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Constructive Commutative Algebra: Projective Modules Over Polynomial Rings and Dynamical Gröbner Bases

✍ Scribed by Ihsen Yengui (auth.)

Publisher
Springer International Publishing
Year
2015
Tongue
English
Leaves
277
Series
Lecture Notes in Mathematics 2138
Edition
1
Category
Library
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✦ Synopsis

The main goal of this book is to find the constructive content hidden in abstract proofs of concrete theorems in Commutative Algebra, especially in well-known theorems concerning projective modules over polynomial rings (mainly the Quillen-Suslin theorem) and syzygies of multivariate polynomials with coefficients in a valuation ring.

Simple and constructive proofs of some results in the theory of projective modules over polynomial rings are also given, and light is cast upon recent progress on the Hermite ring and Gröbner ring conjectures. New conjectures on unimodular completion arising from our constructive approach to the unimodular completion problem are presented.

Constructive algebra can be understood as a first preprocessing step for computer algebra that leads to the discovery of general algorithms, even if they are sometimes not efficient. From a logical point of view, the dynamical evaluation gives a constructive substitute for two highly nonconstructive tools of abstract algebra: the Law of Excluded Middle and Zorn's Lemma. For instance, these tools are required in order to construct the complete prime factorization of an ideal in a Dedekind ring, whereas the dynamical method reveals the computational content of this construction. These lecture notes follow this dynamical philosophy.

✦ Table of Contents

Front Matter....Pages I-VII
Introduction....Pages 1-8
Projective Modules Over Polynomial Rings....Pages 9-103
Dynamical Gröbner Bases....Pages 105-171
Syzygies in Polynomial Rings Over Valuation Domains....Pages 173-205
Exercises....Pages 207-220
Detailed Solutions to the Exercises....Pages 221-253
Back Matter....Pages 255-274

✦ Subjects

Commutative Rings and Algebras; Mathematical Logic and Foundations; Symbolic and Algebraic Manipulation

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