Timeline for Why do we use analytic coordinates to characterize singularity?
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Jun 10, 2014 at 13:23 | vote | accept | CommunityBot | ||
| Jun 10, 2014 at 1:33 | comment | added | user76758 | @user125763: trivial residue field extension at the marked points (so ensures induced map on completions from the etale morphism is an isomorphism). | |
| Jun 9, 2014 at 20:20 | comment | added | bananastack | @user76758: thanks! can I ask what residually trivial mean? | |
| Jun 9, 2014 at 19:43 | comment | added | user76758 | @user125763: It also follows from Artin approximation (the earlier paper of Artin...) that if $(X,x)$ and $(X',x')$ are pointed schemes of finite type over a field or excellent Dedekind domain $R$ and $f:\mathscr{O}_{X,x}^{\wedge} \simeq \mathscr{O}_{X',x'}^{\wedge}$ is an $R$-isomorphism then there is a common residually trivial pointed etale neighborhood $(X'',x'')$ of $(X,x)$ and $(X',x')$ which induces an isomorphism between those completed local rings that agrees with $f$ modulo whatever power of the maximal ideals you like. Same for arbitrary excellent schemes via Popescu's approx. thm. | |
| Jun 9, 2014 at 16:22 | comment | added | Francesco Polizzi | For formal coordinates the answer is yes. This follows from Corollary 1.6 of M. Artin's paper On the solutions of analytic equations.. | |
| Jun 9, 2014 at 15:58 | comment | added | bananastack | dear Francesco: could one use étale or formal coordinates instead? | |
| Jun 9, 2014 at 15:32 | history | edited | Francesco Polizzi | CC BY-SA 3.0 |
added 111 characters in body
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| Jun 9, 2014 at 15:13 | history | answered | Francesco Polizzi | CC BY-SA 3.0 |