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2 produkter
2 produkter
Häftad, Engelska, 2026
1 635 kr
Skickas inom 5-8 vardagar
This volume consists of edited lecture notes from two events at the International Centre for Theoretical Physics (ICTP) in Trieste, Italy, organized in honor of Melvin Hochster and Craig Huneke. The two events were the online International Graduate course on Tight Closure and Its Applications, spread over summer 2022 and the May 2023 School on Commutative Algebra and Algebraic Geometry in Prime Characteristics. The unifying basis of these events was the theory of tight closure, the brainchild of Melvin Hochster and Craig Huneke in the late 1980s, which has had a dramatic effect on the field of commutative algebra, giving unified proofs and strong generalizations of many major theorems in commutative algebra, and stimulating much research, including recent proofs of longstanding conjectures. The aim of the two events as well as of this volume is to provide training in the foundations of commutative algebra and algebraic geometry in prime characteristic and to present some of the exciting recent developments in and beyond tight closure. The lecture notes for the online school also come with exercises and solutions. The topics in the volume include characteristic p methods, test ideals, direct summands, singularities, Hilbert-Kunz multiplicity, Briancon-Skoda theorems, big Cohen-Macaulay algebras, the localization problem, uniform Artin-Rees results, vector bundles and tight closure, singularity invariants. The intended audience is graduate students learning the material as well as researchers wanting the latest advances in one reference.
- Nyhet
Del 309 - Graduate Texts in Mathematics
Commutative Algebra
Inbunden, Engelska, 2026
717 kr
Skickas inom 11-20 vardagar
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.