Timeline for The SGA1 version of Riemann's Existence Theorem is about analytic spaces. How does one relate it to topological covering spaces?
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| Nov 29, 2015 at 5:54 | comment | added | nfdc23 | And yes, I was also pointing out that the discussion there in SGA1 is focusing on finite etale maps, rather than general etale maps (but I forgot to require in my first comment that the finite etale $X$-space has surjective structure map onto $X$, in order for it to match with covering spaces having finite fibers on the topological side). | |
| Nov 29, 2015 at 5:53 | comment | added | nfdc23 | There is no good notion of "unramified" for morphisms of general (locally) ringed spaces, as it involves a notion of $\Omega^1$ specific to each setting (schemes, formal schemes, rigid-analytic spaces, complex-analytic spaces, adic spaces, etc.). In each setting it is a hard theorem that "flat + unramified" (+ locally of finite presentation in some cases) is equivalent to other concepts (such as an infinitesimal criterion, or local analytic isomorphism in the complex-analytic case), and those are collectively called "etale". My first comment proves the categorical equivalence you want. | |
| Nov 29, 2015 at 3:36 | comment | added | Amy | How do you see the relationship, if you at all think that it's relevant, between flat unramified finite surjections among analytic spaces, and maps that are covering spaces? Are these really equivalent categories, or am I fundamentally misunderstanding something? | |
| Nov 29, 2015 at 3:35 | comment | added | Amy | @nfdc23: You said that "etale cover" means "surjective local analytic isomorphism", but I'm somewhat confused about what you meant. I was assuming that etale is defined for locally ringed spaces as being flat and unramified, and that SGA1 uses this definition of etale when it talks about finite etale surjection of analytic spaces. Are you saying that the definition I had in mind has the interpretation of being a "surjective local isomorphism"? Or are you just pointing out that in SGA1 he's talking about finite etale surjections, and not just etale maps? | |
| Nov 28, 2015 at 21:00 | comment | added | nfdc23 | @BenWebster: The rest of my comment gives the proof! | |
| Nov 28, 2015 at 20:59 | comment | added | Ben Webster♦ | @nfdc23 Why is that first sentence (of your first comment) correct? It seems like that's the whole point of the question. | |
| Nov 28, 2015 at 20:59 | comment | added | nfdc23 | @Amy: An "etale cover" is not the notion you intend; you need to impose a condition akin to being a covering space, whereas "etale cover" means "surjective local analytic isomorphism". So you can focus on "finite etale" maps (where "finite" in the analytic setting is classified via coherent sheaves of algebras, in the style of the algebraic setting once one has the theory of coherent analytic sheaves in hand, as is taken for granted in that discussion in SGA1; the book "Coherent Analytic Sheaves" provides such background, as do Seminaire Cartan lectures by Houzel). | |
| Nov 28, 2015 at 20:57 | answer | added | Ben Webster♦ | timeline score: 1 | |
| Nov 28, 2015 at 20:55 | comment | added | nfdc23 | @WilliamChen: For any complex-analytic space $X$, the functor $(Y \rightarrow X) \rightsquigarrow (|Y|\rightarrow|X|)$ from finite etale $X$-spaces to finite-degree covering spaces of the topological space $|X|$ is an equivalence. (By "etale" I mean "local analytic isomorphism".) Full faithfulness implies the rest (as it permits to work locally on $X$!), and an $X$-morphism $Y' \rightarrow Y$ is a section to the finite etale $p_2:Y' \times_X Y\rightarrow Y'$, so a clopen in $Y'\times_X Y$ that maps bijectively onto $Y'$. Since $|\cdot|$ commutes with analytic fiber products (!), we're done. | |
| Nov 28, 2015 at 20:41 | comment | added | Ben Webster♦ | @WilliamChen That's totally correct. | |
| Nov 28, 2015 at 20:39 | comment | added | Will Chen | @Amy Even when $X$ is singular, a topological cover is a local homeomorphism, so surely you should still be able to lift the complex structure on $X$ to one on $Y$? Is this wrong? | |
| Nov 28, 2015 at 20:34 | comment | added | Amy | True. I guess there are two issues I'm still unclear about. One is it is somewhat unclear to me why if $Y\rightarrow X$ is a priori a topological cover, the resulting map of analytic spaces is etale. But I guess that's a somewhat easy technical proof... The second issue is -- does this hold for $X$ singular? In that situation your argument no longer holds... | |
| Nov 28, 2015 at 20:29 | comment | added | pro | If $X$ is, say, a complex manifold, and $Y \to X$ is a topological cover, then $Y$ inherits a complex manifold structure. | |
| Nov 28, 2015 at 20:24 | review | First posts | |||
| Nov 28, 2015 at 20:44 | |||||
| Nov 28, 2015 at 20:20 | history | asked | Amy | CC BY-SA 3.0 |