Synopses & Reviews
Interest in commutative algebra has surged over the past decades. In order to survey and highlight recent developments in this rapidly expanding field, the Centre de Recerca Matematica in Bellaterra organized a ten-days Summer School on Commutative Algebra in 1996. Lectures were presented by six high-level specialists, L. Avramov (Purdue), M.K. Green (UCLA), C. Huneke (Purdue), P. Schenzel (Halle), G. Valla (Genova) and W.V. Vasconcelos (Rutgers), providing a fresh and extensive account of the results, techniques and problems of some of the most active areas of research. The present volume is a synthesis of the lectures given by these authors. Research workers as well as graduate students in commutative algebra and nearby areas will find a useful overview of the field and recent developments in it. Reviews "All six articles are at a very high level; they provide a thorough survey of results and methods in their subject areas, illustrated with algebraic or geometric examples." - Acta Scientiarum Mathematicarum Avramov lecture: "... it contains all the major results [on infinite free resolutions], it explains carefully all the different techniques that apply, it provides complete proofs (...). This will be extremely helpful for the novice as well as the experienced." - Mathematical reviews Huneke lecture: "The topic is tight closure, a theory developed by M. Hochster and the author which has in a short time proved to be a useful and powerful tool. (...) The paper is extremely well organized, written, and motivated." - Zentralblatt MATH Schenzel lecture: "... this paper is an excellent introduction to applications of local cohomology." - Zentralblatt MATH Valla lecture: "... since he is an acknowledged expert on Hilbert functions and since his interest has been so broad, he has done a superb job in giving the readers a lively picture of the theory." - Mathematical reviews Vasconcelos lecture: "This is a very useful survey on invariants of modules over noetherian rings, relations between them, and how to compute them." - Zentralblatt MATH
Review
From reviews: "All six articles are at a very high level; they provide a thorough survey of results and methods in their subject areas, illustrated with algebraic or geometric examples." - Acta Scientiarum Mathematicarum Avramov lecture: "Although this text is not presented as a textbook, it contains all the major results in its area, it explains carefully all the different techniques that apply, it provides complete--albeit short-- proofs (some new, some with a new twist) of most of the results and references for the rest, and it presents numerous relevant questions. This will be extremely helpful for the novice as well as the experienced." - Mathematical reviews Huneke lecture: The topic is tight closure, a theory developed by Melvin Hochster and the author which has in a short time proved to be a useful and powerful tool. This paper is intended to be an introduction to the theory, and it is also a good source of examples and insight into the more recent developments that can be of interest to the more experienced.(...) The paper is extremely well organized, written, and motivated. - Zentralblatt MATH Schenzel lecture: "... this paper is an excellent introduction to applications of local cohomology." - Zentralblatt MATH "This interesting survey would be useful for both the novice and the expert." - Mathematical reviews Valla lecture: "But since he is an acknowledged expert on Hilbert functions and since his interest has been so broad, he has done a superb job in giving the readers a lively picture of the theory. (...) It will serve as a useful reference source for people interested in Hilbert functions." - Mathematical reviews Vasconcelos lecture: "This is a very useful survey on invariants of modules over noetherian rings, relations between them, and how to compute them." - Zentralblatt MATH
Review
From reviews:
"All six articles are at a very high level; they provide a thorough survey of results and methods in their subject areas, illustrated with algebraic or geometric examples." - Acta Scientiarum Mathematicarum
Avramov lecture: "Although this text is not presented as a textbook, it contains all the major results in its area, it explains carefully all the different techniques that apply, it provides complete--albeit short-- proofs (some new, some with a new twist) of most of the results and references for the rest, and it presents numerous relevant questions. This will be extremely helpful for the novice as well as the experienced." - Mathematical reviews
Huneke lecture: The topic is tight closure, a theory developed by Melvin Hochster and the author which has in a short time proved to be a useful and powerful tool. This paper is intended to be an introduction to the theory, and it is also a good source of examples and insight into the more recent developments that can be of interest to the more experienced.(...) The paper is extremely well organized, written, and motivated. - Zentralblatt MATH
Schenzel lecture: "... this paper is an excellent introduction to applications of local cohomology." - Zentralblatt MATH
"This interesting survey would be useful for both the novice and the expert." - Mathematical reviews
Valla lecture: "But since he is an acknowledged expert on Hilbert functions and since his interest has been so broad, he has done a superb job in giving the readers a lively picture of the theory. (...) It will serve as a useful reference source for people interested in Hilbert functions." - Mathematical reviews
Vasconcelos lecture: "This is a very useful survey on invariants of modules over noetherian rings, relations between them, and how to compute them." - Zentralblatt MATH
Synopsis
Based on the Summer School on Commutative Algebra at the Centre de Recerca Matematica in Bellaterra, this volume offers an overview of research in the field. Readers will find a fresh and extensive account of the results, techniques and problems discussed.
Table of Contents
Preface.- Infinite Free Resolutions.- Generic Initial Ideals.- Tight Closure, Parameter Ideals, and Geometry.- On the Use of Local Cohomology in Algebra and Geometry.- Problems and Results on Hilbert Functions of Graded Algebras.- Cohomological Degrees of Graded Modules.