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–M... Read more

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. * deceased. 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. Less