On Liu's book Algebraic Geometry and Arithmetic Curves, he gives a general definition of the projective scheme $\mathbb{P}^n_A$, for $A$ a ring, as $Proj (B)$, for $B=A[x_0,\dots,x_n]$.
Later, he uses a canonical morphism $\pi:\mathbb{P}^n_A\rightarrow \text{Spec }A$. My question is about the nature of this morphism: is $\pi$ induced by the morphism $\text{Spec }(A[x_0,\dots,x_n])\rightarrow \text{Spec }A$, which is induced by the inclusion $A\rightarrow A[x_0,\dots,x_n]$?
Thank you.
Recall that $\mathbb{P}^n_A$ is the gluing of affine $n$-spaces over $A$ (namely $\mathrm{Spec} \bigl( A[\frac{x_0}{x_i},\dotsc,\frac{x_n}{x_i}]\bigr) \cong \mathbb{A}^n_A$), and observe that (by the very construction) the gluing isomorphisms at the intersections are morphisms over $A$. It follows that $\mathbb{P}^n_A$ is also an $A$-scheme.