Example 10.133.5. Let R \to S be a ring map and let N be an S-module. Observe that \text{Diff}^1(S, N) = \text{Der}_ R(S, N) \oplus N. Namely, if D : S \to N is a differential operator of order 1 then \sigma _ D : S \to N defined by \sigma _ D(g) := D(g) - gD(1) is an R-derivation and D = \sigma _ D + \lambda _{D(1)} where \lambda _ x : S \to N is the linear map sending g to gx. It follows that P^1_{S/R} = \Omega _{S/R} \oplus S by the universal property of \Omega _{S/R}.
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