A Term of Commutative Algebra

Allen Altman, Steven Kleiman

 

 

 

A Term of Commutative Algebra

A Term of Commutative Algebra
Allen Altman – Simons Rock
Steven Kleiman – Massachusetts Institute of Technology

ISBN-10: 0-9885572-1-5
ISBN-13: 978-0-9885572-1-5
237 Pages

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There is no shortage of books on Commutative Algebra, but the present book is different. Most books are monographs, with extensive coverage. There is one notable exception: Atiyah and Macdonald’s 1969 classic. It is a clear, concise, and efficient textbook, aimed at beginners, with a good selection of topics. So it has remained popular. However, its age and flaws do show. So there is need for an updated and improved version, which the present book aims to be.

 

Contents

1  Rings and Ideals
2  Prime Ideals
3  Radicals
4  Modules
5  Exact Sequences
6  Direct Limits
7  Filtered Direct Limits
8  Tensor Products
9  Flatness
10  Cayley–Hamilton Theorem
11  Localization of Rings
12  Localization of Modules
13  Support
14  Krull–Cohen–Seidenberg Theory
15  Noether Normalization
Appendix: Jacobson Rings
16  Chain Conditions
17  Associated Primes
18  Primary Decomposition
19  Length
20  Hilbert Functions
Appendix: Homogeneity
21  Dimension
22  Completion
23  Discrete Valuation Rings
24  Dedekind Domains
25  Fractional Ideals
26  Arbitrary Valuation Rings

 

Features

 

- Comprises twenty-six sections; each represents a single lecture, and is self-contained.


- “Grew out of a course of lectures” based primarily on Atiyah and Macdonald’s book, but has been offered a number of times, and has evolved over the years, influenced by other publications and the reactions of the students.


- Exercises are integrated into the development, and complete solutions are given at the end of the book.


- Exercises are designed to provide a means for students to check, solidify, and expand their understanding of the material. The exercises are intentionally not difficult, tricky, or involved. Rarely do they introduce new techniques, although many statements are used afterwards.


- Students are encouraged to try to solve each exercise before looking up its solution. If they become stuck, then they should review the relevant material; if they remain stuck, then they should study the solution, making sure they can eventually solve the exercise on their own. However, students should read the given solution, even if they think they already know it, just to make sure; also, some exercises provide enlightening alternative solutions.


- Instructors are encouraged to examine their students, possibly orally at a blackboard, on a small randomly chosen subset of exercises that have been assigned for the students to write up in their own words over the course of the term.

 

Authors

Allen Altman
Dr. Allen Altman has taught at the University of California at San Diego; the
Universidad Simón Bolívar in Caracas, Venezuela, where he helped to establish the mathematics program; MIT; the University of Oslo (Norway); the University of Pernambuco (Brazil); and the Institute of Pure and Applied Mathematics in Rio de Janeiro (Brazil). He has been awarded Fulbright, National Science Foundation, and Woodrow Wilson fellowships. Dr. Altman's publications include regular contributions to Mathematical Reviews and articles in Transactions of the American Mathematical Society, Communications in Algebra, Advances in Mathematics, Compositio Mathematica, Bulletin of the American Mathematical Society, and American Journal of Mathematics.

Steve Kleiman
Steve Kleiman received the S.B. from MIT in 1961, and the M.A. and Ph.D. from Harvard in 1962 and 65. Oscar Zariski was his doctoral advisor. He went to Columbia University as a Ritt instructor, joining their faculty from 1966-69. He subsequently joined the MIT mathematics faculty, professor in 1976. Professor Kleiman concentrates on problems in algebraic geometry and commutative algebra. A recipient of the Sloan and Guggenheim fellowships, he received the Honorary Doctorate of Science from the University of Copenhagen in 1989, and was elected Foreign Member of the Norwegian Academy of Sciences and Letters, and of the Royal Danish Academy of Sciences and Letters in 2002.