This is problem 7.3.O in Vakil's AG notes (July 31, 2023 version). I am not sure how to define the map locally.
Problem 7.3.O. Let $B$ be a ring and $X$ a $B$-scheme. Suppose $f_0,\cdots,f_n$ are n+1 functions on $X$ with no common zeros, then show that $[f_0,\cdots,f_n]$ gives a morphism of $B$-schemes $X\to\mathbb{P}_B^n$.
My approach. If I understand correctly, by values of a global section $f$ at a point $x\in X$, Vakil means the image of $f$ under the composition $$\Gamma(X,\mathcal{O}_X)\to\mathcal{O}_{X,p}\to\kappa(\mathfrak{p})$$ Now I want to glue morphisms defined on the open affine covering $\{U_i\cong\operatorname{Spec}A_i\}_{i\in I}$ described by these global sections $f_1,\cdots,f_n\in\Gamma(X,\mathcal{O}_X)$. So I start by defining the scheme morphism $f_i:U_i\to \mathbb{P}^n_B$: $$\mathfrak{p}\to [f_0(\mathfrak{p}),\cdots,f_n(\mathfrak{p})]=(f_i(\mathfrak{p})x_j-f_j(\mathfrak{p})x_i| i,j=0,1,\cdots,n)$$ I think this should be fine, since these $f_i$'s do not have common zero. But I am not sure what to do with their values $f_i(\mathfrak{p})\in \kappa(\mathfrak{p})$, as $\kappa(\mathfrak{p})\nsubseteq B$. If I could solve this issue, then I can define a homogeneous ideal corresponding to the homogeneous coordinate $[f_0(\mathfrak{p}),\cdots,f_n(\mathfrak{p})]$. And I can keep going and possibly prove that they agree on overlaps, hence glue to a morphism.
Any help is sincerely appreciated!