By John H. Palmieri

We observe the instruments of reliable homotopy thought to the learn of modules over the mod $p$ Steenrod algebra $A^{*}$. extra accurately, enable $A$ be the twin of $A^{*}$; then we examine the class $\mathsf{stable}(A)$ of unbounded cochain complexes of injective co modules over $A$, within which the morphisms are cochain homotopy periods of maps. This type is triangulated. certainly, it's a sturdy homotopy classification, to be able to use Brown representability, Bousfield localization, Brown-Comenetz duality, and different homotopy-theoretic instruments to check it. One concentration of cognizance is the analogue of the good homotopy teams of spheres, which during this surroundings is the cohomology of $A$, $\mathrm{Ext}_A^{**}(\mathbf{F}_p,\mathbf{F}_p)$. We even have nilpotence theorems, periodicity theorems, a convergent chromatic tower, and several effects.

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