Compute normal sheaf from equations

Question

Assume that we are given $X\subset Y\subset Z$ locally complete intersections, given by $X: f_1,\dots, f_r$, $Y: g_1,\dots, g_s$ and we assume that $Z$ is some ambient $\mathbb A^n$ or $\mathbb P^n$. We can then write down the exact sequence of ideal sheaves $$0\to I_{Y/Z}\to I_{X/Z}\to I_{X/Y}\to 0$$ and over an open set $U$, $$I_{Y/Z}(U)=(g_1,\dots, g_s)\subset k[x_1,\dots,x_n],\\ I_{X/Z}(U)=(f_1,\dots,f_r)\subset k[x_1,\dots,x_n],\\ I_{X/Y}(U)=(f_1,\dots,f_r)\subset\frac{k[x_1,\dots,x_n]}{(g_1,\dots,g_r)}.$$ This seems to be very easy. The exact sequence gives rise to an exact sequence of normal sheaves $$0\to N_{X/Y}\to N_{X/Z}\to N_{Y/Z}$$ which doesn't seem to be that well understood. Can't we just deduce what the maps are on the normal sheaves just because we know them on the ideal sheaves? At least I would expect it to be possible. Unfortunately, I don't really know how to translate the explicit description for the ideal sheaves to one for the normal sheaves... Can anyone help me with that?

Answer 1

Let me deal with the affine case. If $X \subset Y$ then $I_Y \subset I_X$. This means there are functions $h_{i,j}$ such that for each $i$ one has $$ g_i = \sum_j h_{i,j} f_j. $$ Next, for complete intersections the normal bundles are trivial: $$ N_{X/Z} = \mathcal{O}_X^{\oplus r}, \qquad N_{Y/Z} = \mathcal{O}_Y^{\oplus s}. $$ The map $N_{X/Z} \to N_{Y/Z}\vert_X$ is then the map $\mathcal{O}_X^{\oplus r} \to \mathcal{O}_X^{\oplus s}$ is then just given by the restriction of the matrix $(h_{i,j})$ to $X$.