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Lecture Notes on Local Rings

Birger Iversen, edited by Holger Andreas Nielsen
World Scientific
Publication Date: 
Number of Pages: 
[Reviewed by
Felipe Zaldivar
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The book under review is a collection of lecture notes by the late Birger Iversen, edited by his colleague Holger Andreas Nielsen. It could be divided into three parts. The first part, Chapters 1 to 4, might be considered as the first part of a second course on commutative algebra, say after a standard first semester using Atiyah-Macdonald. The first goal of this part is to show that a regular local ring is a unique factorization domain, and the second goal is the Cohen structure theorem for local rings: Any complete Noetherian local ring is a quotient of a complete regular local ring.

The second part, Chapters 5 to 8, is devoted to local cohomology, local duality, dualizing modules and complexes. The highlights are Grothendieck’s local duality theorem, and duality for Cohen-Macaulay modules.

The third, and last part of the book, Chapters 9 to 11, is devoted to some special topics. Chapter nine considers generalizations of some module notions to complexes of modules: amplitude, depth and dimension, including the proof of the tensor product dimension formula and depth inequalities. The second half of this chapter studies Serre’s conditions on the depth of localized modules. Chapter 10 is introduces Serre’s still-open conjectures on the intersection multiplicity for finitely generated modules over a regular local ring. This chapter introduces these two conjectures, examines some consequences of them and concludes with versions of Serre’s conjectures in the graded case. After some preliminaries on the rank of a linear map and McCoy’s theorem, the last chapter starts with the Eisenbud-Buchsbaum criterion for the acyclicity of a complex \(L_{\bullet}\otimes M\) in terms of the depth of \(M\) for the characteristic ideals of the partial Euler characteristics of the finite complex \(L_{\bullet}\) of finitely generated free modules. Next, it introduces the Fitting ideals of a finitely presented module and the characteristic ideals of a module that admits a finite resolution \(L_{\bullet}\) by finitely generated free modules. The last two sections introduce the MacRae invariant for elementary modules and prove that it is an integral ideal.

The potential reader must be aware that some parts of the notes are rather unpolished, and sometimes without warning some facts of homological algebra are taken for granted. There are very few examples and no sets of problems. Also, there are some misprints, for example in page 114, first line, for the definition of the dualizing complex. Regardless of these considerations, this is a very nice text on some important topics on commutative ring theory.

Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is

  • 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