Homological methods in commutative algebra
Ferretti, Andrea
Homological methods in commutative algebra - Rhode Island : American Mathematical Society, 2023. - xviii, 411p. - Graduate studies in mathematics, volume 234 1065-7339 ; .
Includes bibliographical references and index.
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 (Graduate Studies in Mathematics, Volume 233).
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.
Graduate students and researchers interested in commutative algebra.
9781470474362
Commutative algebra
Algebra, Homological
Commutative algebra -- Instructional exposition (textbooks, tutorial papers, etc.).
Commutative algebra -- Homological methods -- Syzygies, resolutions, complexes.
Commutative algebra -- Homological methods -- Homological functors on modules (Tor, Ext, etc.).
Commutative algebra -- Homological methods -- Local cohomology.
Commutative algebra -- Local rings and semilocal rings -- Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.).
512.71 / FER-H
Homological methods in commutative algebra - Rhode Island : American Mathematical Society, 2023. - xviii, 411p. - Graduate studies in mathematics, volume 234 1065-7339 ; .
Includes bibliographical references and index.
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 (Graduate Studies in Mathematics, Volume 233).
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.
Graduate students and researchers interested in commutative algebra.
9781470474362
Commutative algebra
Algebra, Homological
Commutative algebra -- Instructional exposition (textbooks, tutorial papers, etc.).
Commutative algebra -- Homological methods -- Syzygies, resolutions, complexes.
Commutative algebra -- Homological methods -- Homological functors on modules (Tor, Ext, etc.).
Commutative algebra -- Homological methods -- Local cohomology.
Commutative algebra -- Local rings and semilocal rings -- Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.).
512.71 / FER-H
