Definition 10.50.13 (00IE)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.50: Valuation rings
Definition 10.50.13 (cite)

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Definition 10.50.13. Let $A$ be a valuation ring.

The totally ordered abelian group $(\Gamma , \geq )$ of Lemma 10.50.12 is called the value group of the valuation ring $A$ .
The map $v : A - \{ 0\} \to \Gamma$ and also $v : K^* \to \Gamma$ is called the valuation associated to $A$ .
The valuation ring $A$ is called a discrete valuation ring if $\Gamma \cong \mathbf{Z}$ .

Comments (2)

Comment #8751 by Anonymous on December 16, 2023 at 08:06

Pedantic point, but do you want to allow fields to be discrete valuation rings? For example, Tag 034X seems to consider fields as discrete valuation rings, but the value group will trivial.

Comment #9326 by Stacks project on May 31, 2024 at 14:49

No. I changed Lemma 10.120.17 in stead.

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