4 - Unstable Cooperations

Summary

This chapter approaches aspects of unstable homotopy theory from the perspective of the preceding chapter: We study the unstable analog of effective descent and thereby embark on an analysis of the algebro–geometric objects that arise from these considerations. This lends a modern perspective on a variety of classical topics: Hopf rings, the Ravenel–Wilson results on the infinite loopspaces associated to BP, Dieudonné theory, Brown–Gitler spectra, and the formal schemes associated to Eilenberg–Mac Lane spaces. We also include a novel discussion of the functorial algebraic geometry of Hopf rings, which gives a reinterpretation of the central Ravenel–Wilson relation and exerts an overall influence on our presentation of the other materials.