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During late June and early July of 1987 a three week program
(dubbed "microprogram") in Commutative Algebra was held at the
Mathematical Sciences Research Institute at Berkeley. The intent of
the microprogram was to survey recent major results and current
trends in the theory of commutative rings, especially commutative
Noetherian rings. There was enthusiastic international
participation. The papers in this volume, some of which are
expository, some strictly research, and some a combination, reflect
the intent of the program. They give a cross-section of what is
happening now in this area. Nearly all of the manuscripts were
solicited from the speakers at the conference, and in most
instances the manuscript is based on the conference lecture. The
editors hope that they will be of interest and of use both to
experts and neophytes in the field. The editors would like to
express their appreciation to the director of MSRI, Professor
Irving Kaplansky, who first suggested the possibility of such a
conference and made the task of organization painless. We would
also like to thank the IVISRI staff who were unfailingly efficient,
pleasant, and helpful during the meeting, and the manager of MSRI,
Arlene Baxter, for her help with this volume. Finally we would like
to express our appreciation to David Mostardi who did much of the
typing and put the electronic pieces together.
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.
Integral closure has played a role in number theory and algebraic
geometry since the nineteenth century, but a modern formulation of
the concept for ideals perhaps began with the work of Krull and
Zariski in the 1930s. It has developed into a tool for the analysis
of many algebraic and geometric problems. This book collects
together the central notions of integral closure and presents a
unified treatment. Techniques and topics covered include: behavior
of the Noetherian property under integral closure, analytically
unramified rings, the conductor, field separability, valuations,
Rees algebras, Rees valuations, reductions, multiplicity, mixed
multiplicity, joint reductions, the Briancon-Skoda theorem,
Zariski's theory of integrally closed ideals in two-dimensional
regular local rings, computational aspects, adjoints of ideals and
normal homomorphisms. With many worked examples and exercises, this
book will provide graduate students and researchers in commutative
algebra or ring theory with an approachable introduction leading
into the current literature.
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. 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.
The main objective of this monograph is to lay the foundations of
tight closure theory for Noetherian rings containing a field of
characteristic 0. It has been more than ten years since the authors
first began work on tight closure. In that time they have published
many articles on the topic. This remarkably potent method has led
to a number of generalizations of old theorems, improved proofs,
and a host of beautiful new results. This monograph will serve as a
marvelous introduction to tight closure for both researchers and
graduate students in commutative algebra.
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