On torsion-freeness of Kähler differential sheaves (Q6644175)

It is well-known that a variety \(X\) (over an algebraically closed field \(k\)) is non-singular if and only if the sheaf of Kähler differentials \(\Omega_{X/k}^1\) is locally free. Therefore it is natural to try to relate more general sheaf-theoretic properties of \(\Omega_X^1\) to geometric properties of \(X\). In this direction, the authors prove: \(\Omega_X^1\) is torsion-free if and only if \(\mathcal I^2 \to j_* j^* \mathcal I^2\) is surjective, where \(\mathcal I\) is the ideal sheaf of the diagonal \(X \subset X \times X\) and \(j\) is the inclusion of the non-singular locus of \(X \times X\) in \(X \times X\).\N\NAs an immediate corollary, \(\Omega_X^1\) is torsion-free if \(X\) is non-singular. This is a weak version of one direction of the statement mentioned in the beginning.