Quasi-separatedness and the diagonal

Let $U,V\subseteq X$ be affine opens mapping into a common affine open $W\subseteq S$. By [F6] the subscheme $P_{UV}:={\mathrm{pr}}_1^{-1}(U)\cap {\mathrm{pr}}_2^{-1}(V)$ is open in $X{\times }_{S}X$ and represents $U{\times }_{W}V$, an affine scheme by [F5]; and by [F7] a point $x\in X$ satisfies ${\mathrm{\Delta }}_{X\mathrm{/}S}(x)\in P_{UV}$ exactly when $x\in U\cap V$, so ${\mathrm{\Delta }}_{X\mathrm{/}S}^{-1}(P_{UV})=U\cap V$.

The subschemes $P_{UV}$ cover $X{\times }_{S}X$: given a point $z$, let $w\in S$ be the common image of ${\mathrm{pr}}_1(z),{\mathrm{pr}}_2(z)$, choose an affine open $W\ni w$, and use [F3] to choose affine opens $U\subseteq f^{-1}(W)$ containing ${\mathrm{pr}}_1(z)$ and $V\subseteq f^{-1}(W)$ containing ${\mathrm{pr}}_2(z)$; then $z\in P_{UV}$.

Assume ${\mathrm{\Delta }}_{X\mathrm{/}S}$ quasi-compact. For affine $U,V$ over a common affine $W$, the scheme $P_{UV}$ is affine by step 1.1 hence quasi-compact by [F4], so ${\mathrm{\Delta }}_{X\mathrm{/}S}^{-1}(P_{UV})=U\cap V$ is quasi-compact by [F1]. Thus $f$ is quasi-separated by [F1].

Each ${\mathrm{\Delta }}_{X\mathrm{/}S}^{-1}(P_k)$ is the intersection of the two affine opens defining $P_k$ by step 1.1, hence quasi-compact by hypothesis.

Assume conversely that the stated intersection condition holds, and let $Q\subseteq X{\times }_{S}X$ be affine open. By steps 1.1 and 2.1, the affine opens $P_{UV}$ cover the target. For each point of $Q$, choose such a $P_{UV}$ containing it; since $P_{UV}$ is affine, its distinguished opens contained in $Q\cap P_{UV}$ form a neighbourhood basis there. These distinguished opens cover $Q$, so [F4] gives a finite subcover $D(a_1),\dots ,D(a_n)$ with $D(a_j)\subseteq P_j$ for corresponding members $P_j=P_{U_j{V}_j}$. By step 2.3, ${\mathrm{\Delta }}_{X\mathrm{/}S}^{-1}(P_j)=U_j\cap V_j$ is quasi-compact. The inverse image of $D(a_j)$ is the distinguished open defined by the pulled-back section ${\mathrm{\Delta }}_{X\mathrm{/}S}^{\mathrm{\#}}(a_j)$ on this quasi-compact scheme; it is quasi-compact because a quasi-compact scheme has a finite affine open cover and the distinguished open restricts to an affine distinguished open on each member of that cover.

The finite distinguished-open cover in step 3.1 pulls back to a finite open cover of ${\mathrm{\Delta }}_{X\mathrm{/}S}^{-1}(Q)$ by quasi-compact opens, hence ${\mathrm{\Delta }}_{X\mathrm{/}S}^{-1}(Q)$ is quasi-compact. Since $Q$ was any affine open of $X{\times }_{S}X$, every such affine open has quasi-compact inverse image under ${\mathrm{\Delta }}_{X\mathrm{/}S}$.

Let $V\subseteq X{\times }_{S}X$ be any quasi-compact open subscheme. Since affine opens form a basis by [F3], $V$ is covered by affine opens contained in $V$, and quasi-compactness of $V$ extracts a finite subcover $Q_1,\dots ,Q_m$; hence ${\mathrm{\Delta }}_{X\mathrm{/}S}^{-1}(V)={\bigcup }_{j=1}^m{\mathrm{\Delta }}_{X\mathrm{/}S}^{-1}(Q_j)$ is a finite union of quasi-compact sets by step 4.1, therefore quasi-compact.

Step 5.1 shows that ${\mathrm{\Delta }}_{X\mathrm{/}S}^{-1}$ of every quasi-compact open is quasi-compact, so ${\mathrm{\Delta }}_{X\mathrm{/}S}$ is quasi-compact by [F1]; together with step 2.2 this proves the equivalence, and the final clause follows because a quasi-compact open subscheme of a scheme is a finite union of affine opens, as used in step 5.1.