Timeline for How does $f_* O_X$ measure ramification and Grothendieck-Riemann-Roch
Current License: CC BY-SA 2.5
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 29, 2010 at 11:09 | vote | accept | Ariyan Javanpeykar | ||
| May 9, 2010 at 17:54 | comment | added | Matthew Morrow | If your varieties are curves then $f_*O_X$, as an $O_Y$-module, locally looks like a finite extension of Dedekind domains $A/B$. And for any prime ideal $p$ in $A$ you can define the ramification index $e_p$ of $A/B$ at $p$. So surely $f_*O_X$ encodes all the ramification data? But maybe this isn't what you are after. In the higher dimensional case, you need to tell me what you mean by ramification (I usually study things over finite characteristic, where higher dimensional ramification theory is mysterious). | |
| May 9, 2010 at 10:39 | comment | added | Ariyan Javanpeykar | Alright. But I'm still fuzzy about my first question. Do you happen to know how one can get the ramification indices from f_* O_X ? | |
| May 8, 2010 at 20:43 | comment | added | Matthew Morrow | Yes, second Chern class sounds right. I did once ask an algebraic topologist about higher dimensional versions of the formula, and the concensus seems to be that there is no problem in theory, but the formula will get very very ugly as the dimension increases. | |
| May 8, 2010 at 20:35 | comment | added | Ariyan Javanpeykar | That's beautiful. If I remember correctly, here it's the second Chern class that comes into play. I would guess, for an n-dimensional Riemann-Hurwitz formula, the n-th Chern class would come into play. Anyway, I like this alot. Thnx. | |
| May 8, 2010 at 19:48 | history | answered | Matthew Morrow | CC BY-SA 2.5 |