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Sifted degrees of the equations of the Rees module and their connection with the Artin–Rees numbers

  • Philippe Gimenez and Francesc Planas-Vilanova
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Commutative Algebra
This chapter is in the book Commutative Algebra

Abstract

Let A be a Noetherian ring, I an ideal of A, and N ⊂ M finitely generated A-modules. The relation type of I with respect to M, denoted by rt ( I ; M ) , is the maximal degree in a minimal generating set of relations of the Rees module R ( I ; M ) = ⨁ n ≥ 0 I n M . It is a well-known invariant that gives a first measure of the complexity of R ( I ; M ) . To help to measure this complexity, we introduce the sifted type of R ( I ; M ) , denoted by st ( I ; M ) , a new invariant which counts the nonzero degrees appearing in a minimal generating set of relations of R ( I ; M ) . Just as the relation type rt ( I ; M / N ) is closely related to the strong Artin–Rees number s ( N , M ; I ) , it turns out that the sifted type st ( I ; M / N ) is closely related to the medium Artin–Rees number m ( N , M ; I ) , a new invariant which lies in between the weak and strong Artin–Rees numbers of ( N , M ; I ) . We illustrate the meaning, interest and mutual connection of m ( N , M ; I ) and st ( I ; M ) with some examples.

Abstract

Let A be a Noetherian ring, I an ideal of A, and N ⊂ M finitely generated A-modules. The relation type of I with respect to M, denoted by rt ( I ; M ) , is the maximal degree in a minimal generating set of relations of the Rees module R ( I ; M ) = ⨁ n ≥ 0 I n M . It is a well-known invariant that gives a first measure of the complexity of R ( I ; M ) . To help to measure this complexity, we introduce the sifted type of R ( I ; M ) , denoted by st ( I ; M ) , a new invariant which counts the nonzero degrees appearing in a minimal generating set of relations of R ( I ; M ) . Just as the relation type rt ( I ; M / N ) is closely related to the strong Artin–Rees number s ( N , M ; I ) , it turns out that the sifted type st ( I ; M / N ) is closely related to the medium Artin–Rees number m ( N , M ; I ) , a new invariant which lies in between the weak and strong Artin–Rees numbers of ( N , M ; I ) . We illustrate the meaning, interest and mutual connection of m ( N , M ; I ) and st ( I ; M ) with some examples.

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/10.1515/9783110999365-011

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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