On canonical Cohen–Macaulay modules

Let (R,m) be a Noetherian local ring which is a homomorphic image of a local Gorenstein ring and let M be a finitely generated R-module of dimension d>0. According to Schenzel (2004) [Sc3], M is called a canonical Cohen–Macaulay module (CCM module for short) if the canonical module K(M) of M is Cohen–Macaulay. We give another characterization of CCM modules. We describe the non-canonical Cohen–Macaulay locus nCCM(M) of M. If d⩽4 then nCCM(M) is closed in Spec(R). For each d⩾5 there are reduced geometric local rings R of dimension d such that nCCM(R) is not stable under specialization.