04030nam a22005415i 4500 978-3-031-77453-9 DE-He213 20251030114605.0 cr nn 008mamaa 241205s2024 sz | s |||| 0|eng d 9783031774539 978-3-031-77453-9 10.1007/978-3-031-77453-9 doi QA169 PBF bicssc MAT002010 bisacsh PBF thema Christensen, Lars Winther. author. (orcid)0000-0002-9360-123X https://orcid.org/0000-0002-9360-123X aut http://id.loc.gov/vocabulary/relators/aut 88588 Derived Category Methods in Commutative Algebra [electronic resource] / by Lars Winther Christensen, Hans-Bjørn Foxby, Henrik Holm. 1st ed. 2024. Cham : Springer Nature Switzerland : Imprint: Springer, 2024. XXIII, 1119 p. online resource. text txt rdacontent computer c rdamedia online resource cr rdacarrier text file PDF rda Springer Monographs in Mathematics, 2196-9922 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. 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. Algebra, Homological. 70150 Commutative algebra. 70151 Commutative rings. 70152 Category Theory, Homological Algebra. 70155 Commutative Rings and Algebras. 70156 Foxby, Hans-Bjørn. author. aut http://id.loc.gov/vocabulary/relators/aut 88589 Holm, Henrik. author. (orcid)0000-0002-6895-6493 https://orcid.org/0000-0002-6895-6493 aut http://id.loc.gov/vocabulary/relators/aut 88590 SpringerLink (Online service) Springer Nature eBook Printed edition: 9783031774522 Printed edition: 9783031774546 Springer Monographs in Mathematics, 2196-9922 73001 https://doi.org/10.1007/978-3-031-77453-9 ZDB-2-SMA ZDB-2-SXMS 65512 65512