Commutative algebra: Constructive methods. Finite projective modules

TL;DR

This book introduces constructive methods in commutative algebra, emphasizing algorithmic content in theorems about finitely generated projective modules and revisiting classical theories with a new, constructive perspective.

Contribution

It provides an explicit algorithmic approach to classical commutative algebra theories, making existence proofs constructive and computational.

Findings

Algorithmic constructions for finitely generated projective modules

Reinterpretation of Galois theory and Dedekind domains constructively

Simplified proofs of classical algebraic theorems

Abstract

This book is an introductory course to basic commutative algebra with a particular emphasis on finitely generated projective modules. We adopt the constructive point of view, with which all existence theorems have an explicit algorithmic content content. In particular, when a theorem affirms the existence of an object -- the solution of a problem -- a construction algorithm of the object can always be extracted from the given proof. We revisit with a new and often simplifying eye several abstract classical theories. In particular, we review theories which did not have any algorithmic content in their general natural framework, such as Galois theory, the Dedekind domains, the finitely generated projective modules or the Krull dimension.