Example 10.119.4 (00PA)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.119: Around Krull-Akizuki
Example 10.119.4 (cite)

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Example 10.119.4. Consider the Noetherian local ring

$R = k[[x, y]]/(y^2)$

It has dimension 1 and it is Cohen-Macaulay. An example of an extension as in Lemma 10.119.3 is the extension

$k[[x, y]]/(y^2) \subset k[[x, z]]/(z^2), \ \ y \mapsto xz$

in other words it is gotten by adjoining $y/x$ to $R$ . The effect of repeating the construction $n > 1$ times is to adjoin the element $y/x^ n$ .

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