A Course In Commutative Algebra

This text provides the theoretical foundation of algebraic geometry and algebraic number theory. Assuming as prerequisite a standard graduate algebra course, this text attempts to reach an advanced level quickly and efficiently.

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Algebra

**Publication date**: 31 Dec 2003

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**Post time**: 26 Jan 2007 07:03:40

A Course In Commutative Algebra

This text provides the theoretical foundation of algebraic geometry and algebraic number theory. Assuming as prerequisite a standard graduate algebra course, this text attempts to reach an advanced level quickly and efficiently.

Terms and Conditions:

Book Excerpts:

This is a text for a basic course in commutative algebra, written in accordance with the following objectives.

The course should be accessible to those who have studied algebra at the beginning graduate level. For general algebraic background, see author's online text "Abstract Algebra: The Basic Graduate Year". This text will be referred to as TBGY.

The idea is to help the student reach an advanced level as quickly and efficiently as possible.

- In Chapter 1, the theory of primary decomposition is developed so as to apply to modules as well as ideals.

- In Chapter 2, integral extensions are treated in detail, including the lying over, going up and going down theorems. The proof of the going down theorem does not require advanced field theory.

- Valuation rings are studied in Chapter 3, and the characterization theorem for discrete valuation rings is proved.

- Chapter 4 discusses completion, and covers the Artin-Rees lemma and the Krull intersection theorem.

- Chapter 5 begins with a brief digression into the calculus of finite differences, which clarifies some of the manipulations involving Hilbert and Hilbert-Samuel polynomials. The main result is the dimension theorem for finitely generated modules over Noetherian local rings. A corollary is Krull’s principal ideal theorem. Some connections with algebraic geometry are established via the study of affine algebras.

- Chapter 6 introduces the fundamental notions of depth, systems of parameters, and M-sequences.

- Chapter 7 develops enough homological algebra to prove, under approprate hypotheses, that all maximal M-sequences have the same length.

- The brief Chapter 8 develops enough theory to prove that a regular local ring is an integral domain as well as a Cohen-Macaulay ring. After completing the course, the student should be equipped to meet the Koszul complex, the Auslander-Buchsbaum theorems, and further properties of Cohen-Macaulay rings in a more advanced course.

Robert B. Ash wrote:(C) Copyright 2003, by Robert B. Ash. Paper or electronic copies for noncommercial use may be made freely without explicit permission of the author. All other rights are reserved.

Book Excerpts:

This is a text for a basic course in commutative algebra, written in accordance with the following objectives.

The course should be accessible to those who have studied algebra at the beginning graduate level. For general algebraic background, see author's online text "Abstract Algebra: The Basic Graduate Year". This text will be referred to as TBGY.

The idea is to help the student reach an advanced level as quickly and efficiently as possible.

- In Chapter 1, the theory of primary decomposition is developed so as to apply to modules as well as ideals.

- In Chapter 2, integral extensions are treated in detail, including the lying over, going up and going down theorems. The proof of the going down theorem does not require advanced field theory.

- Valuation rings are studied in Chapter 3, and the characterization theorem for discrete valuation rings is proved.

- Chapter 4 discusses completion, and covers the Artin-Rees lemma and the Krull intersection theorem.

- Chapter 5 begins with a brief digression into the calculus of finite differences, which clarifies some of the manipulations involving Hilbert and Hilbert-Samuel polynomials. The main result is the dimension theorem for finitely generated modules over Noetherian local rings. A corollary is Krull’s principal ideal theorem. Some connections with algebraic geometry are established via the study of affine algebras.

- Chapter 6 introduces the fundamental notions of depth, systems of parameters, and M-sequences.

- Chapter 7 develops enough homological algebra to prove, under approprate hypotheses, that all maximal M-sequences have the same length.

- The brief Chapter 8 develops enough theory to prove that a regular local ring is an integral domain as well as a Cohen-Macaulay ring. After completing the course, the student should be equipped to meet the Koszul complex, the Auslander-Buchsbaum theorems, and further properties of Cohen-Macaulay rings in a more advanced course.

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About The Author(s)

Robert B. Ash (1935-2015) is Professor Emeritus of Mathematics in the Department of Mathematics at the University of Illinois. He was an Assistant Professor in Electrical Engineering at Columbia University and then a Visiting Assistant Professor at the University of California, Berkeley, before coming to the University of Illinois Department of Mathematics in 1963. He retired in 1990. Bob wrote thirteen textbooks but he was most proud of the eight books he posted on his web page during his retirement that were available for free downloading.

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