The Stacks project

Remark 31.21.10. In the situation of Lemma 31.21.9 parts (1), (2), (3) are not equivalent to “$j \circ i$ and $j$ are regular immersions at $z$ and $y$”. An example is $X = \mathbf{A}^1_ k = \mathop{\mathrm{Spec}}(k[x])$, $Y = \mathop{\mathrm{Spec}}(k[x]/(x^2))$ and $Z = \mathop{\mathrm{Spec}}(k[x]/(x))$.