Let R be a commutative Noetherian ring, K a nonzero finitely generated suitable R-module, and I an ideal of R. It is shown that if (R, đȘ) is local, then đȘ is G K -perfect if and only if K is a canonical module for R. Furthermore, if I is integrally closed and G K â dim R I < â, then K đ is a canonical R đ-module for every đ â Ass R R/I whenever K satisfies Serre's condition (S 1) or grade K I > 0. Finally, it is shown that if CM â dim R I < â, then R đ is CohenâMacaulay for every đ â Ass R R/I.