Weakly G K -Perfect and Integral Closure of Ideals

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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, 𝔪) is local, then 𝔪 is G K -perfect if and only if K is a canonical module for R. Furthermore, if I is integrally closed and G K − dim R I < ∞, then K 𝔭 is a canonical R 𝔭-module for every 𝔭 ∈ Ass R R/I whenever K satisfies Serre's condition (S 1) or grade K I > 0. Finally, it is shown that if CM − dim R I < ∞, then R 𝔭 is Cohen–Macaulay for every 𝔭 ∈ Ass R R/I.