Timeline for What is the formality behind passing from Number Fields to Number Rings
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 22, 2019 at 9:34 | comment | added | Daniel Loughran | The morphism $Spec(\mathcal O_K) \to Spec(\mathbb Z)$ is finite. But a finite morphism between normal schemes is smooth iff it is etale iff it is unramified. | |
| Jan 22, 2019 at 1:24 | comment | added | Tim Campion | @DanielLoughran I'm suddenly confused: a smooth morphism need not be unramified -- after all, smooth + unramified = etale, right? | |
| Jan 21, 2019 at 19:06 | vote | accept | CommunityBot | moved from User.Id=30211 by developer User.Id=481663 | |
| Jan 21, 2019 at 18:27 | answer | added | Tim Campion | timeline score: 5 | |
| Jan 20, 2019 at 22:40 | comment | added | user30211 | The unique regular model, please. | |
| Jan 20, 2019 at 21:21 | comment | added | Daniel Loughran | A reference for what exactly? Non-smoothness, or the unique regular model? | |
| Jan 20, 2019 at 17:47 | comment | added | user30211 | Could you provide a reference for this? | |
| Jan 20, 2019 at 17:15 | comment | added | Daniel Loughran | It is not smooth over $Spec(\mathbb{Z})$ due to ramification. But it is the unique regular model. | |
| Jan 20, 2019 at 16:38 | comment | added | Tim Campion | I'm trying to invoke standard ideas about normal schemes. Basically by definition, $Spec(\mathcal O_K)$ is the unique normalization of $Spec(K)$. But I'm suddenly doubtful of my claim that this is equivalent to smoothness for dimension 1 schemes. This is true for schemes defined over a perfect field, but I'm not sure it's true for schemes defined over $Spec(\mathbb Z)$. Maybe somebody who actually knows this stuff can weigh in. | |
| Jan 20, 2019 at 16:11 | comment | added | user30211 | Wow, that is perfect. I was not aware of this. Could you please provide a reference for reading about this? | |
| Jan 20, 2019 at 16:01 | comment | added | Tim Campion | Are you aware of the geometric interpretation? A field corresponds to a birational equivalence class of schemes. In the case of a number field, the ring of integers corresponds to the unique smooth (or equivalently normal) affine representative of this equivalence class. | |
| Jan 20, 2019 at 6:58 | history | edited | user30211 | CC BY-SA 4.0 |
added 57 characters in body
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| Jan 20, 2019 at 6:53 | history | asked | user30211 | CC BY-SA 4.0 |