Homological methods in commutative algebra

• Verfasst von: Ferretti, Andrea [VerfasserIn]
• Titel: Homological methods in commutative algebra
• Verf.angabe: Andrea Ferretti
• Verlagsort: Providence, Rhode Island
• Verlag: American Mathematical Society
• E-Jahr: 2023
• Jahr: [2023]
• Umfang: 1 Online-Ressource (xviii, 411 Seiten)
• Illustrationen: Illustrationen
• Gesamttitel/Reihe: Graduate studies in mathematics ; 234
• Fussnoten: Description based on publisher supplied metadata and other sources
• ISBN: 978-1-4704-7435-5
• Schlagwörter: (s)Kommutative Algebra / (s)Homologische Algebra / (s)Abelsche Kategorie / (s)Spektralsequenz / (s)Freier Modul / (s)Koszul-Komplex / (s)Cohen-Macaulay-Ring
• Datenträger: Online-Ressource
• Sprache: eng
• Bibliogr. Hinweis: Erscheint auch als : Druck-Ausgabe: Ferretti, Andrea, 1981 - : Homological methods in commutative algebra. - Providence, Rhode Island : American Mathematical Society, 2023. - xviii, 411 Seiten
• Sach-SW: Commutative algebra -- Instructional exposition (textbooks, tutorial papers, etc.)
• Sach-SW: Commutative algebra -- Homological methods -- Syzygies, resolutions, complexes
• Sach-SW: Commutative algebra -- Homological methods -- Homological functors on modules (Tor, Ext, etc.)
• Sach-SW: Commutative algebra -- Homological methods -- Local cohomology
• Sach-SW: Commutative algebra -- Local rings and semilocal rings -- Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
• K10plus-PPN: 1870716701
• K10plus-PPN:
  Universitätsbibliothek
• Lokale URL UB: Zum Volltext

Abstract

This book develops the machinery of homological algebra and its applications to commutative rings and modules. It assumes familiarity with basic commutative algebra, for example, as covered in the author's book, Commutative Algebra.The first part of the book is an elementary but thorough exposition of the concepts of homological algebra, starting from categorical language up to the construction of derived functors and spectral sequences. A full proof of the celebrated Freyd-Mitchell theorem on the embeddings of small Abelian categories is included.The second part of the book is devoted to the application of these techniques in commutative algebra through the study of projective, injective, and flat modules, the construction of explicit resolutions via the Koszul complex, and the properties of regular sequences. The theory is then used to understand the properties of regular rings, Cohen-Macaulay rings and modules, Gorenstein rings and complete intersections.Overall, this book is a valuable resource for anyone interested in learning about homological algebra and its applications in commutative algebra. The clear and thorough presentation of the material, along with the many examples and exercises of varying difficulty, make it an excellent choice for self-study or as a reference for researchers.

Abstract

Cover -- Title page -- Contents -- Preface -- Conventions -- Chapter 1. Categories -- 1.1. Categories and functors -- 1.2. Sets and classes -- 1.3. Natural transformations -- 1.4. Limits -- 1.5. Adjoint pairs -- 1.6. Exercises -- Chapter 2. Abelian Categories -- 2.1. Additive categories -- 2.2. Abelian categories -- 2.3. Sheaves -- 2.4. Sites -- 2.5. Standard lemmas -- 2.6. Projectives and injectives -- 2.7. Essential extensions -- 2.8. The Freyd-Mitchell theorem -- 2.9. Exercises -- Chapter 3. Derived Functors -- 3.1. Categories of complexes -- 3.2. Derived functors -- 3.3. Computing derived functors -- 3.4. The Ext and Tor functors -- 3.5. The limⁿ functors -- 3.6. Exercises -- Chapter 4. Spectral Sequences -- 4.1. Exact couples -- 4.2. Filtered complexes -- 4.3. Double complexes -- 4.4. Simple applications -- 4.5. The Grothendieck spectral sequence -- 4.6. The Ischebeck spectral sequences -- 4.7. Exercises -- Chapter 5. Projective and Injective Modules -- 5.1. Projective and free modules -- 5.2. Projective dimension -- 5.3. Injective modules -- 5.4. Injective dimension -- 5.5. Global dimension of rings -- 5.6. Free resolutions -- 5.7. Stably free modules -- 5.8. The Quillen-Suslin theorem -- 5.9. Exercises -- Chapter 6. Flatness -- 6.1. Flat modules -- 6.2. Flat morphisms -- 6.3. Faithful flatness -- 6.4. Criteria for flatness -- 6.5. Flatness and freeness -- 6.6. Exercises -- Chapter 7. Koszul Complexes and Regular Sequences -- 7.1. The Koszul complex -- 7.2. Regular sequences -- 7.3. Depth -- 7.4. The Auslander-Buchsbaum formula -- 7.5. Multiplicities revisited -- 7.6. Macaulay resultants -- 7.7. Computing the resultant -- 7.8. Exercises -- Chapter 8. Regularity -- 8.1. Regular local rings -- 8.2. Regularity and global dimension -- 8.3. Change of rings -- 8.4. Regularity and factorization -- 8.5. Normal rings.