Timeline for A question about Dedekind schemes and proper morphisms
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 21, 2020 at 16:09 | comment | added | Lao-tzu | Let us continue this discussion in chat. | |
| Jun 21, 2020 at 15:59 | comment | added | Asvin | Sorry, I think I misread your comment. I think the element we found in X(S) maps to the given point in X(K) under the canonical map. I also don't think we need to use X_S(S) to X_S(K) but currently our proof works only in this case. For instance, if you allow the affine line with double origin as a dedekind scheme and take X to be the affine line over it, injectivity is no longer true in the absolute sense. (I might be making mistakes since I am tired, please check everything I'm saying) | |
| Jun 21, 2020 at 15:39 | comment | added | Asvin | No that's right. This is why everything should really be relative to the structure maps to S which answers the other question you had. | |
| Jun 21, 2020 at 15:10 | comment | added | Lao-tzu | I think again about your answer, I think you found some element in 𝑋(𝑆) maps to 𝑋(𝐾), but may not under the canonical map 𝑋(𝑆)→𝑋(𝐾) induced by the inclusion of the generic point of $S$, e.g. that may happen if the image point $y$ of ${\rm Spec}\ K\to X\to S$ is not the generic point of $S$. Am I wrong somewhere? | |
| Jun 20, 2020 at 21:33 | comment | added | Asvin | Hi, I added details. Let me know if you have questions. | |
| Jun 20, 2020 at 21:32 | history | edited | Asvin | CC BY-SA 4.0 |
added 1257 characters in body
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| Jun 20, 2020 at 21:23 | comment | added | Lao-tzu | I used some results on extending rational morphisms and schematically density to show $X_S(S)\to X_S(K)$ is a bijection, but I still don't know if you are meant $X(S)\to X(K)$. Even so, I'm still wondering how to use the valuative criterion to fill in the finitely many missing point; I can't make it work. It would be grateful if you can also add that to your answer. | |
| Jun 20, 2020 at 18:12 | comment | added | Lao-tzu | OK, thanks, I will try to work it out myself and tell you if I still stuck. | |
| Jun 20, 2020 at 18:11 | comment | added | Asvin | I think it's useful to try and work it out yourself. Anyway, I will try and write more in a hour or two if you are still stuck (busy right now). | |
| Jun 20, 2020 at 18:09 | comment | added | Lao-tzu | OK, I think by "spread out" you mean extending morphisms as in Proposition 10.52 of Görtz-Wedhorn. | |
| Jun 20, 2020 at 18:02 | comment | added | Lao-tzu | Thanks! Injectivity agree! For surjectivity, I don't know how to "spread out" and "fill in the finitely many missing point", could you give more details? And do you use $X(S)\to X(K)$ or $X_S(S)\to X_S(K)$? | |
| Jun 20, 2020 at 17:57 | history | answered | Asvin | CC BY-SA 4.0 |