Higher derivations and universal differential operators (Q1923949)

Let \(k\) be a commutative ring and \(R\) a commutative \(k\)-algebra. Let \(\delta_r: R\to \Omega_r\) be the universal object for \(k\)-derivation of \(R\) in \(R\)-modules \(N\) of order \(\leq r\), and let \(\Delta_s: M\to j_s (M)\) be the universal object for \(k\)-derivation of an \(R\)-module \(M\) in \(R\)-modules \(N\) of order \(\leq s\). Then we have the canonical \(R\)-linear map \(\varphi_R: \Omega_{r+s} \to j_r (\Omega_s)\) satisfying \(\varphi \delta_{r+s} = \Delta_r \delta_s\). The main result is theorem 1 which states that \(\ker (\varphi) = 0\) if \(r=s=1\), which gives rise to a split exact sequence \[ \to\Omega_2 \to j_1(\Omega_1) \to\bigwedge^2 \Omega_1 \to 0. \] Examples show that the main result is no longer true if \(r\) or \(s>1\).