Synopses & Reviews
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. David Eisenbud and Jessica Sidman give an introduction to the geometry of syzygies, addressing the basic question of relating the geometry of a projective variety with an embedding into projective space to the minimal free resolution of its coordinate ring over the polynomial ring of ambient projective space. Melvin Hochster presents an introduction to the theory of tight closure. Graham Leuschke adds a supporting paper on examples of tight closure and how to compute it. Rob Lazarsfeld and Manuel Blickle discuss the theory of multiplier ideals and how they can be used in commutative algebra. Bernard Teissier presents ideas related to resolution of singularities, complemented by Ana Bravo's paper on canonical subalgebra bases.
Synopsis
This book is based on lectures by six internationally known experts presented at the 2002 MSRI introductory workshop on commutative algebra. They focus on the interaction of commutative algebra with other areas of mathematics, including algebraic geometry, group cohomology and representation theory, and combinatorics, with all necessary background provided. Short complementary papers describing work at the research frontier are also included. The unusual scope and format make the book invaluable reading for graduate students and researchers interested in commutative algebra and its various uses.
Synopsis
This 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. Six short complementary papers describing work at the research frontier are also included.
Table of Contents
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.