Trends in Commutative Algebra

by Edited by Luchezar L. Avramov , Mark Green , Craig Huneke , Karen E. Smith , Bernd Sturmfels

• ISBN13: 9780521831956
• ISBN10: 0521831954
• Format: Hardcover
• Copyright: 2004-12-13
• Publisher: Cambridge University Press

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