Hulls and Husks

Let X be a normal scheme and F a coherent sheaf on X . The reflexive hull or double dual of F is the sheaf F ∗∗ := HomX(HomX(F,OX),OX). The natural map F → F ∗∗ kills the torsion subsheaf of F and the support of coker[F → F ] has codimension ≥ 2. This establishes a functor from the category of coherent sheaves on X to the category of reflexive coherent sheaves on X . The same construction works as long as X satisfies Serre’s condition S2, but otherwise the double dual is not S2. One can, however, define a natural functor from the category of quasi coherent sheaves on X to the category of quasi coherent sheaves on X that are S2 as sheaves over their support (17). We denote it by F → F [∗∗] and call it the hull of F . The main question we address in this section is the behaviour of the hull in families. The motivating example is the following theorem which describes all possible base changes that create a flat sheaf out of a non-flat sheaf.