# When is the reflexive hull of the conormal sheaf locally free?

Let $$X\subseteq Y$$ be a (singular) complex analytic subspace or closed subscheme with defining ideal $$I$$, where $$Y$$ is smooth. Stalks of the structure sheaf of $$X$$ are assumed to be integral noetherian domains. The conormal sheaf $$I/I^2$$ of $$X$$ in $$Y$$ is locally free if and only if $$X$$ is a local complete intersection.

I am wondering if there are well known conditions on $$X$$ such that the double dual $$(I/I^2)^{**}$$, or equivalently, when the normal sheaf $$(I/I^2)^*$$ is locally free. The subspace $$X$$ may also be assume to be normal, if necessary.

I guess that it should be a less restrictive assumption than local complete intersection if only the dual of the conormal sheaf is locally free -as the dual is already a reflexive sheaf. But I have not be able to find a nice condition that ensures this.

• I do not whether this helps. The local complete intersection condition can be reinterpreted as the sheaf of Kahler differentials has projective dimension at most one. Similarly the condition on double dual you mention can be reinterpreted as the Kahler differentials modulo torsion has projective dimension at most one. Apr 4 at 16:35
• @Mohan True. I asked the question to better understand how close to smoothness the assumption is. Differentials mod torsion having projective dimension at most one sounds rather close to smoothness. But I have not been able to reformulate that into something more tangible/geometric. Apr 4 at 18:31