Interactions between Homotopy Theory and Algebra.

Intro -- 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. -- 1. Motivation via the derived category. -- 2. Why consider spectra? -- 3. How to construct spectra (Step 1). -- 4. The smash product (Step 2). -- 5. Brave new rings. -- 6. Some algebraic uses of ring spectra. -- 7. Local ring spectra. -- References -- Rational Homotopy Theory: A Brief Introduction -- Introduction -- 1. Foundations -- 2. Sullivan models -- 3. Commutative algebra and rational homotopy theory -- References -- André-Quillen homology of commutative algebras -- 1. Introduction -- 2. Kähler differentials -- 3. Simplicial algebras -- 4. Simplicial resolutions -- 5. The cotangent complex -- 6. Basic properties -- 7. André-Quillen homology and the Tor functor -- 8. Locally complete intersection homomorphisms -- 9. Regular homomorphisms -- References -- Local cohomology in commutative algebra, homotopy theory, and group cohomology -- First steps in brave new commutative algebra -- 1. Introduction -- Part 1. Localization and completion for ideals.

2. Algebraic definitions: Local and Cech cohomology and homology -- 3. Homotopical analogues of the algebraic definitions -- 4. Completion at ideals and Bousfield localization -- 5. Localization away from ideals and Bousfield localization -- 6. Chromatic filtrations for MU-modules. -- 7. Completion theorems and their duals. -- Part 2. Morita equivalences and Gorenstein rings. -- 8. The context, and some examples. -- 9. Morita equivalences. -- 10. Matlis lifts. -- 11. The Gorenstein condition. -- References -- Cotorsion pairs and model categories -- 1. Cotorsion pairs -- 2. Relation between cotorsion pairs and model categories -- 3. Cofibrant generation -- 4. Monoidal structure -- 5. Standard examples -- 6. Gorenstein rings -- 7. Gillespie's work -- References -- Coherent sheaves on an elliptic curve -- 1. Introduction -- 2. Modules over Dedekind domains -- 3. Coherent sheaves on projective varieties -- 4. Coherent sheaves on an elliptic curve -- References -- Lectures on the Cohomology of Finite Groups -- 1. Introduction -- 2. Preliminaries -- 3. Minimal resolutions -- 4. Computations and further structure -- 5. Cohomology and actions of finite groups -- References.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.