Full Description
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.
