Timeline for Etale coverings of certain open subschemes in Spec O_K
Current License: CC BY-SA 2.5
4 events
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| May 1, 2010 at 11:53 | comment | added | Georges Elencwajg | As to your last question: yes, there is a vast generalization of the above to higher dimensional varieties, due to Grauert-Remmert. You can read about it in SGA1 in the very interesting Exposé 12 by Mrs. M.Raynaud (not to be confused with Mr. Michel Raynaud ! ). | |
| May 1, 2010 at 11:19 | comment | added | Georges Elencwajg | Yes, dear Ariyan, you're right: "ramified" here means "POSSIBLY ramified". (This is abuse of language.) And a morphism $F:(Y,\pi) \to (Y', \pi ') $ in the category $\mathcal RevRam (X;D)$ gets sent to its restriction $F _0 : Y_0=\pi^{-1} (X_0) \to Y'_0=(\pi ')^{-1} (X_0)$, a morphism in the category $\mathcal Rev (X_0)$. Amusingly, I had asked myself in how much detail I should post my answer. I opted for terseness for the sake of fluidity of style, but maybe fluidity turned into sloppiness... | |
| Apr 29, 2010 at 21:37 | comment | added | Ariyan Javanpeykar | 1. Shouldn't the objects of RevRam(X,D) be finite $\textbf{possibly}$ ramified coverings of $X$? 2. I'm guessing a morphism in RevRam(X,D) from $(Y,\pi)$ to $(Y^\prime,\pi^\prime)$ is a commutative diagram and similarly for Rev$(X_0)$. 3. How does the restriction functor act on morphisms? Given $(Y,\pi)$ in RevRam(X,D), I send this to $(\pi^{-1}(X_0),\pi|_{\pi^{-1}(X_0)}$, right? I don't see how I get a morphism in Rev$(X_0)$, though. Unless I demand something extra for morphisms in RevRam(X,D). I think this is really interesting. If true, can't one show eq. of cat. for sm. proj. varieties? | |
| Apr 28, 2010 at 22:46 | history | answered | Georges Elencwajg | CC BY-SA 2.5 |