I'm reading Hartshorne Chapter 2.7.
Let $X$ be a scheme over $A$ and $f:X\rightarrow \mathbb P_A^n$ be an $A$ morphism. Define $\mathcal L = f^*(\mathcal O(1))$ and $s_i = f^*(x_i)$. Then $\mathcal L$ is an invertible sheaf and $s_i$ generate the sheaf $\mathcal L$.
I have 2 questions:
1.Why $\mathcal L$ is an invertible sheaf. In general, is that true that for any $f:X\rightarrow Y$ and locally free sheaf $\mathcal F$, we have $f^*\mathcal F$ is still locally free on $X$?
2.Why $s_i$ generate the sheaf $\mathcal L$? In general, let $f:X\rightarrow Y$ be any morphism and $\mathcal F$ be any sheaf on $Y$ and $s$ be a section of $\mathcal F$. Then, is there any relation between $f^*(s)_x$ and $s_x$?