Timeline for Finite unramified analytic coverings vs finite etale coverings
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19 events
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| Dec 30, 2010 at 2:30 | vote | accept | Hugo Chapdelaine | ||
| Dec 30, 2010 at 2:30 | history | bounty ended | Hugo Chapdelaine | ||
| Dec 29, 2010 at 15:53 | answer | added | Hugo Chapdelaine | timeline score: 1 | |
| Dec 29, 2010 at 5:33 | history | edited | Hugo Chapdelaine | CC BY-SA 2.5 |
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| Dec 29, 2010 at 5:07 | history | edited | Hugo Chapdelaine | CC BY-SA 2.5 |
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| Dec 27, 2010 at 22:53 | history | bounty started | Hugo Chapdelaine | ||
| Dec 27, 2010 at 16:33 | history | edited | Hugo Chapdelaine | CC BY-SA 2.5 |
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| Dec 26, 2010 at 3:59 | history | edited | Hugo Chapdelaine | CC BY-SA 2.5 |
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| Dec 26, 2010 at 0:56 | comment | added | BCnrd | The coherent sheaf $\mathcal{F}$ is a sheaf of algebras, not just a sheaf of modules. You also need the "analytic Spec" functor from coherent sheaves of algebras to finite analytic maps, and its compatibility with analytification of the analogous algebraic construction. (To express the compatibility, which has nothing to do with projectivity or properness, one first sets up the map over $\overline{X}^{\rm{an}}$ which is to be an isomorphism, and verifies it via a computation on completed stalks.) It's too unpleasant to get into more detail this way; look at Houzel's Seminaire Cartan exposes. | |
| Dec 26, 2010 at 0:54 | answer | added | Sándor Kovács | timeline score: 5 | |
| Dec 25, 2010 at 23:16 | comment | added | Hugo Chapdelaine | Dear BCnrd, <<So apply GAGA to get coherent sheaf of algebras on $\overline{X}$>> Ok so far so good. Then you wrote: <<then finite $\overline{X}$-scheme which must recover $\overline{Y}$>> Could you explain this in greater details. This I think is what I was looking for. So how do you get this finite $\overline{X}$-scheme structure on $\overline{Y}$ using as the only input data the coherent sheaf $\mathcal{F}$ on $\overline{X}$? If you could make this statement precise so that I can see (2) is used I guess this would answer my question :) | |
| Dec 25, 2010 at 22:51 | comment | added | BCnrd | Dear Hugo: My comment about compatibility of alg. structures is indeed a triviality; I just said it to make precise the statement you really want. Anyway, indeed existence of $\overline{f}$ is where all difficulties lie. Granting that, the argument is quite simple (assuming you admit a good general theory of coherent analytic sheaves). Namely, as I said above, finite analytic maps correspond to coherent sheaves of algebras. So apply GAGA to get coherent sheaf of algebras on $\overline{X}$, then finite $\overline{X}$-scheme which must recover $\overline{Y}$. Finite over proj. is proj. QED | |
| Dec 25, 2010 at 22:34 | history | edited | Hugo Chapdelaine | CC BY-SA 2.5 |
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| Dec 25, 2010 at 22:32 | comment | added | Hugo Chapdelaine | It would be nice to have some kind of "hands on" description of this coherent sheaf $\mathcal{F}^{an}$ since after all it comes from the push forward of a map which is finite unramified outside some analytic divisor. | |
| Dec 25, 2010 at 22:32 | comment | added | Hugo Chapdelaine | <<Anyway, the equality $\pi_1(X) = \pi_1(X^{\rm{an}})$ is very deep>> Well you see here I take for granted the existence of the map $\overline{f}$ so there is no need to use Grauert and Remmert constructions which I think guarantee the existence of such a map $\overline{f}$ compatible with $f$. Personally I think that the existence of the map $\overline{f}$ is deeper than what I'm asking for. So here the whole point is I take the existence of this map as granted! I think that from there the argument should be "clever homological algebra" but I cannot figure it out by myself! | |
| Dec 25, 2010 at 22:30 | comment | added | Hugo Chapdelaine | Hi BCnrd, <<And do you want that in such cases the alg. structure is also unique?>> Yes but I think that this follows from my setup. Once you know that $\overline{Y}$ is projective and by this I mean of course that this projective structure is compatible with the analyitc structure which is given on $\overline{Y}$ then we know that this projective structure is unique, this follows from the corollary on p. 30 of Serre's GAGA paper. <<you must intend to be a finite analytic morphism: proper with finite fibers.>> Yes of course I want that! I will add it | |
| Dec 25, 2010 at 21:25 | comment | added | BCnrd | Surely you want the alg. structure on $Y$ to be compatible with that on $X$ via the analytic $f$. And do you want that in such cases the alg. structure is also unique? (Recall that algebraizations, when they exist, are generally not unique in the non-compact case.) How much of the theory of coherent analytic sheaves are you willing to use? (For example, you must intend $\overline{f}$ to be a finite analytic morphism: proper with finite fibers. And finite analytic morphisms are "classified" by coherent sheaves of algebras...) Anyway, the equality $\pi_1(X) = \pi_1(X^{\rm{an}})$ is very deep. | |
| Dec 25, 2010 at 19:30 | history | edited | Hugo Chapdelaine | CC BY-SA 2.5 |
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| Dec 25, 2010 at 18:06 | history | asked | Hugo Chapdelaine | CC BY-SA 2.5 |