Derived Category Methods in Commutative Algebra
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Beschreibung
Derived category methods entered commutative algebra in the latter half of the 1960s, providing, among other things, a framework for a clear formulation of Grothendieck’s Local Duality Theorem. Since then, their impact on the field has steadily grown and continues to expand.
This book guides readers familiar with rings and modules through the construction of the associated derived category and its triangulated functors. In this context, it develops theories of categorical equivalences for subcategories and homological invariants of objects. The second half of the book focuses on applications to commutative Noetherian rings.
The book can be used as a text for graduate courses, both introductory and advanced, and is intended to serve as a reference for researchers in commutative algebra and related fields. To accommodate readers new to homological algebra, it offers a significantly higher level of detail than most existing texts on the subject.
Derived category methods entered commutative algebra in the latter half of the 1960s, providing, among other things, a framework for a clear formulation of Grothendieck’s Local Duality Theorem. Since then, their impact on the field has steadily grown and continues to expand.
This book guides readers familiar with rings and modules through the construction of the associated derived category and its triangulated functors. In this context, it develops theories of categorical equivalences for subcategories and homological invariants of objects. The second half of the book focuses on applications to commutative Noetherian rings.
The book can be used as a text for graduate courses, both introductory and advanced, and is intended to serve as a reference for researchers in commutative algebra and related fields. To accommodate readers new to homological algebra, it offers a significantly higher level of detail than most existing texts on the subject.
Provides a comprehensive and self-contained introduction to derived categories and their applications in ring theory Includes more than 60 years’ worth of results on the homological properties of modules over commutative rings Serves researchers and graduate students alike with more than 300 examples, 1100 exercises, and complete proofs
Autor*in
Lars Winther Christensen
Themen in »Derived Category Methods in Commutative Algebra«
Local duality Dualizing complex Finitistic dimension Cohen-Macaulay Gorenstein Global dimension Homological algebra Homological dimension Local cohomology Unbounded derived category Projective resolutions Injective resolutions Matlis duality Depth
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"Derived category methods in commutative algebra is a masterful achievement: rigorous, accessible, and pedagogically superb. It should be read by graduate students seeking a solid foundation in derived categories and by researchers requiring a comprehensive reference. ... All in all, I identify it as a well-written, readable book that algebra practitioners will enjoy having on their shelves." (Hossein Faridian, Mathematical Reviews, May, 2026)
“The voluminous outcome is the present book of 1119 pages with 299 references and 25 pages of Index. … To this end they included to each section exercises in order to perform elementary verifications and computations … .The book is a very useful, up-to-date reference book for working mathematicians in Commutative Algebra using homological methods.” (Peter Schenzel, zbMATH 1559.13001, 2025)
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