I'm reading proposition 5.1.31(b) in Qing Liu's book. I use the version in his errata which can be found here: https://www.math.u-bordeaux.fr/~qliu/Book/Errata3/pages169-171.pdf.
Proposition 5.1.31(b). Let $Y = \operatorname{Proj} A[T_0,\dots,T_d]$ and $X$ be an $A$-scheme. For any invertible sheaf $\mathcal L$ on $X$ generated by $s_0,\dots,s_d$, there is a morphism $f:X\rightarrow Y$ and an isomorphism $\rho:\mathcal L\cong f^*\mathcal O_Y$ such that $\rho(X)(s_i) = f^*T_i$. Moreover, $f(X)$ is contained in a hyperplane if and only if $s_0,\dots,s_d$ is not free over $A$.
Qing first construct morphism $f_i:X_{s_i}\rightarrow D_+(T_i)$ and then glue them together.
My question:
In the construction of isomorphism $\mathcal L|_{X_{s_i}}\rightarrow f^*\mathcal O_Y(1)|_{D_+(T_i)}$, Qing just constructs a module map between their global sections. But $X_{s_i}$ may not be affine. In the non-affine case, morphism between sheaves is not determined by the module map. Also, I think the computation there is doubtable.
It seems that Qing only proves the second part in a set-theoretic way. And he only proves it for rational points not arbitrary point. Is there any scheme-theoretic way to state this part and prove it? My attempt is: $f:X\rightarrow Y$ factors through a closed immersion $H\rightarrow Y$ where $H = \operatorname{Proj}(A[T_0,\dots,T_d]/I)$ where $I$ is a linear homogeneous ideal. But I don't know how to prove it.
