Relative projective space from standard charts

Fix $n\ge 0$. For $i\in \{0,\dots ,n\}$ let $B_i=\mathbb{Z}[t_0,\dots ,t_n]\mathrm{/}(t_i-1)\cong \mathbb{Z}[x_{\mathrm{\ell }}^{(i)}:\mathrm{\ell }\mathrm{\ne }i],$ where $x_{\mathrm{\ell }}^{(i)}$ denotes the class of $t_{\mathrm{\ell }}\mathrm{/}{t}_i$, so that $U_i:=\mathrm{Spec}{B}_i$ is an affine scheme with coordinates $x_{\mathrm{\ell }}^{(i)}$ for $\mathrm{\ell }\mathrm{\ne }i$. For $i\mathrm{\ne }j$ let $D_j^{(i)}\subseteq U_i$ be the distinguished open where $x_j^{(i)}$ is invertible. Over it the formula $x_{\mathrm{\ell }}^{(i)}=\frac{t_{\mathrm{\ell }}}{t_i}=\frac{t_{\mathrm{\ell }}\mathrm{/}{t}_j}{t_i\mathrm{/}{t}_j}=\frac{x_{\mathrm{\ell }}^{(j)}}{x_i^{(j)}}$ for $\mathrm{\ell }\mathrm{\ne }i,j$, together with $x_j^{(i)}=1\mathrm{/}{x}_i^{(j)}$, defines a $\mathbb{Z}$-algebra isomorphism $(B_i{)}_{x_j^{(i)}}⟶(B_j{)}_{x_i^{(j)}},x_{\mathrm{\ell }}^{(i)}\mapsto x_{\mathrm{\ell }}^{(j)}\mathrm{/}{x}_i^{(j)}(\mathrm{\ell }\mathrm{\ne }i,j),x_j^{(i)}\mapsto 1\mathrm{/}{x}_i^{(j)},$ and these morphisms identify the open subschemes $D_j^{(i)}\subseteq U_i$ and $D_i^{(j)}\subseteq U_j$ by A principal localization identifies its spectrum with a distinguished open. This is the reciprocal of the analogous formula with $i,j$ interchanged, and on a triple overlap $U_i\cap U_j\cap U_k$ both composites send $x_{\mathrm{\ell }}^{(i)}$ to $x_{\mathrm{\ell }}^{(k)}\mathrm{/}{x}_i^{(k)}$, so the identity and cocycle conditions of Gluing affine schemes along compatible open isomorphisms hold and the affine schemes $U_i$ glue to a scheme, denoted ${\mathbb{P}}_{\mathbb{Z}}^n$, whose open subschemes $U_i$ form an affine cover and are its standard charts.

For an arbitrary base scheme $S$ define ${\mathbb{P}}_S^n=S{\times }_{\mathrm{Spec}\mathbb{Z}}{\mathbb{P}}_{\mathbb{Z}}^n,$ with structure morphism the projection to $S$ (Existence of all scheme fibre products, Schemes and morphisms over a base), and call the open subschemes $U_i^S=S{\times }_{\mathrm{Spec}\mathbb{Z}}{U}_i$ the standard charts over $S$; each is affine over $S$, and they form an open cover of ${\mathbb{P}}_S^n$. When $S=\mathrm{Spec}A$ is affine, each $U_i^S$ is the affine scheme $\mathrm{Spec}A[x_{\mathrm{\ell }}^{(i)}:\mathrm{\ell }\mathrm{\ne }i]$. Because base change preserves open immersions and fibre products, the transition isomorphisms on $U_i^S\cap U_j^S$ are the base changes of the displayed ones, so the charts and their overlaps commute with base change.

For $n=0$ there is one chart $U_0=\mathrm{Spec}\mathbb{Z}[t_0]\mathrm{/}(t_0-1)\cong \mathrm{Spec}\mathbb{Z}$, no gluing takes place, and ${\mathbb{P}}_{\mathbb{Z}}^0=\mathrm{Spec}\mathbb{Z}$, so ${\mathbb{P}}_S^0\cong S$. For $S=\mathrm{\varnothing }$ the product is empty, so ${\mathbb{P}}_{\mathrm{\varnothing }}^n=\mathrm{\varnothing }$. For $n\ge 1$ the constructions for the pairs $(i,j)$ and $(j,i)$ are reciprocal as displayed, and the case $i=j$ is the identity on $U_i$.