Commutative Algebra: Constructive Methods: Finite Projective Modules

Henri Lombardi, Claude Quitté

Springer, Jul 22, 2015 - Mathematics - 996 pages

Translated from the popular French edition, this book offers a detailed introduction to various basic concepts, methods, principles, and results of commutative algebra. It takes a constructive viewpoint in commutative algebra and studies algorithmic approaches alongside several abstract classical theories. Indeed, it revisits these traditional topics with a new and simplifying manner, making the subject both accessible and innovative.

The algorithmic aspects of such naturally abstract topics as Galois theory, Dedekind rings, Prüfer rings, finitely generated projective modules, dimension theory of commutative rings, and others in the current treatise, are all analysed in the spirit of the great developers of constructive algebra in the nineteenth century.

This updated and revised edition contains over 350 well-arranged exercises, together with their helpful hints for solution. A basic knowledge of linear algebra, group theory, elementary number theory as well as the fundamentals of ring and module theory is required. Commutative Algebra: Constructive Methods will be useful for graduate students, and also researchers, instructors and theoretical computer scientists.

Contents

• Chapter I Examples: 1
• Chapter IIThe Basic LocalGlobal Principle and Systems of Linear Equations: 15
• Chapter III The Method of Undetermined Coefficients: 76
• Chapter IV Finitely Presented Modules: 173
• Chapter VFinitely Generated Projective Modules 1: 239
• VI Strictly Finite Algebras and Galois Algebras: 295
• VII The Dynamic Method: 378
• Chapter VIIIFlat Modules: 435
• Chapter IX Local Rings or Just About: 477
• Chapter X Finitely Generated Projective Modules 2: 523
• Chapter XI Distributive Lattices LatticeGroups: 609
• Chapter XII Prüfer and Dedekind Rings: 669
• Chapter XIII Krull Dimension: 734
• Chapter XIV The Number of Generators of a Module: 797
• Chapter XV The LocalGlobal Principle: 834
• Chapter XVI Extended Projective Modules: 885
• Chapter XVII Suslins Stability Theorem the Field Case: 929
• Annex: 947
• Tables of Theorems: 963
• Bibliography: 973
• Index: 984

About the author

Henri Lombardi is a researcher in constructive mathematics, real algebra and algorithmic complexity. Since 2003, with Marie-Françoise Roy and Thierry Coquand, he has developed the international group MAP (Mathematics, Algorithms, Proofs). He has published Épistémologie mathématique, (Ellipse, 2011), and Méthodes matricielles. Introduction à la complexité algébrique, (Springer, 2003, in collaboration with Jounaïdi Abdeljaoued).

Claude Quitté is a researcher in effective commutative algebra, computer algebra and computer science. He has published Algorithmique algébrique, (Masson, 1991) in collaboration with Patrice Naudin.

Henri Lombardi and Claude Quitté also published together with Maria-Gema Díaz-Toca the book Modules sur les anneaux commutatifs (Calvage & Mounet, 2014).

Bibliographic information

• Title: Commutative Algebra: Constructive Methods: Finite Projective Modules; Volume 20 of Algebra and Applications
• Authors: Henri Lombardi, Claude Quitté
• Edition: illustrated
• Publisher: Springer, 2015
• ISBN: 940179944X, 9789401799447
• Length: 996 pages
• Subjects: Computers / Data Science / General; Mathematics / Algebra / Abstract; Mathematics / Algebra / General; Mathematics / Algebra / Linear; Mathematics / Discrete Mathematics