An unramified morphism has an open diagonal

Chart description. Let $U=\mathrm{Spec}B\subseteq X$ be affine over an affine open $V=\mathrm{Spec}A\subseteq S$, and write $C=B{\otimes }_{A}B$, $J=\ker \mu$ for the multiplication $\mu \text{\hspace{0.17em}⁣}:C\to B$. By [F3] the diagonal restricts to the morphism ${\mathrm{\Delta }}_U\text{\hspace{0.17em}⁣}:U\to U{\times }_{V}U=\mathrm{Spec}C$ induced by $\mu$, a closed immersion with ideal $J$. The elements $1\otimes b-b\otimes 1$ for $b$ in a generating set of $B$ over $A$ generate $J$ as a $C$-module, so if $B$ is a finitely generated $A$-algebra, $J$ is a finitely generated $C$-module; and by [F4] $J\mathrm{/}{J}^2\cong {\mathrm{\Omega }}_{B\mathrm{/}A}$.

Unramified implies finite generation and $\mathrm{\Omega }=0$ on charts. If $f$ is unramified then by [F1] it is locally of finite type and ${\mathrm{\Omega }}_{X\mathrm{/}S}=0$, so $B$ is a finitely generated $A$-algebra and ${\mathrm{\Omega }}_{B\mathrm{/}A}=0$ on every chart; hence $J$ is a finitely generated $C$-module with $J=J^2$ by [F4]. Applying [F5] inside the ring $C$ to the ideal $J$ and the $C$-module $J$, we find $e\in J$ with $(1-e)J=0$; then $e^2=e$, since $e\in J$, and $J=eC$: indeed $j=ej\in eC$ for $j\in J$ and $eC\subseteq J$ as $J$ is an ideal.

Conversely, assume $f$ locally of finite type and $\mathrm{\Delta }$ an open immersion. Let $U=\mathrm{Spec}B$ be an affine chart over $V=\mathrm{Spec}A$ as in step 1.1. Since ${\mathrm{\Delta }}^{-1}(U{\times }_{V}U)\supseteq U$, the restriction ${\mathrm{\Delta }}_U\text{\hspace{0.17em}⁣}:U\to U{\times }_{V}U$ is again an open immersion, and by [F3] it is also the closed immersion induced by the surjection $\mu \text{\hspace{0.17em}⁣}:C\to B$ with kernel $J$. Its image is therefore open and closed in $\mathrm{Spec}C$.

The chart diagonal is an open immersion when $f$ is unramified. With $e$ as in step 2.1, $J=eC$ has radical $\sqrt{(e)}$, so the image $V(J)$ of the closed immersion ${\mathrm{\Delta }}_U$ equals $V(e)=D(1-e)$, which is open in $\mathrm{Spec}C$ by [F7]. Moreover $C\mathrm{/}J=C\mathrm{/}eC\cong C\mathrm{/}(e)\cong C_{1-e}$ is exactly the ring of the open subscheme $D(1-e)$, and ${\mathrm{\Delta }}_U$ is the morphism $C\to C\mathrm{/}J$ over $\mathrm{Spec}C$; hence ${\mathrm{\Delta }}_U$ identifies $U$ isomorphically with the open subscheme $D(1-e)$ of $U{\times }_{V}U$, so ${\mathrm{\Delta }}_U$ is an open immersion.

The image is a principal open. Let ${\mathrm{\Delta }}_U(U)=V(I)$ for the radical ideal $I$ of the closed image and $\mathrm{Spec}C\setminus {\mathrm{\Delta }}_U(U)=V(K)$. Since these closed sets are complementary, $I+K=C$ and $IK\subseteq \mathrm{nil}(C)$ because $V(IK)=\mathrm{Spec}C$. Choose $i\in I$ and $k\in K$ with $i+k=1$, and choose $m\ge 1$ with $(ik{)}^m=0$. Expanding $1=(i+k{)}^{2m-1}$, every monomial is divisible by either $i^m$ or $k^m$, so $1=a{i}^m+b{k}^m$ for some $a,b\in C$. Put $e=a{i}^m$; then $1-e=b{k}^m$ and $e(1-e)=ab(ik{)}^m=0$, hence $e^2=e$. Since $e\in I$ and $1-e\in K$, we have ${\mathrm{\Delta }}_U(U)=V(I)\subseteq V(e)=D(1-e)\subseteq \mathrm{Spec}C\setminus V(K)={\mathrm{\Delta }}_U(U)$, so ${\mathrm{\Delta }}_U(U)=D(1-e)$.

$\mathrm{\Delta }$ is an open immersion when $f$ is unramified. The affine charts $U$ of step 3.1 cover $X$, and for each of them $\mathrm{\Delta }{\mathrm{\mid }}_U={\mathrm{\Delta }}_U$ is an isomorphism onto the open subset $D(1-e_U)$ of $U{\times }_{V}U\subseteq X{\times }_{S}X$. The diagonal is injective on points, since $\mathrm{\Delta }(x)=(x,x)$ determines $x$, and an open immersion is exactly a morphism which is locally on the source an isomorphism onto an open subscheme and injective on points; by [F6] the local isomorphisms glue to an isomorphism of $X$ with the open subscheme $\mathrm{\Delta }(X)={\bigcup }_U\mathrm{\Delta }(U)$ of $X{\times }_{S}X$. Hence $\mathrm{\Delta }$ is an open immersion.

The conormal module vanishes. The open immersion ${\mathrm{\Delta }}_U$ identifies $U$ with the open subscheme $D(1-e)$, whose ring is $C\mathrm{/}(e)$ by [F7]; since the structure map of ${\mathrm{\Delta }}_U$ is $C\to C\mathrm{/}J$, the two descriptions of the same ring map give $J=(e)$. Hence $J^2=(e^2)=(e)=J$, so $J\mathrm{/}{J}^2=0$ and [F4] gives ${\mathrm{\Omega }}_{B\mathrm{/}A}=0$. As the charts cover $X$ and $B$ was an arbitrary chart, [F7] gives ${\mathrm{\Omega }}_{X\mathrm{/}S}=0$; with $f$ locally of finite type, [F1] makes $f$ unramified.

Conclusion. Steps 2.1, 3.1 and 4.1 prove that an unramified $f$ has open diagonal, and steps 2.2, 3.2 and 4.2 prove that a locally finite type $f$ with open diagonal is unramified. No separatedness assumption was made: the diagonal is used as a closed immersion only on affine charts, where the multiplication $B{\otimes }_{A}B\to B$ is surjective, and the open condition comes from the idempotent splitting $J=eC$.