I'm wondering if there is a characterization of schemes over a a field $k$ whose dualizing complex is a perfect complex in terms of singularities. E.g. on a proper Cohen-Macauley scheme over a field, the dualizing complex is a sheaf, when is this sheaf have a finite locally free resolution? (Or this never happens?)

Recall a complex is called a perfect complex if it is locally quasi-isomorphic to a bounded complex of locally free sheaves.

I'm wondering if there is a characterization of schemes over a a field $k$ whose dualizing complex is a perfect complex in terms of singularities. E.g. on a proper Cohen-Macauley scheme over a field, the dualizing complex is a sheaf, when is this sheaf have a finite locally free resolution? (Or this never happens unless the scheme is Gorenstein?)

Recall a complex is called a perfect complex if it is locally quasi-isomorphic to a bounded complex of locally free sheaves.