Cohen Macaulay Rings

Author : Winfried Bruns,H. Jürgen Herzog
Publisher : Cambridge University Press
Page : 471 pages
File Size : 40,6 Mb
Release : 1998-06-18
Category : Mathematics
ISBN : 9780521566742

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Cohen-Macaulay Rings by Winfried Bruns,H. Jürgen Herzog Pdf

In the last two decades Cohen-Macaulay rings and modules have been central topics in commutative algebra. This book meets the need for a thorough, self-contained introduction to the homological and combinatorial aspects of the theory of Cohen-Macaulay rings, Gorenstein rings, local cohomology, and canonical modules. A separate chapter is devoted to Hilbert functions (including Macaulay's theorem) and numerical invariants derived from them. The authors emphasize the study of explicit, specific rings, making the presentation as concrete as possible. So the general theory is applied to Stanley-Reisner rings, semigroup rings, determinantal rings, and rings of invariants. Their connections with combinatorics are highlighted, e.g. Stanley's upper bound theorem or Ehrhart's reciprocity law for rational polytopes. The final chapters are devoted to Hochster's theorem on big Cohen-Macaulay modules and its applications, including Peskine-Szpiro's intersection theorem, the Evans-Griffith syzygy theorem, bounds for Bass numbers, and tight closure. Throughout each chapter the authors have supplied many examples and exercises which, combined with the expository style, will make the book very useful for graduate courses in algebra. As the only modern, broad account of the subject it will be essential reading for researchers in commutative algebra.

Author : Yūji Yoshino
Publisher : Cambridge University Press
Page : 191 pages
File Size : 54,7 Mb
Release : 1990-06-28
Category : Mathematics
ISBN : 9780521356947

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Maximal Cohen-Macaulay Modules Over Cohen-Macaulay Rings by Yūji Yoshino Pdf

The purpose of these notes is to explain in detail some topics on the intersection of commutative algebra, representation theory and singularity theory. They are based on lectures given in Tokyo, but also contain new research. It is the first cohesive account of the area and will provide a useful synthesis of recent research for algebraists.

Author : Eben Matlis
Publisher : Lecture Notes in Mathematics
Page : 178 pages
File Size : 44,8 Mb
Release : 1973-06-04
Category : Mathematics
ISBN : UOM:39015017317077

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One-Dimensional Cohen-Macaulay Rings by Eben Matlis Pdf

Author : Winfried Bruns,Udo Vetter
Publisher : Springer
Page : 246 pages
File Size : 46,8 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540392743

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Determinantal Rings by Winfried Bruns,Udo Vetter Pdf

Determinantal rings and varieties have been a central topic of commutative algebra and algebraic geometry. Their study has attracted many prominent researchers and has motivated the creation of theories which may now be considered part of general commutative ring theory. The book gives a first coherent treatment of the structure of determinantal rings. The main approach is via the theory of algebras with straightening law. This approach suggest (and is simplified by) the simultaneous treatment of the Schubert subvarieties of Grassmannian. Other methods have not been neglected, however. Principal radical systems are discussed in detail, and one section is devoted to each of invariant and representation theory. While the book is primarily a research monograph, it serves also as a reference source and the reader requires only the basics of commutative algebra together with some supplementary material found in the appendix. The text may be useful for seminars following a course in commutative ring theory since a vast number of notions, results, and techniques can be illustrated significantly by applying them to determinantal rings.

Author : Graham J. Leuschke,Roger Wiegand
Publisher : American Mathematical Soc.
Page : 390 pages
File Size : 42,5 Mb
Release : 2012-05-02
Category : Mathematics
ISBN : 9780821875810

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Cohen-Macaulay Representations by Graham J. Leuschke,Roger Wiegand Pdf

This book is a comprehensive treatment of the representation theory of maximal Cohen-Macaulay (MCM) modules over local rings. This topic is at the intersection of commutative algebra, singularity theory, and representations of groups and algebras. Two introductory chapters treat the Krull-Remak-Schmidt Theorem on uniqueness of direct-sum decompositions and its failure for modules over local rings. Chapters 3-10 study the central problem of classifying the rings with only finitely many indecomposable MCM modules up to isomorphism, i.e., rings of finite CM type. The fundamental material--ADE/simple singularities, the double branched cover, Auslander-Reiten theory, and the Brauer-Thrall conjectures--is covered clearly and completely. Much of the content has never before appeared in book form. Examples include the representation theory of Artinian pairs and Burban-Drozd's related construction in dimension two, an introduction to the McKay correspondence from the point of view of maximal Cohen-Macaulay modules, Auslander-Buchweitz's MCM approximation theory, and a careful treatment of nonzero characteristic. The remaining seven chapters present results on bounded and countable CM type and on the representation theory of totally reflexive modules.

Author : M. Aigner
Publisher : Springer Science & Business Media
Page : 259 pages
File Size : 47,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9789401012201

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Higher Combinatorics by M. Aigner Pdf

It is general consensus that Combinatorics has developed into a full-fledged mathematical discipline whose beginnings as a charming pastime have long since been left behind and whose great signifi cance for other branches of both pure and applied mathematics is only beginning to be realized. The last ten years have witnessed a tremendous outburst of activity both in relatively new fields such as Coding Theory and the Theory of Matroids as well as in' more time honored endeavors such as Generating Functions and the Inver sion Calculus. Although the number of text books on these subjects is slowly increasing, there is also a great need for up-to-date surveys of the main lines of research designed to aid the beginner and serve as a reference for the expert. It was the aim of the Advanced Study Institute "Higher Combinatorics" in Berlin, 1976, to help fulfill this need. There were five sections: I. Counting Theory, II. Combinatorial Set Theory and Order Theory, III. Matroids, IV. Designs and V. Groups and Coding Theory, with three principal lecturers in each section. Expanded versions of most lectures form the contents of this book. The Institute was designed to offer, especially to young researchers, a comprehen sive picture of the most interesting developments currently under way. It is hoped that these proceedings will serve the same purpose for a wider audience.

Author : Eben Matlis
Publisher : Springer
Page : 168 pages
File Size : 47,9 Mb
Release : 2006-11-15
Category : Mathematics
ISBN : 9783540469230

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One-Dimensional Cohen-Macaulay Rings by Eben Matlis Pdf

Author : R. Y. Sharp
Publisher : Cambridge University Press
Page : 371 pages
File Size : 55,8 Mb
Release : 2000
Category : Mathematics
ISBN : 9780521646239

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Steps in Commutative Algebra by R. Y. Sharp Pdf

Introductory account of commutative algebra, aimed at students with a background in basic algebra.

Author : Hideyuki Matsumura
Publisher : Cambridge University Press
Page : 338 pages
File Size : 48,8 Mb
Release : 1989-05-25
Category : Mathematics
ISBN : 0521367646

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Commutative Ring Theory by Hideyuki Matsumura Pdf

This book explores commutative ring theory, an important a foundation for algebraic geometry and complex analytical geometry.

Author : David Eisenbud,Irena Peeva
Publisher : Springer
Page : 107 pages
File Size : 43,8 Mb
Release : 2016-03-08
Category : Mathematics
ISBN : 9783319264370

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Minimal Free Resolutions over Complete Intersections by David Eisenbud,Irena Peeva Pdf

This book introduces a theory of higher matrix factorizations for regular sequences and uses it to describe the minimal free resolutions of high syzygy modules over complete intersections. Such resolutions have attracted attention ever since the elegant construction of the minimal free resolution of the residue field by Tate in 1957. The theory extends the theory of matrix factorizations of a non-zero divisor, initiated by Eisenbud in 1980, which yields a description of the eventual structure of minimal free resolutions over a hypersurface ring. Matrix factorizations have had many other uses in a wide range of mathematical fields, from singularity theory to mathematical physics.

Author : Yuji Yoshino
Publisher : Unknown
Page : 0 pages
File Size : 54,7 Mb
Release : 1990
Category : Electronic
ISBN : 0524356947

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Cohen-Macaulay Modules Over Cohen-Macaulay Rings by Yuji Yoshino Pdf

Author : Melvin Hochster
Publisher : American Mathematical Soc.
Page : 86 pages
File Size : 40,5 Mb
Release : 1975
Category : Mathematics
ISBN : 9780821816745

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Topics in the Homological Theory of Modules Over Commutative Rings by Melvin Hochster Pdf

Contains expository lectures from the CBMS Regional Conference in Mathematics held at the University of Nebraska, June 1974. This book deals mainly with developments and still open questions in the homological theory of modules over commutative (usually, Noetherian) rings.

Author : Craig Huneke,Irena Swanson
Publisher : Cambridge University Press
Page : 446 pages
File Size : 45,6 Mb
Release : 2006-10-12
Category : Mathematics
ISBN : 9780521688604

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Integral Closure of Ideals, Rings, and Modules by Craig Huneke,Irena Swanson Pdf

Ideal for graduate students and researchers, this book presents a unified treatment of the central notions of integral closure.

Author : David Eisenbud
Publisher : Springer Science & Business Media
Page : 254 pages
File Size : 44,5 Mb
Release : 2006-10-28
Category : Mathematics
ISBN : 9780387264561

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The Geometry of Syzygies by David Eisenbud Pdf

First textbook-level account of basic examples and techniques in this area. Suitable for self-study by a reader who knows a little commutative algebra and algebraic geometry already. David Eisenbud is a well-known mathematician and current president of the American Mathematical Society, as well as a successful Springer author.

Author : Maurice Auslander,Mark Bridger
Publisher : American Mathematical Soc.
Page : 150 pages
File Size : 43,9 Mb
Release : 1969
Category : Commutative rings
ISBN : 9780821812945

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Stable Module Theory by Maurice Auslander,Mark Bridger Pdf

The notions of torsion and torsion freeness have played a very important role in module theory--particularly in the study of modules over integral domains. Furthermore, the use of homological techniques in this connection has been well established. It is the aim of this paper to extend these techniques and to show that this extension leads naturally to several new concepts (e.g. k-torsion freeness and Gorenstein dimension) which are useful in the classification of modules and rings.