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Author: Alan B. Evans
Publisher:
ISBN:
Category : Algebra, Universal
Languages : en
Pages : 122
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Book Description
Author: Alan B. Evans
Publisher:
ISBN:
Category : Algebra, Universal
Languages : en
Pages : 122
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Book Description
Author: Winfried Bruns
Publisher: Cambridge University Press
ISBN: 0521566746
Category : Mathematics
Languages : en
Pages : 471
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Book Description
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: Winfried Bruns
Publisher: Springer
ISBN: 3540392742
Category : Mathematics
Languages : en
Pages : 246
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Book Description
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: Eben Matlis
Publisher: Springer
ISBN: 3540469230
Category : Mathematics
Languages : en
Pages : 168
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Book Description
Author: Birger Iversen
Publisher: World Scientific
ISBN: 9814603678
Category : Mathematics
Languages : en
Pages : 224
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Book Description
The content in Chapter 1-3 is a fairly standard one-semester course on local rings with the goal to reach the fact that a regular local ring is a unique factorization domain. The homological machinery is also supported by Cohen-Macaulay rings and depth. In Chapters 4-6 the methods of injective modules, Matlis duality and local cohomology are discussed. Chapters 7-9 are not so standard and introduce the reader to the generalizations of modules to complexes of modules. Some of Professor Iversen's results are given in Chapter 9. Chapter 10 is about Serre's intersection conjecture. The graded case is fully exposed. The last chapter introduces the reader to Fitting ideals and McRae invariants.
Author: Shirō Gotō
Publisher: American Mathematical Soc.
ISBN: 0821825844
Category : Mathematics
Languages : en
Pages : 149
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Book Description
At first, this volume was intended to be an investigation of symbolic blow-up rings for prime ideals defining curve singularities. The motivation for that has come from the recent 3-dimensional counterexamples to Cowsik's question, given by the authors and Watanabe: it has to be helpful, for further researches on Cowsik's question and a related problem of Kronecker, to generalize their methods to those of a higher dimension. However, while the study was progressing, it proved apparent that the framework of Part I still works, not only for the rather special symbolic blow-up rings but also in the study of Rees algebras R(F) associated to general filtrations F = {F[subscript]n} [subscript]n [subscript][set membership symbol][subscript bold]Z of ideals. This observation is closely explained in Part II of this volume, as a general ring-theory of Rees algebras R(F). We are glad if this volume will be a new starting point for the further researchers on Rees algebras R(F) and their associated graded rings G(F).
Author:
Publisher: American Mathematical Soc.
ISBN: 9780821872499
Category : Mathematics
Languages : en
Pages : 312
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Book Description
This is an introduction to local cohomology which takes cognizance of the breadth of its interactions with other areas of mathematics. The text covers topics such as the number of defining equations of algebraic sets, connectedness properties of algebraic sets, and connections to sheaf cohomology and to de Rham cohomology.
Author: Jan R. Strooker
Publisher: Cambridge University Press
ISBN: 0521315263
Category : Mathematics
Languages : en
Pages : 325
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Book Description
This book presents an account of several conjectures arising in commutative algebra from the pioneering work of Serre and Auslander-Buchsbaum. The approach is via Hochster's 'Big Cohen-Macaulay modules', though the complementary view point of Peskine-Szpiro and Roberts, who study the homology of certain complexes, is not neglected. Various refinements of Hochster's construction, obtained in collaboration with Bartijn, are included. A special feature is a long chapter written by Van den Dries which explains how a certain type of result can be 'lifted' from prime characteristic to characteristic zero. Though this is primarily a research monograph, it does provide introductions to most of the topics treated. Non-experts may therefore find it an appealing guide into an active area of algebra.
Author: Melvin Hochster
Publisher: American Mathematical Soc.
ISBN: 0821825569
Category : Mathematics
Languages : en
Pages : 105
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Book Description
This book uses a powerful new technique, tight closure, to provide insight into many different problems that were previously not recognized as related. The authors develop the notion of weakly Cohen-Macaulay rings or modules and prove some very general acyclicity theorems. These theorems are applied to the new theory of phantom homology, which uses tight closure techniques to show that certain elements in the homology of complexes must vanish when mapped to well-behaved rings. These ideas are used to strengthen various local homological conjectures. Initially, the authors develop the theory in positive characteristic, but it can be extended to characteristic 0 by the method of reduction to characteristic $p$. The book would be suitable for use in an advanced graduate course in commutative algebra.
Author: Lawrence Joel Marx
Publisher:
ISBN:
Category :
Languages : en
Pages : 126
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Book Description
