**Memoirs of the American Mathematical Society**

1994;
134 pp;
Softcover

MSC: Primary 13; 14;

Print ISBN: 978-0-8218-2584-6

Product Code: MEMO/110/526

List Price: $42.00

Individual Member Price: $25.20

**Electronic ISBN: 978-1-4704-0105-4
Product Code: MEMO/110/526.E**

List Price: $42.00

Individual Member Price: $25.20

# The Cohen-Macaulay and Gorenstein Rees Algebras Associated to Filtrations

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*Shiro Goto; Koji Nishida*

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

# Table of Contents

## The Cohen-Macaulay and Gorenstein Rees Algebras Associated to Filtrations

- Contents v6 free
- Preface vii8 free
- Part I: The Cohen-Macaulay symbolic Rees algebras for curue singularities 110 free
- Part II: Filtrations and the Gorenstein property of the associated Rees algebras 6978
- 1 Introduction 6978
- 2 Preliminaries 7988
- 3 Proof of Theorem (1.1) 8695
- 4 Proof of Theorems (1.3) and (1.5) 91100
- 5 The Gorenstein property of Rees algebras R(F) and the condition (S[sub(2)]) for A 101110
- 6 Graded rings R† 107116
- 7 Examples for R[sub(s)](p) 119128
- 8 Normalized Rees algebras R(I) 123132
- 9 Bad example 127136
- References 130139