Trends in Commutative Algebra

In 2002, an introductory workshop was held at the Mathematical Sciences Research Institute in Berkeley to survey some of the many 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.

Preface; 1. Commutative algebra in the cohomology of groups Dave Benson; 2. Modules and cohomology over group algebras Srikanth Iyengar; 3. An informal introduction to multiplier ideals Manuel Blickle and Robert Lazarsfeld; 4. Lectures on the geometry of syzygies David Eisenbud, with a chapter by Jessica Sidman; 5. Commutative algebra of n points in the plane Mark Haiman, with an appendix by Ezra Miller; 6. Tight closure theory and characteristic p methods Melvin Hochster, with an appendix by Graham J. Leuschke; 7. Monomial ideals, binomial ideals, polynomial ideals Bernard Teissier; 8. Some facts about canonical subalgebra bases Ana Bravo.

This 2005 book is based on lectures by six internationally known experts presented at the 2002 MSRI introductory workshop on commutative algebra. These focus on the interaction of commutative algebra with other areas of mathematics, including algebraic geometry, group cohomology and representation theory, and combinatorics.