Universal Kähler differential module

Let $A\stackrel{\phi }{\to }B$ be a homomorphism of commutative rings and let ${\mathrm{Der}}_A(B,-)$ be the derivation functor of Derivation of an algebra. A Kähler differential module for $A\to B$ is a pair $({\mathrm{\Omega }}_{B\mathrm{/}A},\mathrm{d})$ consisting of a $B$-module ${\mathrm{\Omega }}_{B\mathrm{/}A}$ and an $A$-derivation $\mathrm{d}\text{\hspace{0.17em}⁣}:B\to {\mathrm{\Omega }}_{B\mathrm{/}A}$ such that for every $B$-module $M$ the assignment

$g⟼g\circ \mathrm{d},{\mathrm{Hom}}_B({\mathrm{\Omega }}_{B\mathrm{/}A},M)⟶{\mathrm{Der}}_A(B,M),$

is a bijection, and such that these bijections are natural in $M$: for every $B$-linear map $h\text{\hspace{0.17em}⁣}:M\to N$ the square

$\begin{array}{ccc}{\mathrm{Hom}}_B({\mathrm{\Omega }}_{B\mathrm{/}A},M)& \stackrel{g\mapsto g\circ \mathrm{d}}{\to }& {\mathrm{Der}}_A(B,M)\\ \downarrow h\circ -& & \downarrow h\circ -\\ {\mathrm{Hom}}_B({\mathrm{\Omega }}_{B\mathrm{/}A},N)& \stackrel{g\mapsto g\circ \mathrm{d}}{\to }& {\mathrm{Der}}_A(B,N)\end{array}$

commutes. In other words, ${\mathrm{\Omega }}_{B\mathrm{/}A}$ represents the covariant functor $M\mapsto {\mathrm{Der}}_A(B,M)$ on $B$-modules, and $\mathrm{d}$ is the universal $A$-derivation of $B$ over $A$; the element $\mathrm{d}b$ is the image of $b$ under it. Whether such a pair exists for a given $A\to B$ is not part of the definition; when it does, the pair is uniquely determined up to a unique compatible isomorphism, as the next paragraph records.

Uniqueness. If $(\mathrm{\Omega },\mathrm{d})$ and $({\mathrm{\Omega }}^{\mathrm{\prime }},{\mathrm{d}}^{\mathrm{\prime }})$ are both Kähler differential modules for the same ring map $A\to B$, the universal property of the first applied to the derivation ${\mathrm{d}}^{\mathrm{\prime }}$ produces a unique $B$-linear $u\text{\hspace{0.17em}⁣}:\mathrm{\Omega }\to {\mathrm{\Omega }}^{\mathrm{\prime }}$ with $u\circ \mathrm{d}={\mathrm{d}}^{\mathrm{\prime }}$, and the property of the second applied to $\mathrm{d}$ produces a unique $B$-linear $v\text{\hspace{0.17em}⁣}:{\mathrm{\Omega }}^{\mathrm{\prime }}\to \mathrm{\Omega }$ with $v\circ {\mathrm{d}}^{\mathrm{\prime }}=\mathrm{d}$. Then $(v\circ u)\circ \mathrm{d}=v\circ {\mathrm{d}}^{\mathrm{\prime }}=\mathrm{d}$ and $(u\circ v)\circ {\mathrm{d}}^{\mathrm{\prime }}={\mathrm{d}}^{\mathrm{\prime }}$, while the identity maps of $\mathrm{\Omega }$ and ${\mathrm{\Omega }}^{\mathrm{\prime }}$ have the same property; the injectivity clause of the universal property applied twice gives $v\circ u={\mathrm{i}\mathrm{d}}_{\mathrm{\Omega }}$ and $u\circ v={\mathrm{i}\mathrm{d}}_{{\mathrm{\Omega }}^{\mathrm{\prime }}}$. So $u$ is an isomorphism with inverse $v$, and $u$ is the only $B$-linear map from $\mathrm{\Omega }$ to ${\mathrm{\Omega }}^{\mathrm{\prime }}$ compatible with the two universal derivations. In particular ${\mathrm{\Omega }}_{B\mathrm{/}A}$ is determined by the ring map $A\to B$ up to canonical isomorphism, which is what justifies writing it as ${\mathrm{\Omega }}_{B\mathrm{/}A}$ without further qualification.

Functoriality in ring maps. Given a commutative square of ring maps $A\to B$, $A^{\mathrm{\prime }}\to B^{\mathrm{\prime }}$, $u:A\to A^{\mathrm{\prime }}$ and $v:B\to B^{\mathrm{\prime }}$, and universal pairs for its two horizontal maps, regard ${\mathrm{\Omega }}_{B^{\mathrm{\prime }}\mathrm{/}{A}^{\mathrm{\prime }}}$ as a $B$-module through $v$. The composite ${\mathrm{d}}^{\mathrm{\prime }}\circ v$ is an $A$-derivation: it is additive, satisfies Leibniz with this module action, and kills $A$ since the square commutes. Universality gives a unique $B$-linear map ${\mathrm{\Omega }}_{B\mathrm{/}A}\to {\mathrm{\Omega }}_{B^{\mathrm{\prime }}\mathrm{/}{A}^{\mathrm{\prime }}}$ sending $\mathrm{d}b$ to ${\mathrm{d}}^{\mathrm{\prime }}(v(b))$. For identity squares this is the identity; for composable squares the composite has the prescribed values on $\mathrm{d}b$ and so equals the map of the composite square by uniqueness. This proves functoriality in ring maps separately from naturality in the target module $M$.