The converse to a theorem of Sharp on Gorenstein modules

Abstract

Let A be a commutative local Noetherian ring with identity of Krull dimension n, m its maximal ideal. Sharp has proved that if A is Cohen-Macauley and a homomorphic image of a Gorenstein local ring, then A has a Gorenstein module M with dim A / m Ext n ⁡ ( A / m , M ) = 1 {\dim _{A/m}}\operatorname {Ext}^n(A/m,M) = 1 . The aim of this note is to prove the converse to this theorem.