Lecture Notes on Local Rings

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.

Contents:

• Dimension of a Local Ring
• Modules over a Local Ring
• Divisor Theory
• Completion
• Injective Modules
• Local Cohomology
• Dualizing Complexes
• Local Duality
• Amplitude and Dimension
• Intersection Multiplicities
• Complexes of Free Modules

Readership: Graduate students and academic researchers with an interest in algebra, commutative algebra, algebra geometry, homological algebra and algebraic number theory.
Key Features:

• Although the proofs are fairly short, the key points give readers the opportunity to supply details for their own satisfaction
• The classical result of Auslander-Buchsbaum on unique factorization in a regular local ring is treated in a context of divisor and Picard groups, and this enlightens and connects to methods from number theory
• This book contains original research of the late Professor Iversen that are not published in this form before