Trends in Commutative Algebra,9780521831956

Trends in Commutative Algebra

by
ISBN13: 9780521831956
ISBN10: 0521831954
Format: Hardcover
Pub. Date: 2004-12-13
Publisher(s): Cambridge University Press
List Price: $142.00

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Summary

In 2002, an introductory workshop was held at the Mathematical Sciences Research Institute in Berkeley to survey some of the many new directions of the commutative algebra field. Six principal speakers each gave three lectures, accompanied by a help session, describing the interaction of commutative algebra with other areas of mathematics for a broad audience of graduate students and researchers. This book is based on those lectures, together with papers from contributing researchers. David Benson and Srikanth Iyengar present an introduction to the uses and concepts of commutative algebra in the cohomology of groups. Mark Haiman considers the commutative algebra of n points in the plane. Ezra Miller presents an introduction to the Hilbert scheme of points to complement Professor Haiman's paper. Further contributors include David Eisenbud and Jessica Sidman; Melvin Hochster; Graham Leuschke; Rob Lazarsfeld and Manuel Blickle; Bernard Teissier; and Ana Bravo.

Table of Contents

Preface ix
Commutative Algebra in the Cohomology of Groups
1(50)
Dave Benson
Introduction
2(1)
Some Examples
2(6)
Group Cohomology
8(4)
Finite Generation
12(3)
Krull Dimension
15(1)
Depth
16(3)
Associated Primes and Steenrod Operations
19(3)
Associated Primes and Transfer
22(1)
Idempotent Modules and Varieties
23(3)
Modules with Injective Cohomology
26(4)
Duality Theorems
30(4)
More Duality Theorems
34(2)
Dual Localization
36(2)
Quasiregular Sequences
38(13)
Appendix: Two-Groups of Order 64 and Their mod 2 Cohomology
43(2)
Acknowledgements
45(1)
References
45(6)
Modules and Cohomology over Group Algebras
51(36)
Srikanth Iyengar
Introduction
51(1)
The Group Algebra
52(4)
Modules over Group Algebras
56(8)
Projective Modules
64(5)
Structure of Projectives
69(5)
Cohomology of Supplemented Algebras
74(3)
Group Cohomology
77(2)
Finite Generation of the Cohomology Algebra
79(8)
References
84(3)
An Informal Introduction to Multiplier Ideals
87(28)
Manuel Blickle
Robert Lazarsfeld
Introduction
87(2)
Definition and Examples
89(6)
The Multiplier Ideal of Monomial Ideals
95(4)
Invariants Arising from Multiplier Ideals and Applications
99(4)
Further Local Properties of Multiplier Ideals
103(4)
Asymptotic Constructions
107(8)
References
112(3)
Lectures on the Geometry of Syzygies
115(38)
David Eisenbud
Jessica Sidman
Hilbert Functions and Syzygies
116(9)
Points in the Plane and an Introduction to Castelnuovo-Mumford Regularity
125(10)
The Size of Free Resolutions
135(6)
Linear Complexes and the Strands of Resolutions
141(12)
References
149(4)
Commutative Algebra of n Points in the Plane
153(28)
Mark Haiman
Ezra Miller
Introduction
153(1)
A Subspace Arrangement
154(7)
A Ring of Invariants
161(7)
A Remarkable Grobner Basis
168(13)
Appendix: Hilbert Schemes of Points in the Plane
172(7)
References
179(2)
Tight Closure Theory and Characteristic p Methods
181(30)
Melvin Hochster
Graham J. Leuschke
Introduction
181(3)
Reasons for Thinking About Tight Closure
184(3)
The Definition of Tight Closure in Positive Characteristic
187(1)
Basic Properties of Tight Closure and the Briancon Skoda Theorem
188(2)
Direct Summands of Regular Rings are Cohen--Macaulay
190(1)
The Ein-Lazarsfeld-Smith Comparison Theorem
191(1)
Extending the Theory to Affine Algebras in Characteristic 0
192(1)
Test Elements
192(2)
Test Elements Using the Lipman-Sathaye Theorem
194(2)
Tight Closure for Submodules
196(3)
Further Thoughts and Questions
199(12)
Appendix: Some Examples in Tight Closure
201(6)
References
207(4)
Monomial Ideals, Binomial Ideals, Polynomial Ideals
211(36)
Bernard Teissier
Introduction
211(2)
Strong Principalization of Monomial Ideals by Toric Maps
213(5)
The Integral Closure of Ideals
218(2)
The Monomial Briancon--Skoda Theorem
220(2)
Polynomial Ideals and Nondegeneracy
222(7)
Resolution of Binomial Varieties
229(4)
Resolution of Singularities of Branches
233(14)
Appendix: Multiplicities, Volumes and Nondegeneracy
237(6)
References
243(4)
Some Facts About Canonical Subalgebra Bases
247
Ana Bravo
Introduction
247(1)
SABGI Bases Versus Grobner Bases
248(1)
When Are SAGBI Bases Finite?
249(1)
Finite SAGBI Bases
250(2)
An Algorithm to Compute SAGBI bases
252(1)
Geometric Interpretation
252(1)
Acknowledgments
253(1)
References
253

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