On torsion-freeness of Kähler differential sheaves

Abstract

Let $X$ be a normal algebraic variety over an algebraically closed field $k$. We prove that the Kähler differential sheaf of $X$ is torsion-free if and only if any regular section of the ideal sheaf of the first order deformation of $X$ inside $X{\times }_{k}X$, defined outside the singular locus of $X{\times }_{k}X$, extends regularly to the singular locus.