Weakly G_K-Perfect and Integral Closure of Ideals

Abstract

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, ) is local, then  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 _ is a canonical R _-module for every  ∊ 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 _ is Cohen–Macaulay for every  ∊ Ass _R R/I.

Key Words:

• Canonical module
• Cohen–Macaulay dimension
• G _K -Dimension
• G _K -Perfect module
• Integral closure
• Suitable module
• Weakly G _K -perfect ideal

2000 Mathematics Subject Classification:

• 13D45
• 13H10
• 13B21

ACKNOWLEDGMENTS

The authors are deeply grateful to the referee for careful reading of the original manuscript and valuable suggestions.

Notes

Communicated by S. Goto.