Existence and generators of Kähler differentials

The pair of [F1] is a $B$-module with an $A$-derivation. The free module $F$ is a $B$-module and $R$ is a $B$-submodule by construction, so ${\mathrm{\Omega }}_{B\mathrm{/}A}=F\mathrm{/}R$ is a $B$-module and $\mathrm{d}$ is a map $B\to {\mathrm{\Omega }}_{B\mathrm{/}A}$. Each of the three relator families lies in $R$, hence vanishes in the quotient: $[b+b^{\mathrm{\prime }}]-[b]-[b^{\mathrm{\prime }}]\in R$ gives $\mathrm{d}(b+b^{\mathrm{\prime }})=\mathrm{d}b+\mathrm{d}{b}^{\mathrm{\prime }}$, the element $[b{b}^{\mathrm{\prime }}]-b[b^{\mathrm{\prime }}]-b^{\mathrm{\prime }}[b]\in R$ gives $\mathrm{d}(b{b}^{\mathrm{\prime }})=b\mathrm{d}{b}^{\mathrm{\prime }}+b^{\mathrm{\prime }}\mathrm{d}b$, and $[\phi (a)]\in R$ gives $\mathrm{d}(\phi (a))=0$. So $\mathrm{d}$ is additive, $A$-constant and satisfies Leibniz, that is, $\mathrm{d}\in {\mathrm{Der}}_A(B,{\mathrm{\Omega }}_{B\mathrm{/}A})$ by [F2].

Every derivation descends to a map out of ${\mathrm{\Omega }}_{B\mathrm{/}A}$. Let $M$ be a $B$-module and $D\in {\mathrm{Der}}_A(B,M)$. Since $F$ is free with basis $\{[b]:b\in B\}$, there is a unique $B$-linear map $G\text{\hspace{0.17em}⁣}:F\to M$ with $G([b])=D(b)$ for all $b\in B$. By [F2] the map $D$ is additive, $A$-constant and satisfies Leibniz, so $G$ kills each relator: $G([b+b^{\mathrm{\prime }}]-[b]-[b^{\mathrm{\prime }}])=D(b+b^{\mathrm{\prime }})-D(b)-D(b^{\mathrm{\prime }})=0$, similarly $G([b{b}^{\mathrm{\prime }}]-b[b^{\mathrm{\prime }}]-b^{\mathrm{\prime }}[b])=D(b{b}^{\mathrm{\prime }})-bD(b^{\mathrm{\prime }})-b^{\mathrm{\prime }}D(b)=0$, and $G([\phi (a)])=D(\phi (a))=0$; here we used that $G$ is $B$-linear, so that $G(b[b^{\mathrm{\prime }}])=bG([b^{\mathrm{\prime }}])=bD(b^{\mathrm{\prime }})$. The three families generate $R$ as a $B$-submodule and [F1] presents ${\mathrm{\Omega }}_{B\mathrm{/}A}=F\mathrm{/}R$, so $G$ factors through a $B$-linear $\bar{G}\text{\hspace{0.17em}⁣}:{\mathrm{\Omega }}_{B\mathrm{/}A}\to M$ with $\bar{G}([b]+R)=D(b)$, that is, $\bar{G}\circ \mathrm{d}=D$.

The construction of step 2.1 is the unique inverse. Let $h\text{\hspace{0.17em}⁣}:{\mathrm{\Omega }}_{B\mathrm{/}A}\to M$ be $B$-linear with $h\circ \mathrm{d}=D$ for some $D\in {\mathrm{Der}}_A(B,M)$. Evaluating on $\mathrm{d}b$ gives $h(\mathrm{d}b)=h([b]+R)=D(b)=\bar{G}(\mathrm{d}b)$ for every $b\in B$, where $\bar{G}$ is the map produced in step 2.1. The classes $[b]+R=\mathrm{d}b$ range over the images of a basis of the free module $F$, so they generate ${\mathrm{\Omega }}_{B\mathrm{/}A}=F\mathrm{/}R$ as a $B$-module, and two $B$-linear maps agreeing on a generating set are equal; hence $h=\bar{G}$. Therefore $g\mapsto g\circ \mathrm{d}$ is a bijection ${\mathrm{Hom}}_B({\mathrm{\Omega }}_{B\mathrm{/}A},M)\to {\mathrm{Der}}_A(B,M)$ for every $B$-module $M$.

Naturality in $M$. Let $t\text{\hspace{0.17em}⁣}:M\to N$ be $B$-linear and let $h\in {\mathrm{Hom}}_B({\mathrm{\Omega }}_{B\mathrm{/}A},M)$. Then $t\circ (h\circ \mathrm{d})=(t\circ h)\circ \mathrm{d}$ as maps $B\to N$, since both sides send $b$ to $t(h(\mathrm{d}b))$, and $t\circ h$ is again $B$-linear, so the assignment of step 3.1 carries $h$ followed by $t$ to the derivation $t\circ (h\circ \mathrm{d})$; this is exactly the commutativity required of the bijections in [F3].

Conclusion. Steps 1.1, 2.1 and 3.1 verify both clauses of [F3] for the pair $({\mathrm{\Omega }}_{B\mathrm{/}A},\mathrm{d})$ of [F1], and step 4.1 verifies naturality, so that pair is a Kähler differential module for $A\to B$. The construction used only the free module on the set $B$ and the submodule generated by the three relator families, so it exists for every ring homomorphism $A\to B$ with no finiteness hypothesis, and step 3.1 exhibits the classes $\mathrm{d}b$ as a generating set of ${\mathrm{\Omega }}_{B\mathrm{/}A}$ over $B$.