Commutative Algebra

Format:

Book

Author/Creator:

Hoffman

Jia, Xiaohong, Author.

Wang, Haohao, Author.

Language:

English

Physical Description:

1 online resource (300 p.) ill

Edition:

1st ed.

Place of Publication:

Mercury Learning and Information 2016

Dulles, Virginia ; Boston, Massachusetts ; New Delhi, [India] : Mercury Learning & Information, 2016.

Summary:

The purpose of this book is twofold: to present some basic ideas in commutative algebra and algebraic geometry and to introduce topics of current research, centered around the themes of Gr bner bases, resultants and syzygies. The presentation of the material combines definitions and proofs with an emphasis on concrete examples. The authors illustrate the use of software such as Mathematica and Singular.The design of the text in each chapter consists of two parts: the fundamentals and the applications, which make it suitable for courses of various lengths, levels, and topics based on the mathematical background of the students. The fundamentals portion of the chapter is intended to be read with minimal outside assistance, and to learn some of the most useful tools in commutative algebra. The applications of the chapter are to provide a glimpse of the advanced mathematical research where the topics and results are related to the material presented earlier. In the applications portion, the authors present a number of results from a wide range of sources without detailed proofs. The applications portion of the chapter is suitable for a reader who knows a little commutative algebra and algebraic geometry already, and serves as a guide to some interesting research topics.This book should be thought of as an introduction to more advanced texts and research topics. Its novelty is that the material presented is a unique combination of the essential methods and the current research results. The goal is to equip readers with the fundamental classical algebra and geometry tools, ignite their research interests, and initiate some potential research projects in the related areas.

Contents:

Front Matter

Half Title

License Page

Title Page

Copyright Page

Contents

Preface

List of Figures

Body Matter

1. Introduction

2. Rings and Ideals

2.1 Rings and Ideals

2.2 Localization of a Ring

2.3 Ideals in a Polynomial Ring

2.4 Grobner Basis of an Ideal

2.5 Elimination and Extension

2.6 Implicitization

2.7 Schemes

2.8 Grobner Basis Applications

3. Modules

3.1 Modules

3.2 Exact Sequences and Commutative Diagrams

3.3 Projective and Injective Modules

3.4 Tensor Product of Modules

3.5 Flatness

3.6 Localization

3.7 Local Property

3.8 Associated Primes

3.9 Primary Decomposition of Modules

3.10 Modules of Finite Length

3.11 Krull Dimension of a Ring

3.12 Dimension of Modules

3.13 Rank of Modules

3.14 Computational Applications

4. Graded and Local Rings and Modules

4.1 Graded Rings and Modules

4.2 Graded Localization

4.3 Graded Associated Primes

4.4 Filtration

4.5 System of Parameters

4.6 Regular and Quasi-regular M-Sequence

4.7 Proj of a Graded Ring

4.8 Graded Free Resolution

4.9 Dimension and Multiplicity

4.10 Applications

5. Homological Method

5.1 Complexes

5.2 Complex of Tor

5.3 Koszul Complex

5.4 Regular Sequences

5.5 Regular Rings

5.6 Complex of Ext

5.7 Exactness Criteria for Complexes

5.8 Local Cohomology

5.9 Applications

Back Matter

BIbliography

Index.

Notes:

Includes bibliographical references and index.

Description based on publisher supplied metadata and other sources.

ISBN:

9781683922643

1683922646

9781944534707

1944534709

1-944534-70-9

OCLC:

1537942312