General Ring Theory/Derivations

Definition (derivation):

Let $S$ be a unital $R$-algebra. Moreover, let $M$ be a module over the algebra $S$. Then a derivation on $S$ over $R$ with values in $M$ is a morphism of $R$-modules $d:S\to M$ such that

$d(st)=sd(t)+td(s)$

for all $s,t\in S$. We denote the $S$-module of all derivations of $S$ over $R$ with values in $M$ by $\operatorname {Der} _{R}(S,M)$.

Definition (universal derivation):

Let $S$ be a unital $R$-algebra. Then the universal derivation of $S$ over $R$ is the derivation $d:S\to \Omega _{S/R}$, where $\Omega _{S/R}$ is the $S$-module that arises by taking the free $S$-module generated by the formal symbols $ds$ (one for each $s\in S$) and dividing out the ideal generated by elements of the form $d(ts)-tds-sdt$ as well as $d(rs)-rd(s)$ (where $r\in R$, $t,s\in S$), and $d$ sends $s$ to $ds$.

Proposition (universal property of the universal derivation):

Let $S$ be a unital $R$-algebra and let $M$ be a module over the algebra $S$. Then for each $\delta \in \operatorname {Der} _{R}(S,M)$, there exists a unique morphism of $S$-modules $f:\Omega _{S/R}\to M$ such that $\delta =f\circ d$. This pairing induces an isomorphism of $R$-modules

$\operatorname {Hom} _{S}(\Omega _{S/R},M)\cong \operatorname {Der} _{R}(S,M)$.

Proof: For uniqueness, note that the universal derivation $d$ is surjective, so $f$ is determined by $\delta =f\circ d$. For existence, note that if $F$ is the free $S$-module with generators $ds$ for all $s\in S$, then there is a morphism $F\to M$ that sends $ds$ to $\delta (s)$, and it factors over the quotient in the definition of $\Omega _{S/R}$ by the properties of $\delta$. $\Box$

Proposition (universal derivation is unique):

The universal derivation of a unital $R$-algebra $S$ together with its target module is uniquely defined.

Proof: This follows immediately from the universal property that defines the corresponding object uniquely. $\Box$