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Consider two (covering) flat morphisms: $$f: \operatorname{Spec} \mathbb{R}[y] \to \operatorname{Spec} \mathbb{R}[x], y^2=x$$ and $$g: \operatorname{Spec} \mathbb{Z}[i] \to \operatorname{Spec} \mathbb{Z}.$$

$f$ is ramified at $(x)$. The fiber over the point $(x-a)$ for $a>0$ has two points $(y\pm\sqrt{a})$, and the fiber over the point $(x-a)$ for $a<0$ has one point $(y^2-a)$ of degree $2$. The two patterns can be described by archimedean topology.

$g$ is ramified at $(2)$. The fiber over the point $(p)$ for $1\pmod4$ has two points $(a\pm bi)$, and the fiber over the point $(p)$ for $3\pmod4$ has one point $(p)$ of degree $2$. The two patterns can be described by 2-adic topology.

It seems that there is a good analogy between them but I can't find appropriate language to describe it.

EDIT: I found a related post. For the morphism $f$, the fiber can be varied continuously in each area $a>0$ and $a<0$, which can explain the number of the points of the fiber is constant in each area (2 and 1 respectively).

My most curious point is that: Is there some analogous interpretation for the morphism $g$?

KReiser
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aerile
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    Did you mean $x=y^2$ ? I don't know how class field theory works in abelian extensions of $k[x]$. My best knowledge is that (no need that the extension is abelian) $Gal(R/k[x])$ is a quotient of the fundamental group of $\overline{k}-a_1,\ldots,a_n$ the ramified points (monodromy group of the covering) – reuns Oct 18 '20 at 05:58
  • They're both integral extensions. https://encyclopediaofmath.org/wiki/Integral_extension_of_a_ring – Qiaochu Yuan Oct 18 '20 at 06:04
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    A formatting tip: using \operatorname{Spec} to format $\operatorname{Spec}$ produces better formatting, and I've made that adjustment for you in this post. You may also wish to include the top-level tag [tag:algebraic-geometry] in the future - this usually gets more exposure than schemes, covering-spaces, or ramification. – KReiser Oct 18 '20 at 06:16
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    I mean I don't know if there's much more to say than you have two morphisms of Dedekind schemes and that you can detect ramification at the completed local ring. – Alex Youcis Oct 18 '20 at 06:21
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    For a nice place to read about this, there is Chapter 1 in Neukirch's book Algebraic Number Theory – Brian Shin Oct 18 '20 at 20:50

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