Stable Homotopy over the Steenrod Algebra.

Intro -- Contents -- List of Figures -- Preface -- Chapter 0. Preliminaries -- 0.1. Grading and other conventions -- 0.2. Hopf algebras -- 0.3. Modules and comodules -- 0.4. Homological algebra -- 0.5. Two small examples -- Chapter 1. Stable homotopy over a Hopf algebra -- 1.1. The category Stable(Γ) -- 1.2. The functor H -- 1.2.1. Remarks on Hopf algebra extensions -- 1.3. Some classical homotopy theory -- 1.4. The Adams spectral sequence -- 1.5. Bousfield classes and Brown-Comenetz duality -- 1.6. Further discussion -- Chapter 2. Basic properties of the Steenrod algebra -- 2.1. Quotient Hopf algebras of A -- 2.1.1. Quasi-elementary quotients of A -- 2.2. P[sup(s)][sub(t)]-homology -- 2.2.1. Miscellaneous results about P[sup(s)][sub(t)]-homology -- 2.3. Vanishing lines for homotopy groups -- 2.3.1. Proof of Theorems 2.3.1 and 2.3.2 when p = 2 -- 2.3.2. Changes necessary when p is odd -- 2.4. Self-maps via vanishing lines -- 2.5. Construction of spectra of specified type -- 2.6. Further discussion -- Chapter 3. Chromatic structure -- 3.1. Margolis' killing construction -- 3.2. A Tate version of the functor H -- 3.3. Chromatic convergence -- 3.4. Further discussion: work of Mahowald and Shick -- 3.5. Further discussion -- Chapter 4. Computing Ext with elements inverted -- 4.1. The q[sub(n)]-based Adams spectral sequence -- 4.2. The Q[sub(n)]-based Adams spectral sequence -- 4.3. A(n) as an A-comodule -- 4.4. 1/2A(n) satisfies the vanishing plane condition -- 4.5. 1/2A(n) generates the expected thick subcategory -- 4.5.1. The proof of Proposition 4.5.7 -- 4.6. Some computations and applications -- 4.6.1. Computation of (Q[sub(n)])[sub(**)](Q[sub(n)]) -- 4.6.2. Eisen's calculation -- 4.6.3. The υ[sub(1)]-inverted Ext of the mod 2 Moore spectrum -- Chapter 5. Quillen stratification and nilpotence -- 5.1. Statements of theorems.

5.1.1. Quillen stratification -- 5.1.2. Nilpotence -- 5.2. Nilpotence and F-isomorphism via the Hopf algebra D -- 5.2.1. Nilpotence: Proof of Theorem 5.1.5 -- 5.2.2. F-isomorphism: Proof of Theorem 5.1.2 -- 5.3. Nilpotence and F-isomorphism via quasi-elementary quotients -- 5.3.1. Nilpotence: Proof of Theorem 5.1.6 -- 5.3.2. F-isomorphism: Proof of Theorem 5.1.3 -- 5.4. Further discussion: nilpotence at odd primes -- 5.5. Further discussion: miscellany -- Chapter 6. Periodicity and other applications of the nilpotence theorems -- 6.1. The periodicity theorem -- 6.2. y-maps and their properties -- 6.3. Properties of ideals -- 6.4. The proof of the periodicity theorem -- 6.5. Computation of some invariants in HD[sub(**)] -- 6.6. Computation of a few Bousfield classes -- 6.7. Ideals and thick subcategories -- 6.7.1. The thick subcategory conjecture -- 6.7.2. Rank varieties -- 6.8. Further discussion: slope supports -- 6.9. Further discussion: miscellany -- Appendix A. An underlying model category -- Appendix B. Steenrod operations and nilpotence in Ext**[sub(Γ)] (k,k) -- B.1. Steenrod operations in Hopf algebra cohomology -- B.2. Nilpotence in Ext over quotients of A: p = 2 -- B.3. Nilpotence in Ext over quotients of A: p odd -- B.3.1. Sketch of proof of Conjecture B.3.4, and other results -- Bibliography -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- K -- L -- M -- N -- P -- Q -- R -- S -- T -- U -- V -- W -- X -- Y -- Z.

Description based on publisher supplied metadata and other sources.

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