Maps out of the module of differentials are the same as derivations.
Lemma 10.131.3. The module of differentials of S over R has the following universal property. The map
\mathop{\mathrm{Hom}}\nolimits _ S(\Omega _{S/R}, M) \longrightarrow \text{Der}_ R(S, M), \quad \alpha \longmapsto \alpha \circ \text{d}
is an isomorphism of functors.
Proof.
By definition an R-derivation is a rule which associates to each a \in S an element D(a) \in M. Thus D gives rise to a map [D] : \bigoplus S[a] \to M. However, the conditions of being an R-derivation exactly mean that [D] annihilates the image of the map in the displayed presentation of \Omega _{S/R} above.
\square
Comments (1)
Comment #1219 by David Corwin on
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