Derived Category Methods in Commutative Algebra / by Lars Winther Christensen, Hans-Bjørn Foxby, Henrik Holm.

Format:

Book

Author/Creator:

Christensen, Lars Winther.

Contributor:

Foxby, Hans-Bjø.

Holm, Henrik.

Series:

Springer Monographs in Mathematics, 2196-9922

Language:

English

Subjects (All):

Algebra, Homological.

Commutative algebra.

Commutative rings.

Category Theory, Homological Algebra.

Commutative Rings and Algebras.

Local Subjects:

Category Theory, Homological Algebra.

Commutative Rings and Algebras.

Physical Description:

1 online resource (1130 pages)

Edition:

1st ed. 2024.

Place of Publication:

Cham : Springer Nature Switzerland : Imprint: Springer, 2024.

Summary:

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.

Contents:

1 Modules

2 Complexes

3 Categorical Constructions

4 Equivalences and Isomorphisms

5 Resolutions

6 The Derived Category

7 Derived Functors

8 Homological Dimensions

9 Gorenstein Homological Dimensions

10 Dualizing Complexes

11 Torsion and Completion

12 A Brief for Commutative Ring Theorists

13 Derived Torsion and Completion

14 Krull Dimension, Depth, and Width

15 Support Theories

16 Homological Invariants over Local Rings

17 Going Local

18 Dualities and Cohen-Macaulay Rings

19 Gorenstein Dimensions and Gorenstein Rings

20 Global Dimension and Regular Rings

APPENDIX A: Acyclicity and Boundedness

APPENDIX B: Minimality

APPENDIX C: Structure of Injective Modules

APPENDIX D: Projective Dimension of Flat Modules

APPENDIX E: Triangulated Categories.

Notes:

Description based on publisher supplied metadata and other sources.

ISBN:

9783031774539

3031774531

OCLC:

1477221134