Birger Iversen,Holger Andreas Nielsen

Lecture Notes on Local Rings

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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
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2,374 printed pages
Original publication
2014

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