CC BY-SA 2.5
13 events
| when toggle format | what | by | license | comment | |
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| Aug 1, 2010 at 10:33 | comment | added | Qing Liu | Fantastic ! EGA IV$_3$, Cor. 11.2.7 is exactly the answer to my question. Thanks ! | |
| Jul 31, 2010 at 20:09 | comment | added | BCnrd | @Q: Yes (assuming you meant flat map to be of finite presentation, and base to be affine). Much of EGA IV$_3$, section 11 is devoted to the thorny issue of descent of flatness through the limit game, the key being a result of Raynaud. I am traveling and don't have EGA on my laptop (and numdam download is too slow here), but should be easy to find required results there; try 11.2. Appendix C of the Thomason-T. article in Grothendieck Festschrift proves the awe-inspiring fact that every qcqs scheme is an invlim of finite type $\mathbf{Z}$-schemes with affine transition maps. | |
| Jul 30, 2010 at 18:46 | comment | added | Qing Liu | @B: there is one point in the limit argument for which I don't see obvious reasons: is it possible to descent a flat scheme $X\to S$ to a flat scheme over something essentially of finite type over $\mathbb Z$ ? | |
| Jul 30, 2010 at 2:59 | comment | added | BCnrd | If one sets up the formulation of the result with a general base, or really local base ring (not just dvr), then limit arguments allow us to reduce to the case of a base essentially of finite type over $\mathbf{Z}$, so Artin approximation can be applied. That is, with the proper formulation one has a structure theorem over any base at all but the real content is the excellent case; all explained in the Freitag-Kiehl book. Once the general case is done, can then specialize to a dvr base, having now avoided any excellence hypotheses on it. | |
| Jul 29, 2010 at 15:42 | comment | added | Qing Liu | But any étale morphism is open, so its image contains the generic points of the components passing through $x=y=0$. As the top curve is irreducible, this is impossible. To answer your original question, yes you have to work étale locally. If $V$ is excellent, using Artin's approximation theorem, for any singuar point $p$ of $X$, there exist two étale morphisms $U\to X$ and $U\to Spec(V[x,y]/(xy-\pi))$ containning $p$ and $(x=y=\pi=0)$ in their images. | |
| Jul 29, 2010 at 15:06 | comment | added | Ricky | I don't think so, for example all open immersions are étale of relative dimension 0, but are not étale cover. In any case I realize that in my question I really ask too much (and your example should work). In the case of field, the guess is true, but only étale locally: Néron Models, pag 246, says exactly this. | |
| Jul 29, 2010 at 15:00 | comment | added | Francesco Polizzi | Since the relative dimension is zero, any étale map is an étale cover, isn't it? | |
| Jul 29, 2010 at 14:42 | comment | added | Ricky | Sorry, I wrote "étale cover" but I meant "étale map", so I think that $\pi_1=0$ is not so interesting. | |
| Jul 29, 2010 at 14:36 | comment | added | Francesco Polizzi | It seems to me unlikely, at least over $\mathbb{C}$. In this case in fact xy=0 is a contractible topological space, so $\pi_1=\pi_1^{alg}=0$, so every étale cover is trivial. | |
| Jul 29, 2010 at 14:23 | comment | added | Ricky | I get your point, it could be that I have to work étale locally. Nevertheless I'm not sure that your example gives an answer: one should prove that there are no Zariski open subset of the nodal curve that are étale covering of $xy=0$. | |
| Jul 29, 2010 at 14:01 | history | undeleted | Francesco Polizzi | ||
| Jul 29, 2010 at 14:00 | history | deleted | Francesco Polizzi | ||
| Jul 29, 2010 at 13:58 | history | answered | Francesco Polizzi | CC BY-SA 2.5 |