The Stacks project

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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View Example 10.119.4 as pdf