ABSTRACT

This chapter explains how the solution of the Ravenel Conjectures by Ethan S. Devinatz, Michael J. Hopkins, D. C. Ravenel, and Jeffrey H. Smith leads to a canonical filtration in stable homotopy theory. It also explains that the chromatic filtration arises canonically from the global structure of the stable homotopy category. The chapter describes the geometric origins of the chromatic filtration through the relation with the stack of formal groups. As expressed in Waldhausen’s vision of brave new algebra, the category Sp of spectra should be thought of as a homotopical enrichment of the derived category DZ of abelian groups. The chapter demonstrates that many geometric structures have homotopical manifestations in the chromatic picture. The final ingredient in the formulation of the asymptotic algebraicity of chromatic homotopy theory is the algebraic model itself.