Stable homotopy over the Steenrod algebra / John H. Palmieri.

Format:

Book

Author/Creator:

Palmieri, John H. (John Harold), 1964- author.

Series:

Memoirs of the American Mathematical Society ; no. 716.

Memoirs of the American Mathematical Society, 0065-9266 ; number 716

Language:

English

Subjects (All):

Homotopy theory.

Steenrod algebra.

Physical Description:

1 online resource (193 p.)

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, 2001.

Language Note:

English

Summary:

This title applys the tools of stable homotopy theory to the study of modules over the mod $p$ Steenrod algebra $A DEGREES{*}$. More precisely, let $A$ be the dual of $A DEGREES{*}$; then we study the category $\mathsf{stable}(A)$ of unbounded cochain complexes of injective comodules over $A$, in which the morphisms are cochain homotopy classes of maps. This category is triangulated. Indeed, it is a stable homotopy category, so we can use Brown representability, Bousfield localization, Brown-Comenetz duality, and other homotopy-theoretic tools to study it. One focus of attention is the analogue of the stable homotopy groups of spheres, which in this setting is the cohomology of $A$, $\mathrm{Ext}_A DEGREES{**}(\mathbf{F}_p, \mathbf{F}_p)$. This title also has nilpotence theorems, periodicity theorems, a convergent chromatic tower, and a nu

Contents:

""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(Î?)""; ""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 Ï?[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""

Notes:

"Volume 151, number 716 (second of 5 numbers)."

Includes bibliographical references and index.

Description based on print version record.

ISBN:

1-4704-0309-9