Commutative Algebra

This text is a direct translation from the Japanese book published in 2011 and includes considerable updates and four new appendices by M. Hashimoto. It provides a concise and readable introduction to modern commutative algebra written by two researchers who have explored the area extensively. The book is also meant to supply the reader with sufficient tools to challenge unsolved questions related to commutative ring theory. Commutative algebra arose from number theory, algebraic geometry, and invariant theory, to describe "ideal numbers," "algebraic functions," and "invariant functions,'' respectively. After the introduction of homological methods, the field acquired independent interest, methods and the status as an area of study in its own right. With the birth of modern commutative algebra, new connections arose with algebraic geometry through projective geometry and singularity theory. Nowadays, commutative algebra is related to many branches of mathematics including those mentioned above as well as to topology, combinatorics, and group theory.

The discussion begins with the basic theory of Noetherian rings, dimension theory, ideal-adic topologies, and completion. Then the techniques of homological algebra are introduced and using these techniques, the readers are introduced to Cohen–Macaulay and Gorenstein rings, which are main themes of the theory. The theory of graded rings and modules is one of the central parts of this text, providing rich families of examples, and many connections with other fields are realized through graded rings. The discussion proceeds with rings closely related to Cohen–Macaulay rings—rings with FLC, and Buchsbaum rings. In the last chapter, the techniques of positive characteristic, originated by Hochster and Huneke, are introduced. Using the Frobenius map, many results are obtained in a beautiful and cohesive manner. The book covers elementary to advanced content in a streamlined way.