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The Cohen-Macaulay and Gorenstein Rees Algebras Associated to Filtrations
Shiro Goto and Koji Nishida
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Memoirs of the American Mathematical Society
1994; 134 pp; softcover
Volume: 110
ISBN-10: 0-8218-2584-4
ISBN-13: 978-0-8218-2584-6
List Price: US$40
Individual Members: US$24
Institutional Members: US$32
Order Code: MEMO/110/526
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This monograph consists of two parts. Part I investigates the Cohen-Macaulay and Gorenstein properties of symbolic Rees algebras for one-dimensional prime ideals in Cohen-Macaulay local rings. Practical criteria for these algebras to be Cohen-Macaulay and Gorenstein rings are described in terms of certain elements in the prime ideals. This framework is generalized in Part II to Rees algebras \(R(F)\) and graded rings \(G(F)\) associated to general filtrations of ideals in arbitrary Noetherian local rings. Goto and Nishida give certain cohomological characterizations for algebras \(R(F)\) to be Cohen-Macaulay or Gorenstein rings in connection with the corresponding ring-theoretic properties of \(G(F)\). In this way, readers follow a history of the development of the ring theory of Rees algebras. The book raises many important open questions.

Readership

Commutative algebraists, algebraic geometers, and specialists working on singularities.

Table of Contents

Part I. The Cohen-Macaulay symbolic Rees algebras for curve singularities
  • Introduction
  • Preliminaries
  • The case of dimension 1
  • The case of dimension 2
  • The Cohen-Macaulay and Gorenstein properties of \(G_s(\mathbf p)\)
  • The Cohen-Macaulay and Gorenstein properties of \(R_s(\mathbf p)\)
  • Examples
  • References
Part II. Filtrations and the Gorenstein property of the associated Rees algebras
  • Introduction
  • Preliminaries
  • Proof of Theorem (1.1)
  • Proof of Theorems (1.3) and (1.5)
  • The Gorenstein property of Rees algebras \(R(F)\) and the condition \((S_2)\) for \(A\)
  • Graded rings \(R^{\dagger }\)
  • Examples for \(R_s(\mathbf p)\)
  • Normalized Rees algebras \(\overline R(I)\)
  • Bad example
  • References
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