Complete Intersection and twist I'm trying to tackle exercise 5.3.3 of Liu's Book, and i have trouble with proving the following fact.
If $X=V_+(f_1,...,f_d)$ is of dimension n-d in $\mathbb P^n_k$ then if we call $Z=V_+(f_1,...,f_{d-1})$, and $I$ the ideal defining $X$ in $\mathcal O_Z $, then $I$ is isomorphic to $O_Z(-\deg f_d)$. Could anyone enlighten me?
Thx
 A: We can write $$X=V_+(f_1,\dots,f_d)=H_1\cap\dots\cap H_d=Z\cap H_d,$$ 
where $H_j=V_+(f_j)\subset \mathbb P^n$ has ideal sheaf $\mathscr I_{H_j}=(f_j)^\sim\subset \mathscr O_{\mathbb P^n}$. The ideal sheaf of $X$ is 
$$\mathscr I_X=\mathscr I_{H_1}+\dots +\mathscr I_{H_d}=(f_1,\dots,f_d)^\sim\subset\mathscr O_{\mathbb P^n}\,\,\,\,\,\,\,\,\,\,\,\,\,\rightsquigarrow\,\,\,\,\,\mathscr O_X=\mathscr O_{\mathbb P^n}/\mathscr I_X,$$
that of $Z$ is 
$$\mathscr I_Z=\mathscr I_{H_1}+\dots +\mathscr I_{H_{d-1}}=(f_1,\dots,f_{d-1})^\sim\subset\mathscr O_{\mathbb P^n}\,\,\,\,\,\rightsquigarrow\,\,\,\,\,\mathscr O_Z=\mathscr O_{\mathbb P^n}/\mathscr I_Z.$$
Let $S$ be the graded ring $k[x_0,\dots,x_n]$, so that $\mathscr O_{\mathbb P^n}=S^\sim$. There is an exact sequence
$$0\to S(-\deg f_d)\to S/(f_1,\dots,f_{d-1})\to S/(f_1,\dots,f_d)\to 0.$$
By applying the (exact) functor $(-)^\sim$, we get
$$0\to I\to \mathscr O_Z\to \mathscr O_X\to 0,$$ the ideal short exact sequence of $X=Z\cap H_d$ inside $Z$. 
