Commutative algebra

• Verfasst von: Ferretti, Andrea [VerfasserIn]
• Titel: Commutative algebra
• Verf.angabe: Andrea Ferretti
• Verlagsort: Providence, Rhode Island
• Verlag: American Mathematical Society
• E-Jahr: 2023
• Jahr: [2023]
• Umfang: 1 Online-Ressource (xvii, 373 Seiten)
• Gesamttitel/Reihe: Graduate studies in mathematics ; 233
• Fussnoten: Description based on publisher supplied metadata and other sources
• ISBN: 978-1-4704-7433-1
• Schlagwörter: (s)Kommutative Algebra / (s)Noetherscher Ring / (s)Faktorisierung / (s)Gröbner-Basis / (s)Ringerweiterung <Mathematik> / (s)Dedekind-Ring / (s)Dimensionstheorie
• Datenträger: Online-Ressource
• Sprache: eng
• Bibliogr. Hinweis: Erscheint auch als : Druck-Ausgabe: Ferretti, Andrea, 1981 - : Commutative algebra. - Providence, Rhode Island : American Mathematical Society, 2023. - xvii, 373 Seiten
• RVK-Notation: SK 230
• Sach-SW: Commutative algebra -- Instructional exposition (textbooks, tutorial papers, etc.)
• Sach-SW: Number theory -- Algebraic number theory: global fields -- Algebraic numbers; rings of algebraic integers
• Sach-SW: Commutative algebra -- Computational aspects and applications -- None of the above, but in this section
• Sach-SW: Number theory -- Instructional exposition (textbooks, tutorial papers, etc.)
• Sach-SW: Algebraic geometry -- Foundations -- Varieties and morphisms
• K10plus-PPN: 1854633619
• K10plus-PPN:
  Universitätsbibliothek
• Lokale URL UB: Zum Volltext

Abstract

This book provides an introduction to classical methods in commutative algebra and their applications to number theory, algebraic geometry, and computational algebra. The use of number theory as a motivating theme throughout the book provides a rich and interesting context for the material covered. In addition, many results are reinterpreted from a geometric perspective, providing further insight and motivation for the study of commutative algebra.The content covers the classical theory of Noetherian rings, including primary decomposition and dimension theory, topological methods such as completions, computational techniques, local methods and multiplicity theory, as well as some topics of a more arithmetic nature, including the theory of Dedekind rings, lattice embeddings, and Witt vectors. Homological methods appear in the author's sequel, Homological Methods in Commutative Algebra.Overall, this book is an excellent resource for advanced undergraduates and beginning graduate students in algebra or number theory. It is also suitable for students in neighboring fields such as algebraic geometry who wish to develop a strong foundation in commutative algebra. Some parts of the book may be useful to supplement undergraduate courses in number theory, computational algebra or algebraic geometry. The clear and detailed presentation, the inclusion of computational techniques and arithmetic topics, and the numerous exercises make it a valuable addition to any library.

Abstract

Cover -- Title page -- Contents -- Preface -- Chapter 1. Basics -- 1.1. Rings and ideals -- 1.2. Quotients -- 1.3. Modules -- 1.4. More constructions with modules -- 1.5. Euclidean rings -- 1.6. Localization -- 1.7. Graded rings and modules -- 1.8. Exercises -- Chapter 2. Finiteness Conditions -- 2.1. Principal ideal domains -- 2.2. Artinian and Noetherian modules -- 2.3. Noetherian rings -- 2.4. Artinian rings -- 2.5. Length -- 2.6. Exercises -- Chapter 3. Factorization -- 3.1. Unique factorization domains -- 3.2. Primary decomposition -- 3.3. Primary decomposition for modules -- 3.4. Factorization in Dedekind rings -- 3.5. The structure of modules over Dedekind rings -- 3.6. Exercises -- Chapter 4. Computational Methods -- 4.1. The resultant -- 4.2. Discriminants -- 4.3. Gröbner bases -- 4.4. More algorithmic operations -- 4.5. Exercises -- Chapter 5. Integral Dependence -- 5.1. Integral extensions -- 5.2. Going up and down -- 5.3. Noether normalization -- 5.4. Integral extensions of Dedekind rings -- 5.5. Exercises -- Chapter 6. Lattice Methods -- 6.1. Additive structure of number rings -- 6.2. Prime extensions in number rings -- 6.3. Prime extensions in Dedekind rings -- 6.4. Galois extensions of Dedekind rings -- 6.5. Discriminant and ramification -- 6.6. Computing prime factorizations -- 6.7. Geometry of ideal lattices -- 6.8. Cyclotomic rings -- 6.9. Exercises -- Chapter 7. Metric and Topological Methods -- 7.1. Absolute values -- 7.2. Valuations and valuation rings -- 7.3. Discrete valuation rings -- 7.4. Direct and inverse limits -- 7.5. Completion of rings and modules -- 7.6. Hensel's lemma -- 7.7. Witt vectors -- 7.8. Exercises -- Chapter 8. Geometric Dictionary -- 8.1. Affine varieties -- 8.2. The Nullstellensatz -- 8.3. The Ax-Grothendieck theorem -- 8.4. Morphisms -- 8.5. Local rings and completions revisited.