Timeline for An unfamiliar (to me) form of Hensel's Lemma
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| Sep 28, 2013 at 12:27 | history | wiki removed | François G. Dorais | ||
| Feb 20, 2010 at 1:56 | history | edited | Wanderer | CC BY-SA 2.5 |
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| Feb 19, 2010 at 18:18 | history | edited | Wanderer | CC BY-SA 2.5 |
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| Feb 19, 2010 at 16:49 | comment | added | Jérôme Poineau | @AS: That's a great answer! Let me just point out that with virtually the same method, you can prove the same results for a larger class of fields (thus proving they are Henselian), namely fraction fields of local rings of analytic spaces. They are not Banach rings but inductive limits of them (for any r>0, take the completion of the ring of functions that converge on the disc of radius r with center at your point). If your space in Archimedean though, you won't have such a precise form of the theorem (you can say exactly what the radii are only thanks to the ultrametric triangle inequality). | |
| Feb 18, 2010 at 20:55 | comment | added | Qing Liu | Pete:for henselian DVR, the proof can be even simpler. In a neighborhood of x\in X(k), choose a regular parameter system f_1,...,f_d of X_k at x; lift them to F_1,..., F_d in O_{X}(U) in an open neighborhood U of x, and consider the closed subscheme V(F_1,...,F_d) of U. Shrinking U if necessary then this closed subscheme is a section of U lifting x (this is a special case of BLR "Néron Models", Cor. 9.1/9. | |
| Feb 18, 2010 at 19:11 | history | edited | Wanderer | CC BY-SA 2.5 |
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| Feb 18, 2010 at 17:05 | comment | added | Qing Liu | Let $y\in Y(k)$. After shrinking $Y$ around $y$, it becomes étale over some affine space $A$ over $S=Spec(R)$ (local structure of smooth morphisms). Let $S\to A$ be a section passing through the image of $y$ in $A$ and consider $Y\times_A S\to S$. It is étale with a section defined by $y$. Now it is one of the equivalent properties of henselian local rings that this section lifts to a section of $Y\times_A S$, which by projection to $Y$ gives a section lifting $y$. Conversely, this lifting property of étale schemes over $R$ characterizes henselian local rings. | |
| Feb 18, 2010 at 17:00 | comment | added | Qing Liu | AS: the surjection of $X(R)\to X(k)$ in the smooth locus can be found in my book (Cor. 6.2.13) for complete local rings $R$. The statement given by Pete is actually $Y(R)\to Y(k)$ is surjective for smooth schemes $Y$ over $R$ (the properness is for $X(K)=X(R)$ and the regularity if for $X(R)\subsetes X_{smooth}(R)$. In general, if $R$ is a local henselian ring with residue field $k$, and $Y$ is smooth over $R$, then the surjectivity holds. The proof comes in the next comment (no space left). | |
| Feb 18, 2010 at 14:03 | history | edited | Wanderer | CC BY-SA 2.5 |
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| Feb 18, 2010 at 13:39 | comment | added | Pete L. Clark | @AS: for instance, it is Lemma 9 in math.uga.edu/~pete/plclarkarxiv7.pdf. If you look there, you will be referred back to a 1985 paper of Jordan and Livne on Shimura curves, which is where I first learned about it (since my thesis work was on Shimura curves). I think the result is known to most people who seriously study curves over DVRs, so it should go back to the 1960s if not earlier. | |
| Feb 18, 2010 at 13:26 | history | edited | Wanderer | CC BY-SA 2.5 |
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| Feb 18, 2010 at 13:24 | comment | added | Wanderer | Hm, I like your last statement (I am not yet an arithmetic geometer :)). Could you give me some reference for it? In fact I expected to find this somewhere in Liu's book, but my first attempt to find it (using the index of the book) has been unsuccessful. | |
| Feb 18, 2010 at 13:11 | comment | added | Pete L. Clark | ...(See my first comment above). As a working arithmetic geometer, the version of HL that I use most often is the following: if K is a complete local field with valuation ring R and residue field k and X/R is a finite-type proper regular flat R-scheme, then the image of the reduction map X(K) = X(R) -> X(k) is precisely the set of smooth points in X(k). (This version is a consequence of Bost's nice formulation.) | |
| Feb 18, 2010 at 13:02 | comment | added | Pete L. Clark | @AS: Yes, it's interesting! [When you say Ribenboim, do you really mean Ribenboim -- as in the paper that Franz cited -- or Roquette, as in the paper that I cited?] I will say though that I do not believe that what you have written is the most general possible Hensel's Lemma, a quick corollary of my belief that the most general possible HL does not exist. For instance, this version of HL applies to a complete valuation ring; in (e.g.) Eisenbud's book there is a form of HL which applies to a complete, m-adically separated local ring. Neither class of rings contains the other... | |
| Feb 18, 2010 at 12:30 | history | edited | Wanderer | CC BY-SA 2.5 |
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| Feb 18, 2010 at 12:06 | history | edited | Wanderer | CC BY-SA 2.5 |
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| Feb 18, 2010 at 11:58 | history | edited | Wanderer | CC BY-SA 2.5 |
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| Feb 18, 2010 at 11:51 | history | edited | Wanderer | CC BY-SA 2.5 |
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| Feb 18, 2010 at 11:45 | history | answered | Wanderer | CC BY-SA 2.5 |