Title Interactions between Homotopy Theory and Algebra

Published Providence : American Mathematical Society, 2007

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Description 1 online resource (352 p.) Series Contemporary Mathematics ; v.436 Contemporary Mathematics Contents ""Contents""; ""Preface""; ""List of Participants""; ""Introductory Lecture Series""; ""Model Categories and Simplicial Methods""; ""1. Model Categories and Resolutions""; ""2. Quillen Functors and Derived Functors""; ""3. Generating New Model Categories""; ""4. Simplicial Algebras and Resolutions in Non-abelian Settings""; ""5. Resolutions in Model Categories""; ""References""; ""Lectures on Local Cohomology""; ""1. Introduction""; ""2. Local Cohomology""; ""3. Injective Modules over Noetherian Rings and Matlis Duality""; ""4. Cohen-Macaulay and Gorenstein rings"" ""5. Vanishing Theorems and the Structure of Hdm(R)""""6. Vanishing Theorems II""; ""7. Appendix 1: Using local cohomology to prove a result of Kalkbrenner and Sturmfels""; ""8. Appendix 2: Bass numbers and Gorenstein Rings""; ""References""; ""Derived categories, resolutions, and Brown representability""; ""Introduction""; ""1. Derived categories""; ""1.1. Additive and abelian categories""; ""1.2. Categories of complexes""; ""1.3. Localization""; ""1.4. An alternative definition""; ""1.5. Extension groups""; ""1.6. Hereditary categories""; ""1. 7. Bounded derived categories""; ""1.8. Notes"" ""2. Triangulated categories""""2.1. The axioms""; ""2.2. The octahedral axiom""; ""2.3. Cohomological functors""; ""2.4. Uniqueness of exact triangles""; ""2.5. K(A) is triangulated""; ""2.6. Notes""; ""3. Localization of triangulated categories""; ""3.1. Quasi-isomorphisms""; ""3.2. D(A) is triangulated""; ""3.3. Triangulated and thick subcategories""; ""3.4. The kernel of a localization""; ""3.5. Verdier localization""; ""3.6. Notes""; ""4. Brown representability""; ""4.1. Coherent functors""; ""4.2. The abelianization of a triangulated category"" ""4.3. The idempotent completion of a triangulated category""""4.4. Homotopy colimits""; ""4.5. Brown representability""; ""4.6. Notes""; ""5. Resolutions""; ""5.1. Injective resolutions""; ""5.2. Projective Resolutions""; ""5.3. Derived functors""; ""5.4. Notes""; ""6. Differential graded algebras and categories""; ""6.1. Differential graded algebras and modules""; ""6.2. Differential graded categories""; ""6.3. Duality""; ""6.4. Injective and projective resolutions""; ""6.5. Compact objects and perfect complexes""; ""6.6. Notes""; ""7. Algebraic triangulated categories"" ""7.1. Exact categories""""7.2. Frobenius categories""; ""7.3. The derived category of an exact category""; ""7.4. The stable category of a Frobenius category""; ""7.5. Algebraic triangulated categories""; ""7.6. The stable homotopy category is not algebraic""; ""7. 7. The differential graded category of an exact category""; ""7.8. Notes""; ""Appendix A. The octahedral axiom""; ""References""; ""Exercises on derived categories, resolutions, and Brown representability""; ""Topics Lecture Series""; ""Spectra for commutative algebraists""; ""0. Introduction."" Notes Description based upon print version of record ""1. Motivation via the derived category."" English Subject Homotopy theory -- Congresses Commutative algebra -- Congresses Commutative algebra Homotopy theory Mathematics. Physical Sciences & Mathematics. Geometry. Genre/Form Conference papers and proceedings Form Electronic book Author Christensen, J.Daniel Dwyer, William G Summer School on Interactions between Homotopy Theory and Algebra Corporate Author Summer School on Interactions between Homotopy Theory and Algebra ISBN 9780821881156 0821881159

Other Titles Contemporary Mathematics Contemporary Mathematics, Volume 436 Interactions between homotopy theory and algebra: Summer School on Interactions between Homotopy Theory and Algebra, University of Chicago, July 26-August 6, 2004, Chicago, Illinois

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