Coleff-Herrera currents, duality, and Noetherian operators

Journal article, 2011

Let I be a coherent subsheaf of a locally free sheaf O(E-0) and suppose that I = O(E-0)/I has pure codimension. Starting with a residue current R obtained from a locally free resolution of I we construct a. vector-valued Coleff-Herrera current it with support on the variety associated to I such that phi is in I if and only if mu phi = 0. Such a current mu can also be derived algebraically from a fundamental theorem of Roos about the bidualizing functor, and the relation between these two approaches is discussed. By a construction due to Bjork one gets Noetherian operators for I from the current mu. The current R also provides an explicit realization of the Dickenstein-Sessa decomposition and other related canonical isomorphisms.