A parameter sequence regular in a hypersurface

For $r=1,2,3$, the coordinate localization $P_r=k[x_1,\dots ,x_r{]}_{(x_1,\dots ,x_r)}$ is Noetherian local. Its dimension is $r$: the coordinate-prime chain has length $r$, while localization cannot increase the dimension $r$ of the polynomial ring. The quotients $P_r\mathrm{/}(x_1,\dots ,x_r{)}^q$ are the degree-$<q$ truncated polynomial rings. Their inverse limit is $k\mathrm{⟦}{x}_1,\dots ,x_r\mathrm{⟧}$, since compatible truncations give exactly one coefficient per monomial and multiplication agrees coefficientwise. By the completion results in the Given paragraph, this series ring is Noetherian local of dimension $r$, with maximal ideal generated by the variables and residue field $k$. It is a domain: the lowest nonzero homogeneous parts of two nonzero series multiply to a nonzero polynomial over the field $k$.

Put $Q=k\mathrm{⟦}x,y,z\mathrm{⟧}$. Its quotient $A=Q\mathrm{/}(xy)$ is Noetherian local. Every prime of $Q$ containing $xy$ contains $x$ or $y$; any chain of such primes lies above one of these two ideals and so has length at most $2$, because $Q\mathrm{/}(x)\cong k\mathrm{⟦}y,z\mathrm{⟧}$ and $Q\mathrm{/}(y)\cong k\mathrm{⟦}x,z\mathrm{⟧}$ both have dimension $2$. Either branch also supplies a chain of length $2$, hence $\dim A=2$.

Suppose $(x+y)g\in (xy)$ in $Q$. Reducing modulo $(x)$ gives $yg\equiv 0$ in the domain $k\mathrm{⟦}y,z\mathrm{⟧}$, so $g\in (x)$; reducing modulo $(y)$ similarly gives $g\in (y)$. Coefficientwise, $(x)\cap (y)=(xy)$ in $Q$, so $g\in (xy)$. Hence $x+y$ is a nonzerodivisor on $A$.

The quotient $A\mathrm{/}(x+y,z)\cong k\mathrm{⟦}x\mathrm{⟧}\mathrm{/}(x^2)$ is nonzero and has only one prime, $(x)$, since $x$ is nilpotent and the quotient by $(x)$ is $k$. Thus $(x+y,z)A$ has radical the maximal ideal $(x,y,z)A$. Together with step 2.1, this makes $x+y,z$ a system of parameters.

After quotienting by $x+y$ one has $A\mathrm{/}(x+y)\cong k\mathrm{⟦}x,z\mathrm{⟧}\mathrm{/}(x^2)\cong k\mathrm{⟦}z\mathrm{⟧}\oplus xk\mathrm{⟦}z\mathrm{⟧}$ as a $k\mathrm{⟦}z\mathrm{⟧}$-module. Multiplication by $z$ is injective on both summands, and the final quotient is nonzero by step 3.1. Therefore $x+y,z$ is a regular sequence on $A$, as claimed.