The Stacks project

Lemma 110.14.1. There exists a local ring $R$ and a regular sequence $x, y, z$ (in the maximal ideal) such that there exists a nonzero element $\delta \in R/zR$ with $x\delta = y\delta = 0$.

Proof. Let $R = k[x, y, z] \oplus E$ where $E$ is the module above considered as a square zero ideal. Then it is clear that $x, y, z$ is a regular sequence in $R$, and that the element $\delta \in E/zE \subset R/zR$ gives an element with the desired properties. To get a local example we may localize $R$ at the maximal ideal $\mathfrak m = (x, y, z, E)$. The sequence $x, y, z$ remains a regular sequence (as localization is exact), and the element $\delta $ remains nonzero as it is supported at $\mathfrak m$. $\square$

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