The Stacks project

Proof. Since $X$ (and hence $Y$) is locally Noetherian all 4 types of regular immersions agree, and moreover we may check whether a morphism is a regular immersion on the level of local rings, see Lemma 31.20.8. The implication (1) $\Rightarrow$ (2) is Lemma 31.21.7. The implication (2) $\Rightarrow$ (3) is Lemma 31.21.6. Thus it suffices to prove that (3) implies (1).

Assume (3). Set $A = \mathcal{O}_{X, x}$. Denote $I \subset A$ the kernel of the surjective map $\mathcal{O}_{X, x} \to \mathcal{O}_{Y, y}$ and denote $J \subset A$ the kernel of the surjective map $\mathcal{O}_{X, x} \to \mathcal{O}_{Z, z}$. Note that any minimal sequence of elements generating $J$ in $A$ is a quasi-regular hence regular sequence, see Lemma 31.20.5. By assumption the conormal sequence

is split exact as a sequence of $A/J$-modules. Hence we can pick a minimal system of generators $f_1, \ldots , f_ n, g_1, \ldots , g_ m$ of $J$ with $f_1, \ldots , f_ n \in I$ a minimal system of generators of $I$. As pointed out above $f_1, \ldots , f_ n, g_1, \ldots , g_ m$ is a regular sequence in $A$. It follows directly from the definition of a regular sequence that $f_1, \ldots , f_ n$ is a regular sequence in $A$ and $\overline{g}_1, \ldots , \overline{g}_ m$ is a regular sequence in $A/I$. Thus $j$ is a regular immersion at $y$ and $i$ is a regular immersion at $z$. $\square$