The Stacks project

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 $\Omega _{S/R}$.