Remark 10.69.8. Let $k$ be a field. Consider the ring

$A = k[x, y, w, z_0, z_1, z_2, \ldots ]/ (y^2z_0 - wx, z_0 - yz_1, z_1 - yz_2, \ldots )$

In this ring $x$ is a nonzerodivisor and the image of $y$ in $A/xA$ gives a quasi-regular sequence. But it is not true that $x, y$ is a quasi-regular sequence in $A$ because $(x, y)/(x, y)^2$ isn't free of rank two over $A/(x, y)$ due to the fact that $wx = 0$ in $(x, y)/(x, y)^2$ but $w$ isn't zero in $A/(x, y)$. Hence the analogue of Lemma 10.68.7 does not hold for quasi-regular sequences.

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