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Z r -graded rings and their canonical modules

  • Margherita Barile and Winfried Bruns
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Commutative Algebra
This chapter is in the book Commutative Algebra

Abstract

In “Cohen–Macaulay rings” Bruns and Herzog define the graded canonical module for Z -graded rings. We generalize the definition to multigradings and prove that the canonical module “localizes.” As an application, we give a divisorial proof of the theorem of Danilov and Stanley on the canonical module of affine normal monoid rings. Along the way, we develop the basic theory of multigraded rings and modules.

Abstract

In “Cohen–Macaulay rings” Bruns and Herzog define the graded canonical module for Z -graded rings. We generalize the definition to multigradings and prove that the canonical module “localizes.” As an application, we give a divisorial proof of the theorem of Danilov and Stanley on the canonical module of affine normal monoid rings. Along the way, we develop the basic theory of multigraded rings and modules.

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/10.1515/9783110999365-013

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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