Orateur
Description
We introduce the notion of an “initial condition” for a module M over a commutative Noetherian local ring (A,m), allowing for a recursive construction of its “solution modules”. If M has zero-dimensional support, such as the residue field of A, we demonstrate that the solution module E(M) is its “linear closure”, turning out to be an injective hull of M. The construction of E(M) for finitely generated M, hence injective hull of M, is explicit and computable, devoid of the need for Zorn’s lemma. As an application, we improve Baer’s criterion for a module N with zero-dimensional support to be injective: If any A-homomorphism from m to N lifts to A, then N is injective. For the case that A is Artinian and M is its residue field, we demonstrate how systems of linear equations are explicitly solved to obtain an injective hull.
