Let $Z \subset Y \subset \mathbb{A^n}$ be a smooth subvarieties of $\mathbb{A^n}$. I'm trying to show that there is an exact sequence of normal bundles.

$0 \rightarrow N_{Z/Y} \rightarrow N_{Z} \rightarrow N_{Y}|_{Z} \rightarrow 0$

It seems obvious, but I can't figure out how things work in algebraic setting.

More precisely, let $I \subset J \subset k[x_1, ...x_n]$ be ideals defining Y and Z. Then,

$N_Z = \mathcal{Hom_Z(J/J^2, O/J)}$

$N_Y|_Z = \mathcal{Hom_Y(I/I^2, O/I) \otimes O/J}$

I need a natural map $N_Z \to N_Y|_Z$. And I think the natural restriction map

$\phi \mapsto \phi|_{I/I^2} \otimes 1$

is a candidates. But it is not well defined, and I stuck.

What's the problem? I appreciate any help.