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Rings with q-torsionfree canonical modules

  • Naoki Endo , Laura Ghezzi , Shiro Goto , Jooyoun Hong , Shin-ichiro Iai , Toshinori Kobayashi , Naoyuki Matsuoka and Ryo Takahashi
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Commutative Algebra
This chapter is in the book Commutative Algebra

Abstract

Let A be a Noetherian local ring with canonical module K A . We characterize A when K A is a torsionless, reflexive, or q-torsionfree module for an integer q ≥ 3 . If A is a Cohen–Macaulay ring, H.-B. Foxby proved in 1974 that the A-module K A is q-torsionfree if and only if the ring A is q-Gorenstein. With mild assumptions, we provide a generalization of Foxby’s result to arbitrary Noetherian local rings admitting the canonical module. In particular, since the reflexivity of the canonical module is closely related to the ring being Gorenstein in low codimension, we also explore quasinormal rings, introduced by W. V. Vasconcelos. We provide several examples as well.

Abstract

Let A be a Noetherian local ring with canonical module K A . We characterize A when K A is a torsionless, reflexive, or q-torsionfree module for an integer q ≥ 3 . If A is a Cohen–Macaulay ring, H.-B. Foxby proved in 1974 that the A-module K A is q-torsionfree if and only if the ring A is q-Gorenstein. With mild assumptions, we provide a generalization of Foxby’s result to arbitrary Noetherian local rings admitting the canonical module. In particular, since the reflexivity of the canonical module is closely related to the ring being Gorenstein in low codimension, we also explore quasinormal rings, introduced by W. V. Vasconcelos. We provide several examples as well.

Chapters in this book

  1. Frontmatter I
  2. Preface V
  3. Contents VII
  4. Wolmer Vasconcelos, a tribute IX
  5. Vasconcelos Bibliography (1964–2024) XVII
  6. The early years: flatness, projectivity, and R-sequences 1
  7. Joint work with Wolmer Vasconcelos. The early years 41
  8. Homological properties of the module of differentials 71
  9. Hilbert coefficients and Sally modules: a survey of Vasconcelos’ contributions 97
  10. Graph rings and ideals: Wolmer Vasconcelos’ contributions 133
  11. Degrees: Vasconcelos contributions 203
  12. The books of Wolmer V. Vasconcelos 241
  13. Finding nilpotents with Wolmer Vasconcelos 261
  14. Rees algebras over residual intersections 279
  15. Structure theorems for Gorenstein ideals of codimension four with small number of generators 297
  16. Sifted degrees of the equations of the Rees module and their connection with the Artin–Rees numbers 333
  17. Reembeddings of special border basis schemes 353
  18. Z r -graded rings and their canonical modules 389
  19. Rings with q-torsionfree canonical modules 405
https://doi.org/10.1515/9783110999365-014

Chapters in this book

  1. Frontmatter I
  2. Preface V
  3. Contents VII
  4. Wolmer Vasconcelos, a tribute IX
  5. Vasconcelos Bibliography (1964–2024) XVII
  6. The early years: flatness, projectivity, and R-sequences 1
  7. Joint work with Wolmer Vasconcelos. The early years 41
  8. Homological properties of the module of differentials 71
  9. Hilbert coefficients and Sally modules: a survey of Vasconcelos’ contributions 97
  10. Graph rings and ideals: Wolmer Vasconcelos’ contributions 133
  11. Degrees: Vasconcelos contributions 203
  12. The books of Wolmer V. Vasconcelos 241
  13. Finding nilpotents with Wolmer Vasconcelos 261
  14. Rees algebras over residual intersections 279
  15. Structure theorems for Gorenstein ideals of codimension four with small number of generators 297
  16. Sifted degrees of the equations of the Rees module and their connection with the Artin–Rees numbers 333
  17. Reembeddings of special border basis schemes 353
  18. Z r -graded rings and their canonical modules 389
  19. Rings with q-torsionfree canonical modules 405
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