TITLE: Hartshorne Page 150, Theorem 7.1
QUESTION [9 upvotes]: Theorem 7.1 (a) says that- If $\phi$: $X \rightarrow \mathbb P_A^n $ is an $A$- morphism, then $\phi^*(\mathcal O(1)) $ is an invertible sheaf on $X$, which is generated by the global sections $s_i=\phi^*(x_i) $, $i$=0,1,...,n.
I do not know how to prove that the  global sections $s_i$ generate $ \phi ^*(\mathcal O(1)) $.
Also I have one more question in the proof of 7.1 (b)-
 While giving the ring homomorphism $A[y_0,...,y_n]$ $\rightarrow$ $\Gamma$($X_i$, $\mathcal   O_X{_i} $), I do not undrstand what is $ s_i / s_j $ . Here $s_i $ and $s_j$ are two global sections of an $ \mathcal O_X $- module. How to undersatnd that their quotient(?) is an element  of $\Gamma$ ($X_i$, $\mathcal O_X{_i}$). 
Can anyone please explain these things.

REPLY [6 votes]: The sections $s_i$ and $s_j$ are sections of an invertible sheaf (not just any old $\mathcal O_X$-module).  By definition $s_i$ is nowhere-zero on $X_i$, and so then $s_i$ forms a basis for the invertible sheaf over $X_i$, i.e. provides a trivialization
of the invertible sheaf over $X_i$.  We may thus write, on $X_i$, the $s_j = f s_i$ for some section $f$ of $\mathcal O_X$ over $X_i$.  For obvious reasons,
one denotes $f$ by $s_j/s_i$.

As for your first question, check that $x_0, \ldots, x_n$ generate $\mathcal O(1)$ on $\mathbb P^n$.  Then, check that this property is preserved under pull-back.