Timeline for Does there exist a classification of covering spaces in algebraic geometry?
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
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| May 15, 2010 at 16:54 | vote | accept | Akela | ||
| May 11, 2010 at 20:29 | comment | added | Charles Staats | See also <a href="mathoverflow.net/questions/21717/… answer</a>, and Ravi Vakhil's comment. | |
| May 11, 2010 at 18:59 | answer | added | Xandi Tuni | timeline score: 4 | |
| May 11, 2010 at 18:54 | history | edited | Akela | CC BY-SA 2.5 |
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| May 11, 2010 at 18:52 | comment | added | BCnrd | An instructive example beyond the "limit of finite etale" case: let $X$ be the nodal plane cubic. The infinite tree $X'$ of projective lines is etale over $X$ with covering group $\mathbf{Z}$, and for each $n \ge 1$ there's a unique $n$-gon intermediate cover $X_n$. This $X'$ is not "ind-finite-etale" over $X$, and is the analytification of the topological universal cover. So one has the "surprise" result that ${\rm{H}}^1(X,\mathbf{Z}) \ne 0$ (etale cohomology) whereas $\pi_1(X)$ has no nonzero cont. homs to $\mathbf{Z}$. Upshot: for non-normal $X$ some funny things happen. | |
| May 11, 2010 at 18:47 | comment | added | BCnrd | In para. 3 you meant "algebraic geometry"? Deep result of Grauert/Remmert: if $X$ is loc. f.type over $\mathbf{C}$ (no smoothness, no properness), analytification is equivalence from category of finite etale covers of $X$ to finite covering spaces of $X(\mathbf{C})$. Best analogue of universal cover: invlim $X'$ of pointed conn'd finite etale covers (for $X$ conn'd of f.type, say). This gigantic scheme is conn'd, no nontrivial conn'd finite etale covers, ${\rm{Aut}}(X'/X)$ is "opposite" to $\pi_1(X,x)$, & conj. classes of closed subgps correspond to conn'd "ind-finite-etale" covers of $X$. | |
| May 11, 2010 at 18:31 | history | asked | Akela | CC BY-SA 2.5 |