---
date: 2022-01-23 02:14
modification date: Sunday 23rd January 2022 02:14:39
title: unramified
aliases: [unramified morphism]

---

Tags: 
#AG 
Refs:
[scheme](Unsorted/scheme.md)

# unramified

Idea: generalized a covering map for Riemann surfaces, which can fail to be topological covering maps due to ramification at a finite set of points.

- A morphism $f: B\to A$ of *rings* is **unramified** iff it is [finite type](Unsorted/finite%20type.md) and the sheaf of [Kahler differentials](Unsorted/algebraic%20de%20Rham%20cohomology.md) $\Omega_{A/B}$ vanishes.
- A morphism $f:X\to Y$ of *schemes* is **unramified** iff there exist affine opens $U \subseteq X, f(U) \subseteq V\subseteq f(X)$ where the induced ring morphism $U = \spec B \to V=\spec A$ is unramified.
	- Equivalently, $f$ is **unramified** iff $f$ is [locally of finite type](Unsorted/finite%20type.md) and $\Omega_{A/B}$ vanishes.
- Equivalently, $f: X\to Y$ is **locally unramified at a point** $y\in Y$ if $\OO_{Y, y} / \mfm_x \OO_{Y, y}$ is a finite separable field extension of $\OO_{X, x}/\mfm_x$ where $x=f(y)\in X$, and is **unmraified** if of [[finite type]] and everywhere locally unramified.
	- Equivalently: if $f:X\to Y$ is locally of finite type, then the induced map on stalks $\OO_{Y, f(x)} \to \OO_{X, x}$ yields a separable extension of residue fields $\kappa(f(x)) \injects \OO_{X, x}/ \mfm_{f(x)}\OO_{X, x}$
- A morphism of local rings $f:A\to B$ is ranumified iff $f(\mfm_A)B = \mfm_B$ and $\kappa(B)$ is a finite separable extension of $\kappa(A)$.
	- Upgrade to schemes by requiring it on stalks: $f\in\Sch(X, Y)$ is unramified if locally of finite type and every induced map $f_x^\sharp: \OO_{Y, f(x)}\to \OO_{X, x}$ is an unramified morphism of local rings.


In terms of Riemann surfaces:
![](attachments/Pasted%20image%2020221111145524.png)
![](attachments/Pasted%20image%2020221111145539.png)

Relation to [number fields](Unsorted/number%20field.md):

- Let $L/K$ with $\OO_L/\OO_K$, so $\iota: \OO_K \injects \OO_L$ corresponds to $\iota^*: \spec \OO_L\to \spec \OO_K$; when is this unramified? Let $q\in \spec \OO_L$ lie above $p\da q \intersect \OO_K$, then $\kappa(q)/\kappa(p)$ is always a finite separable extension since rings of integers of number fields are 1-dimensional and thus $p,q$ are either zero or maximal. In the latter case, $\kappa(p) = \OO_K/p, \kappa(q) = \OO_L/q$, and $\OO_k/p \to\OO_L/q$ is an integral extension of perfect fields and thus finite and separable.
	- Punchline: $\OO_K\to\OO_L$ is unramified iff for each $p = q \intersect \OO_K$, the map on localizations satisfies $p \OO_{L, q} = q$, so the ramification index of $p$ in $L$ is 1.

# Examples

![](attachments/Pasted%20image%2020221111145910.png)