The Stacks project

Proof. After shrinking $Y$ and $X$ we may assume that $i$ and $j$ are closed immersions. Denote $\mathcal{I} \subset \mathcal{O}_ X$ the ideal sheaf of $Y$ and $\mathcal{J} \subset \mathcal{O}_ X$ the ideal sheaf of $Z$. The conormal sequence is $0 \to \mathcal{I}/\mathcal{I}\mathcal{J} \to \mathcal{J}/\mathcal{J}^2 \to \mathcal{J}/(\mathcal{I} + \mathcal{J}^2) \to 0$. Let $z \in Z$ and set $y = i(z)$, $x = j(y) = j(i(z))$. Choose $f_1, \ldots , f_ n \in \mathcal{I}_ x$ which map to a basis of $\mathcal{I}_ x/\mathfrak m_ z\mathcal{I}_ x$. Extend this to $f_1, \ldots , f_ n, g_1, \ldots , g_ m \in \mathcal{J}_ x$ which map to a basis of $\mathcal{J}_ x/\mathfrak m_ z\mathcal{J}_ x$. This is possible as we have assumed that the sequence of conormal sheaves is split in a neighbourhood of $z$, hence $\mathcal{I}_ x/\mathfrak m_ x\mathcal{I}_ x \to \mathcal{J}_ x/\mathfrak m_ x\mathcal{J}_ x$ is injective.

Proof of (1). By Lemma 31.20.5 we can find an affine open neighbourhood $U$ of $x$ such that $f_1, \ldots , f_ n, g_1, \ldots , g_ m$ forms a quasi-regular sequence generating $\mathcal{J}$. Hence by Algebra, Lemma 10.69.5 we see that $g_1, \ldots , g_ m$ induces a quasi-regular sequence on $Y \cap U$ cutting out $Z$.

Proof of (2). Exactly the same as the proof of (1) except using More on Algebra, Lemma 15.30.12.

Proof of (3). By Lemma 31.20.5 (applied twice) we can find an affine open neighbourhood $U$ of $x$ such that $f_1, \ldots , f_ n$ forms a Koszul-regular sequence generating $\mathcal{I}$ and $f_1, \ldots , f_ n, g_1, \ldots , g_ m$ forms a Koszul-regular sequence generating $\mathcal{J}$. Hence by More on Algebra, Lemma 15.30.14 we see that $g_1, \ldots , g_ m$ induces a Koszul-regular sequence on $Y \cap U$ cutting out $Z$. $\square$