Lecture notes on local rings / Birger Iversen ; edited by Holger Andreas Nielsen.

Format:

Book

Author/Creator:

Iversen, Birger, author.

Contributor:

Nielsen, Holger Andreas, editor.

Language:

English

Subjects (All):

Local rings.

Injective modules (Algebra).

Intersection homology theory.

Physical Description:

1 online resource (224 p.)

Place of Publication:

Singapore : World Scientific, 2014.

Language Note:

English

Summary:

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

Contents:

Preface; Contents; 1. Dimension of a Local Ring; 1.1 Nakayama's lemma; 1.2 Prime ideals; 1.3 Noetherian modules; 1.4 Modules of finite length; 1.5 Hilbert's basis theorem; 1.6 Graded rings; 1.7 Filtered rings; 1.8 Local rings; 1.9 Regular local rings; 2. Modules over a Local Ring; 2.1 Support of a module; 2.2 Associated prime ideals; 2.3 Dimension of a module; 2.4 Depth of a module; 2.5 Cohen-Macaulay modules; 2.6 Modules of finite projective dimension; 2.7 The Koszul complex; 2.8 Regular local rings; 2.9 Projective dimension and depth; 2.10 -depth; 2.11 The acyclicity theorem

2.12 An example3. Divisor Theory; 3.1 Discrete valuation rings; 3.2 Normal domains; 3.3 Divisors; 3.4 Unique factorization; 3.5 Torsion modules; 3.6 The first Chern class; 3.7 Regular local rings; 3.8 Picard groups; 3.9 Dedekind domains; 4. Completion; 4.1 Exactness of the completion functor; 4.2 Separation of the -adic topology; 4.3 Complete filtered rings; 4.4 Completion of local rings; 4.5 Structure of complete local rings; 5. Injective Modules; 5.1 Injective modules; 5.2 Injective envelopes; 5.3 Decomposition of injective modules; 5.4 Matlis duality; 5.5 Minimal injective resolutions

5.6 Modules of finite injective dimension5.7 Gorenstein rings; 6. Local Cohomology; 6.1 Basic properties; 6.2 Local cohomology and dimension; 6.3 Local cohomology and depth; 6.4 Support in the maximal ideal; 6.5 Local duality for Gorenstein rings; 7. Dualizing Complexes; 7.1 Complexes of injective modules; 7.2 Complexes with finitely generated cohomology; 7.3 The evaluation map; 7.4 Existence of dualizing complexes; 7.5 The codimension function; 7.6 Complexes of flat modules; 7.7 Generalized evaluation maps; 7.8 Uniqueness of dualizing complexes; 8. Local Duality; 8.1 Poincaré series

8.2 Grothendieck's local duality theorem8.3 Duality for Cohen-Macaulay modules; 8.4 Dualizing modules; 8.5 Locally factorial domains; 8.6 Conductors; 8.7 Formal fibers; 9. Amplitude and Dimension; 9.1 Depth of a complex; 9.2 The dual of a module; 9.3 The amplitude formula; 9.4 Dimension of a complex; 9.5 The tensor product formula; 9.6 Depth inequalities; 9.7 Condition Sr of Serre; 9.8 Factorial rings and condition Sr; 9.9 Condition S r; 9.10 Specialization of Poincaré series; 10. Intersection Multiplicities; 10.1 Introduction to Serre's conjectures; 10.2 Filtration of the Koszul complex

10.3 Euler characteristic of the Koszul complex10.4 A projection formula; 10.5 Power series over a field; 10.6 Power series over a discrete valuation ring; 10.7 Application of Cohen's structure theorem; 10.8 The amplitude inequality; 10.9 Translation invariant operators; 10.10 Todd operators; 10.11 Serre's conjecture in the graded case; 11. Complexes of Free Modules; 11.1 McCoy's theorem; 11.2 The rank of a linear map; 11.3 The Eisenbud-Buchsbaum criterion; 11.4 Fitting's ideals; 11.5 The Euler characteristic; 11.6 McRae's invariant; 11.7 The integral character of McRae's invariant

Bibliography

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

Description based on print version record.

ISBN:

981-4603-66-X