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Part 1. Localization and completion for ideals. 2 2. Algebraic definitions: Local and Čech cohomology and homology 3 3. Homotopical analogues of the algebraic definitions 10 4. Completion at ideals and Bousfield localization 13 5. Localization away from ideals and Bousfield localization 15 6. Chromatic filtrations for M U -modules. 16 7. Completion theorems and their duals. 19 Part 2. Morita equivalences and Gorenstein rings. 21 8. The context, and some examples. 21 9. Morita equivalences. 24doi:10.1090/conm/436/08412 fatcat:7ns4vgxsxja5zdhuxkw6paipwy