I recently posted a very similar question, but I hid the question I really wanted answered in it. I'm posting this to make that question explicit.
Let $A$ be a nonzero commutative ring with unit. Define $\mathbb P_A^n$ to be the scheme $\operatorname {Proj} A[T_0,\dots,T_n]$, where the grading on the polynomial ring is by degree.
I would like to show that the only $n$ for which $\mathbb P_A^n$ is affine is $n=0$. This is exercise $3.11$ in chapter $2$ of Liu's book. I can show that for any $\mathbb P_A^n$, the sheaf of global sections is $A$. (This is the hint given in the book.) So it is easy to see that if $\mathbb P_A^n$ is affine, the only possibility is that it is isomorphic to $\operatorname{Spec} A$. One can directly verify that they are isomorphic for $n=0$. I would like to show that for larger $n$, $\mathbb P_A^n$ is not affine.
One way to do this is to proceed as in the question I linked in show that the $\mathbb P_A^n$ of different dimensions are non-isomorphic. Unfortunately, this seems to require cohomology theory, and Liu poses this problem before discussing cohomology.
Another way is to note that if $\mathbb P_A^n$ is affine, it is isomorphic to $\operatorname{Spec} A$ as a scheme over $A$. One can then use a base change argument, as sketched in a comment to the answer on the question I linked. I like this solution, but again, Liu has presented this problem before discussing base change.
Is there another more elementary way to show $\mathbb P_A^n$ is not affine for $n>1$, using the fact about global sections I noted above? I normally hate questions of the form "Can you prove $X$ with your hands tied behind your back?", so it pains me a little to be asking one, but I really have no idea what Liu intended. If it not possible to do this in a more elementary way, I would appreciate if someone experienced could confirm that. Thank you.
Take one of the affine pieces, say $D_+(T_0) \cong \mathbb{A}_A^n$. The inclusion $D_+(T_0) \subset \mathbb{P}_A^n$ would then be the morphism of affine schemes corresponding to the ring homomorphism $A \to A[T_1/T_0, \dots, T_n/T_0]$. If $A$ is non-zero then there should be some issue here.