The relative projective-space diagonal is closed

Since separatedness is local on the base by [F2], it suffices to treat $S=\mathrm{Spec}A$ affine, the general case following by base change along an affine open cover of $S$ using the compatibility in [F1]; for $S=\mathrm{\varnothing }$ the scheme ${\mathbb{P}}_S^n$ is empty and the claim is automatic.

Over $S=\mathrm{Spec}A$, fix charts $U_i=\mathrm{Spec}{B}_i$, $U_j=\mathrm{Spec}{B}_j$ of [F1]. Then $U_i{\times }_S{U}_j\cong \mathrm{Spec}(B_i{\otimes }_A{B}_j)$ by [F4], with $B_i{\otimes }_A{B}_j\cong A[x_{\mathrm{\ell }}^{(i)},y_m^{(j)}]$ for $\mathrm{\ell }\mathrm{\ne }i$, $m\mathrm{\ne }j$.

By [F6] the inverse image ${\mathrm{\Delta }}^{-1}(U_i{\times }_S{U}_j)$ inside ${\mathbb{P}}_S^n$ is $U_i\cap U_j$, and the restriction of $\mathrm{\Delta }$ to $U_i{\times }_S{U}_j$ is the morphism $U_i\cap U_j\to U_i{\times }_S{U}_j$ whose two composites with the projections are the inclusions.

Suppose $i\mathrm{\ne }j$. By [F1] the intersection $U_i\cap U_j$ is $D(x_j^{(i)})=\mathrm{Spec}A[x_{\mathrm{\ell }}^{(i)}{]}_{(x_j^{(i)})}$, which is affine. On that overlap the transition coordinates satisfy $y_i^{(j)}=(x_j^{(i)}{)}^{-1}$ and $y_m^{(j)}=x_m^{(i)}(x_j^{(i)}{)}^{-1}$ for $m\mathrm{\ne }i,j$. Thus the morphism of step 3.1 corresponds to the ring map $A[x_{\mathrm{\ell }}^{(i)},y_m^{(j)}]\to A[x_{\mathrm{\ell }}^{(i)}{]}_{(x_j^{(i)})}$ with those images and $x_{\mathrm{\ell }}^{(i)}\mapsto x_{\mathrm{\ell }}^{(i)}$. This map is surjective because $y_i^{(j)}$ maps to $(x_j^{(i)}{)}^{-1}$.

Suppose $i=j$. Then $U_i\cap U_i=U_i$ is affine and the ring map is $A[x_{\mathrm{\ell }}^{(i)},y_m^{(i)}]\to A[x_{\mathrm{\ell }}^{(i)}]$ with $y_m^{(i)}\mapsto x_m^{(i)}$, again surjective.

For $i\mathrm{\ne }j$ put $J=(x_j^{(i)}{y}_i^{(j)}-1,y_m^{(j)}-x_m^{(i)}{y}_i^{(j)}(m\mathrm{\ne }i,j))$ in the source ring of step 4.1. Every generator of $J$ maps to zero under that step's coordinate map. Conversely, quotienting by $J$ makes $x_j^{(i)}$ invertible with inverse $y_i^{(j)}$ and expresses every other $y$-generator as $x_m^{(i)}{y}_i^{(j)}$, so the quotient is precisely $A[x_{\mathrm{\ell }}^{(i)}{]}_{(x_j^{(i)})}$; hence $J$ is the kernel. For $i=j$ the kernel of step 4.2 is $(y_m^{(i)}-x_m^{(i)}(m\mathrm{\ne }i))$. The mixed-chart generators are, up to sign, $x_i{y}_j-x_j{y}_i$ and $x_i{y}_m-x_m{y}_i$ after $x_i=y_j=1$; the same-chart generators are $x_i{y}_m-x_m{y}_i$ after $x_i=y_i=1$. Thus these are exactly the ideals of the diagonal on the chart products.

By [F3] applied to the affine base $S=\mathrm{Spec}A$ and the affine open cover $\{U_i\}$ of ${\mathbb{P}}_S^n$, steps 4.1, 4.2 and 5.1 show that $\mathrm{\Delta }$ is a closed immersion; the empty base and the case $n=0$, where ${\mathbb{P}}_S^0=S$ and $\mathrm{\Delta }$ is an isomorphism, are included.

By [F5] the morphism ${\mathbb{P}}_S^n\to S$ is separated, which completes the proof.