I am working on Exercise 6.3.N. on Ravi Vakil's notes (on morphism from $S$ scheme to $\mathbb{P}_B^n$) and I would like some assistance. The exercise states: Let $B$ be a ring. If $X$ is a $B$-scheme, and $f_0, ..., f_n$ are $n+1$ functions on $X$ with no common zeros, then show that $[f_0, ..., f_n]$ gives a morphism of $B$-schemes $X \rightarrow \mathbb{P}_B^n$.
I would appreciate any hint, comments, etc. Thank you very much!
Edit (Not by OP) (10/06/2021). In the 2017 version of the notes, this is exercise 6.3.M.
If I denote the homogeneous coordinates on $\mathbf{P}^n_B$ by $x_0, \dots, x_n$ then $X_{f_i}$ should map into the affine open $D(x_i) \subset \mathbf{P}^n_B$. To specify this restriction you just need to give a $B$-homomorphism $B[x_0/x_i, \dots, x_n/x_i] \to \Gamma(X_{f_i}, \mathscr{O}_X)$. What's the right formula? Then check that these agree on overlaps.