eBook ISBN:  9781470403096 
Product Code:  MEMO/151/716.E 
List Price:  $66.00 
MAA Member Price:  $59.40 
AMS Member Price:  $39.60 
eBook ISBN:  9781470403096 
Product Code:  MEMO/151/716.E 
List Price:  $66.00 
MAA Member Price:  $59.40 
AMS Member Price:  $39.60 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 151; 2001; 172 ppMSC: Primary 55; 18; 16;
We apply the tools of stable homotopy theory to the study of modules over the mod \(p\) Steenrod algebra \(A^{*}\). More precisely, let \(A\) be the dual of \(A^{*}\); 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, BrownComenetz duality, and other homotopytheoretic 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 \(A\), \(\mathrm{Ext}_A^{**}(\mathbf{F}_p,\mathbf{F}_p)\). We also have nilpotence theorems, periodicity theorems, a convergent chromatic tower, and a number of other results.
ReadershipGraduate students and research mathematicians interested in algebraic topology.

Table of Contents

Chapters

0. Preliminaries

1. Stable homotopy over a Hopf algebra

2. Basic properties of the Steenrod algebra

3. Chromatic structure

4. Computing Ext with elements inverted

5. Quillen stratification and nilpotence

6. Periodicity and other applications of the nilpotence theorems


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We apply the tools of stable homotopy theory to the study of modules over the mod \(p\) Steenrod algebra \(A^{*}\). More precisely, let \(A\) be the dual of \(A^{*}\); 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, BrownComenetz duality, and other homotopytheoretic 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 \(A\), \(\mathrm{Ext}_A^{**}(\mathbf{F}_p,\mathbf{F}_p)\). We also have nilpotence theorems, periodicity theorems, a convergent chromatic tower, and a number of other results.
Graduate students and research mathematicians interested in algebraic topology.

Chapters

0. Preliminaries

1. Stable homotopy over a Hopf algebra

2. Basic properties of the Steenrod algebra

3. Chromatic structure

4. Computing Ext with elements inverted

5. Quillen stratification and nilpotence

6. Periodicity and other applications of the nilpotence theorems