Projective space is Proj of a polynomial ring

The chart rings. The opens $D_{+}(x_i)$ cover: a homogeneous prime containing all $x_i$ contains $S_{+}=(x_0,\dots ,x_n)$ and is not in Proj. There is a graded isomorphism $S[x_i^{-1}]\cong A[y_{\mathrm{\ell }}:\mathrm{\ell }\mathrm{\ne }i][z,z^{-1}],x_i\mapsto z,x_{\mathrm{\ell }}\mapsto y_{\mathrm{\ell }}z,$ with $\mathrm{deg}z=1$ and $\mathrm{deg}{y}_{\mathrm{\ell }}=0$; the inverse sends $z$ to $x_i$ and $y_{\mathrm{\ell }}$ to $x_{\mathrm{\ell }}\mathrm{/}{x}_i$. Taking degree zero gives $S_{(x_i)}=A[x_{\mathrm{\ell }}\mathrm{/}{x}_i:\mathrm{\ell }\mathrm{\ne }i]$ with no relations among these variables beyond those of $A$. Thus $D_{+}(x_i)\cong U_i$ over $\mathrm{Spec}A$.

The overlap ring. For $i\mathrm{\ne }j$ the element $\tau =x_j\mathrm{/}{x}_i=x_j^{(i)}\in S_{(x_i)}$ has the property that the distinguished open $D(\tau )\subseteq D_{+}(x_i)$ is $D_{+}(x_i{x}_j)$, because a point of $D_{+}(x_i)$ has $x_j$ outside its homogeneous prime exactly when the degree-zero chart element $\tau =x_j\mathrm{/}{x}_i$ is outside the corresponding prime; thus $D(\tau )=D_{+}(x_i)\cap D_{+}(x_j)=D_{+}(x_i{x}_j)$. Localising the chart ring gives $S_{(x_i{x}_j)}=(S[(x_i{x}_j{)}^{-1}]{)}_0\cong (S_{(x_i)}{)}_{\tau }$, and symmetrically $(S_{(x_j)}{)}_{{\tau }^{\mathrm{\prime }}}\cong S_{(x_i{x}_j)}$ with ${\tau }^{\mathrm{\prime }}=x_i\mathrm{/}{x}_j$, so both $D_{+}(x_i)$ and $D_{+}(x_j)$ contain a copy of the same affine scheme $D_{+}(x_i{x}_j)$ as their overlap.

The case $n=0$. For $n=0$ we have $S=A[x_0]$, the single chart $D_{+}(x_0)$ covers by step 1.1 and $S_{(x_0)}=A$, so $\mathrm{Proj}S=D_{+}(x_0)=\mathrm{Spec}A$; by [F2] the glued ${\mathbb{P}}_A^0$ has the single chart $U_0=\mathrm{Spec}A$ and no gluing, so it too is $\mathrm{Spec}A$, and the identification is canonical.

Matching the transition maps on overlaps. Under the identifications of step 1.1 and step 1.2, the transition isomorphism between the two copies of $D_{+}(x_i{x}_j)$ inside $D_{+}(x_i)$ and $D_{+}(x_j)$ is induced by the inclusions $S_{(x_i)}\hookrightarrow S_{x_i{x}_j}$ and $S_{(x_j)}\hookrightarrow S_{x_i{x}_j}$, followed by the unique comparison of the two localisations; explicitly it sends $x_{\mathrm{\ell }}^{(i)}=x_{\mathrm{\ell }}\mathrm{/}{x}_i$ to $(x_{\mathrm{\ell }}\mathrm{/}{x}_j)\mathrm{/}(x_i\mathrm{/}{x}_j)=x_{\mathrm{\ell }}^{(j)}\mathrm{/}{x}_i^{(j)}$ expressed in the localisation $A[x_{\mathrm{\ell }}^{(j)}{]}_{x_i^{(j)}}$, that is, $x_{\mathrm{\ell }}^{(i)}\mapsto x_{\mathrm{\ell }}^{(j)}\mathrm{/}{x}_i^{(j)}$ and $x_j^{(i)}\mapsto 1\mathrm{/}{x}_i^{(j)}$. This is exactly the transition isomorphism in the gluing data of ${\mathbb{P}}_A^n$ [F2], on the same affine charts; the cocycle condition of the two gluing systems therefore agrees, so the isomorphisms of step 1.1 glue to an isomorphism of $\mathrm{Spec}A$-schemes $\mathrm{Proj}S\to {\mathbb{P}}_A^n$ by [F5].

Base change. Let $A\to B$ be a ring homomorphism. Then $S{\otimes }_{A}B=B[x_0,\dots ,x_n]$ with $\mathrm{deg}{x}_i=1$, and by [F3], [L1] one has $S_{(x_i)}{\otimes }_{A}B\cong (S{\otimes }_{A}B{)}_{(x_i)}=B[x_{\mathrm{\ell }}^{(i)}:\mathrm{\ell }\mathrm{\ne }i]$; the transition formulas of step 2.2 are identities in localized polynomial rings over the integers, so they base change to the same formulas over $B$, and the gluing data of $\mathrm{Proj}(A[x_0,\dots ,x_n]){\times }_{\mathrm{Spec}A}\mathrm{Spec}B$ and of $\mathrm{Proj}(B[x_0,\dots ,x_n])$ coincide; unicity of gluing [F5] gives the commutative square of the statement.

Conclusion. Step 2.2 gives the canonical isomorphism $\mathrm{Proj}A[x_0,\dots ,x_n]\cong {\mathbb{P}}_A^n$ over $\mathrm{Spec}A$, step 2.1 the case $n=0$, and step 3.1 the compatibility with $A\to B$; all constructions use only the inherited affine gluing and associated-sheaf interfaces, so the Axiom of Choice [A1] is inherited and not used again. [A1, step 2.1, step 2.2, step 3.1] \qed