Preface

Summary

Over the last fifteen years, commutative algebra has experienced a striking evolution. During this period the outlook of the subject has been altered, new connections to other areas have been established, and powerful techniques have been developed. To foster further development a year-long program on commutative algebra was held at MSRI during the 2002–03 academic year, starting with an introductory workshop on September 9–13, 2002. This workshop concentrated on the interplay and growing connections between commutative algebra and other areas, such as algebraic geometry, the cohomology of groups, and combinatorics.

Six main speakers each gave a series of three talks during the week: David Benson, David Eisenbud, Mark Haiman, Melvin Hochster, Rob Lazarsfeld, and Bernard Teissier. The workshop was very well attended, with more than 120 participants. Every series of main talks was supplemented by a discussion/talk session presented by a young researcher: Manuel Blickle, Ana Bravo, Srikanth Iyengar, Graham Leuschke, Ezra Miller, and Jessica Sidman. Each of these speakers has contributed a paper, or in some cases a combined paper, in this volume.

David Benson spoke on the cohomology of groups, presenting some of the many questions which are unanswered and which have a close relationship to modern commutative algebra. He gave us many convincing reasons for working in the “graded” commutative case, where signs are introduced when commuting elements of odd degree. Srikanth Iyengar gives background information for Benson’s notes.

David Eisenbud spoke on a classical subject in commutative algebra: free resolutions. In his paper with a chapter by Jessica Sidman, he visits this classic territory with a different perspective, by drawing close ties between graded free resolutions and the geometry of projective varieties. He leads us through recent developments, including Mark Green’s proof of the linear syzygy conjecture.

Mark Haiman lectured on the commutative algebra of n points in the plane. This leads quite rapidly to the geometry of the Hilbert scheme, and to substantial combinatorial questions (and answers) which can be phrased in terms of common questions in commutative algebra such as asking about the Cohen–Macaulay property for certain Rees algebras. Ezra Miller writes an appendix about the Hilbert scheme of n points in the plane.