Timeline for Understanding the completed stalk of the dualizing sheaf of a family of nodal curves at a node
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 14 at 18:22 | comment | added | user267839 | You mentioned that the description of local sections $\omega_{C}(U)$ in terms of residues on "normalision side" is "well known". Do you know a reference discussing this or could you briefly sketch the arguments involved to see it? | |
| May 3, 2017 at 18:06 | vote | accept | stupid_question_bot | ||
| Feb 19, 2017 at 1:03 | history | edited | stupid_question_bot | CC BY-SA 3.0 |
added 16 characters in body
|
| Feb 19, 2017 at 1:02 | answer | added | stupid_question_bot | timeline score: 3 | |
| Feb 18, 2017 at 22:57 | comment | added | stupid_question_bot | @nfdc23 Do you have a reference for the statement "formation of the dualizing sheaf is compatible with etale localization on the source"? Does this mean that if $u : U\rightarrow C$ is etale, then $u^*\omega_{C/S}$ is canonically isomorphic to $\omega_{U/S}$? | |
| Feb 18, 2017 at 21:45 | comment | added | nfdc23 | The completion has the merit of living over the ring whereas the standard model I described only has a pointed etale neighborhood in common. It is indeed convenient to be able to go back and forth (usually with justification by Artin approximation if a more elementary method doesn't come to mind). In deJong's paper on alterations he goes between these etale-local situations and completions, depending on the context. The formation of the dualizing sheaf is compatible with etale localization on the source; that should reduce whatever you need to computing with the "standard model" I mentioned. | |
| Feb 18, 2017 at 20:54 | comment | added | stupid_question_bot | @nfdc23 That is a good point, and I suppose this implies that by 'completion' I should always mean the 'completion of the strict henselization of the local ring' - ie, the completion of the etale local ring. I use completions because this paper always uses completions (this is Chapter 4 of Bertin-Romagny's "Champs de Hurwitz"), though they seem to speak of completions and etale local rings interchangeably, which is a bit annoying for me, though I suppose it is somewhat justified by your comment. Perhaps they use completions because it is easier to write power series than etale local functions? | |
| Feb 18, 2017 at 2:08 | comment | added | nfdc23 | Your description of the completed local ring is incorrect since $P$ may not have the same residue field as $s$ (though $k(P)/k(s)$ is always separable; not obvious!). Is there a reason you use completions instead of an etale neighborhood? Using Artin approximation, the book of Freitag & Kiehl on the Weil conjectures shows (Prop. 2.7, 2.8 in Ch. III) that there exists a residually trivial local-etale neighborhood ${\rm{Spec}}(A) \rightarrow {\rm{Spec}}(O_{S,s})$ and $a \in \mathfrak{m}_A$ such that $({\rm{Spec}}(A[x,y]/(xy-a), (\mathfrak{m}_A, (x,y)))$ shares an etale neighborhood with $(C,P)$. | |
| Feb 17, 2017 at 19:48 | history | asked | stupid_question_bot | CC BY-SA 3.0 |