In exercise §II.III.VIII. of Algebraic geometry and arithmetic curves by Qing.Liu. one is asked the following:
Let $X$ be a scheme over a ring $A.$ Let $f_0, \cdots,f_n\in\mathcal O_X(X)$ be such that the $f_{i,x}$ generate the unit ideal of $\mathcal O_{X,x}$ for every $x\in X.$ Show that $X=\cup_i X_{f_i}$ and that we have a morphism $f:X\rightarrow\mathbb P_A^n$ such that $f^{-1}(D_+(T_i))=X_{f_i}$ and that $f|_{X_{f_i}}$ is induced by the map $g:A[T_i^{-1}T_j]_j\rightarrow\mathcal O_X(X_{f_i})$ sending $T_i^{-1}T_j$ to $f_i^{-1}f_j.$ If $A=k$ is a field and $x\in X(k),$ determine $f(x)$ as well.
That $X=\cup_i X_{f_i}$ is easy, and I think the morphism $f$ comes
from the invertibility of $f_i$ in $\mathcal O_X(X_{f_i}),$
from the fact that morphisms into an affine scheme are in bijection with ring-homomorphisms of the rings of global sections,
from the composition of the above deduced map with the inclusion $D_+(T_i)\rightarrow\mathbb P_A^n,$
and finally from the compatibility of these maps.
But how could I show the compatibility of these maps? I am now confused by those entangling relations between morphisms and between schemes, and cannot see how to verify this compatibility.
Also, to determine $f(x),$ first identiy $\mathbb P_k^n(k)$ with $\mathbb P(k^{n+1}).$ Then, for $x\in X(k)\cap X_{f_i},$ we find $f(x)=(\alpha_0,\cdots,\alpha_n),$ where $\alpha_j$ is the image of $f_i^{-1}f_j$ in $k=k(x).$
Any hint or answer is welcomed.
P.S. Please forgive my cumbersomeness in this area, and help me improve upon it, as I am quite unfamiliar with algebraic geometry. Sincere thanks in advance. :)
For the compatibility note that $D_+(f_i) \cap D_+(f_j)=D_+(f_if_j)$ and also $\mathcal{O}_X(X_{f_i}) \cap \mathcal{O}_X(X_{f_j})=\mathcal{O}_X(X_{f_if_j})$.
Now you need to check that if we localize $g_i : A[T_i^{-1}T_k] \to \mathcal{O}_X(X_{f_i})$ at $\frac{T_j}{T_i}$ and $g_j : A[T_j^{-1}T_l] \to \mathcal{O}_X(X_{f_j})$ at $\frac{T_i}{T_j}$ we obtain the same map $A[(T_iT_j)^{-1}T_kT_l] \to \mathcal{O}_X(X_{f_if_j})$.