Commutative Algebra: Durham 1981

doi:10.1017/CBO9781107325517

This book is concerned with the research conducted in the late 1970s and early 1980s in the theory of commutative Neotherian rings. It consists of articles by invited speakers at the Symposium of Commutative Algebra held at the University of Durham in July 1981; these articles are all based on lectures delivered at the Symposium. The purpose of this book is to provide a record of at least some aspects of the Symposium, which several of the world leaders in the field attended. Several articles are included which provide surveys, incorporating historical perspective, details of progress made and indications of possible future lines of investigation. The book will be of interest to scholars of commutative and local algebra.

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Contents

Frontmatter
pp i-iv
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Contents
pp v-vi
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Preface
pp vii-viii
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Addresses of Contributors
pp ix-x
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List of Participants
pp xi-xii
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PART I - THE LOCAL HOMOLOGICAL CONJECTURES, BIG COHEN-MACAULAY MODULES, AND RELATED TOPICS
pp 1-1
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The syzygy problem: a new proof and historical perspective
pp 2-11
- By E.G. Evans, University of Illinois, Phillip Griffith, University of Illinois
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The theory of homological dimensions of complexes
pp 12-17
- By Hans-Bjørn Foxby, Københavns Universitet
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Complexes of injective modules
pp 18-31
- By Hans-Bjørn Foxby, Københavns Universitet
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The local homological conjectures
pp 32-54
- By Melvin Hochster, University of Michigan
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The rank of a module
pp 55-60
- By G. Horrocks, University of Newcastle upon Tyne
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Modules of generalized fractions and balanced big Cohen-Macaulay modules
pp 61-82
- By R. Y. Sharp, University of Sheffield, H. Zakeri, University of Sheffield
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Sur la théorie des complexes parfaits
pp 83-90
- By L. Szpiro, École Normale Supérieure
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PART II - DETERMINANTAL IDEALS, FINITE FREE RESOLUTIONS, AND RELATED TOPICS
pp 91-91
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Some exact complexes and filtrations related to certain special young diagrams
pp 92-108
- By Kaan Akin, Brandeis University, David A. Buchsbaum, Brandeis University
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The canonical module of a determinantal ring
pp 109-120
- By Winfried Bruns, Universität Osnabrück–Abteilung Vechta–Driverstrasse 22
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The MacRae invariant
pp 121-128
- By Hans-Bjørn Foxby, Københavns Universitet
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Finite free resolutions arid some basic concepts of commutative algebra
pp 129-138
- By D.G. Northcott, University of Sheffield
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PART III - MULTIPLICITY THEORY, HILBERT AND POINCARÉ SERIES, ASSOCIATED GRADED RINGS, AND RELATED TOPICS
pp 139-139
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Blowing-up of Buchsbaum rings
pp 140-162
- By Shiro Goto, Nihon University
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Necessary conditions for an analytic algebra to be strict
pp 163-169
- By J. Herzog, Universitätsstrasse 3
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Multiplicities, hilbert functions and degree functions
pp 170-178
- By D. Rees, University of Exeter
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Finiteness conditions in commutative algebra and solution of a problem of Vasconcelos
pp 179-203
- By Jan-Erik Roos, University of Stockholm
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On the use of graded lie algebras in the theory of local rings
pp 204-230
- By Jan-Erik Roos, University of Stockholm
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Reductions, local cohomology and hilbert functions of local rings
pp 231-241
- By Judith D. Sally, Northwestern University
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FURTHER PROBLEMS
pp 242-250
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Commutative Algebra

Durham 1981

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Contents

Frontmatter
pp i-iv

Contents
pp v-vi

Preface
pp vii-viii

Addresses of Contributors
pp ix-x

List of Participants
pp xi-xii

PART I - THE LOCAL HOMOLOGICAL CONJECTURES, BIG COHEN-MACAULAY MODULES, AND RELATED TOPICS
pp 1-1

The syzygy problem: a new proof and historical perspective
pp 2-11

The theory of homological dimensions of complexes
pp 12-17

Complexes of injective modules
pp 18-31

The local homological conjectures
pp 32-54

The rank of a module
pp 55-60

Modules of generalized fractions and balanced big Cohen-Macaulay modules
pp 61-82

Sur la théorie des complexes parfaits
pp 83-90

PART II - DETERMINANTAL IDEALS, FINITE FREE RESOLUTIONS, AND RELATED TOPICS
pp 91-91

Some exact complexes and filtrations related to certain special young diagrams
pp 92-108

The canonical module of a determinantal ring
pp 109-120

The MacRae invariant
pp 121-128

Finite free resolutions arid some basic concepts of commutative algebra
pp 129-138

PART III - MULTIPLICITY THEORY, HILBERT AND POINCARÉ SERIES, ASSOCIATED GRADED RINGS, AND RELATED TOPICS
pp 139-139

Blowing-up of Buchsbaum rings
pp 140-162

Necessary conditions for an analytic algebra to be strict
pp 163-169

Multiplicities, hilbert functions and degree functions
pp 170-178

Finiteness conditions in commutative algebra and solution of a problem of Vasconcelos
pp 179-203

On the use of graded lie algebras in the theory of local rings
pp 204-230

Reductions, local cohomology and hilbert functions of local rings
pp 231-241

FURTHER PROBLEMS
pp 242-250

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Commutative Algebra

Durham 1981

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