3 - Finite Spectra

Summary

This chapter introduces the basic results of chromatic homotopy theory while adhering to the language of algebraic geometry. Our approach centers on the study of descent in the setting of stable homotopy theory; Quillen’s theorem from the preceding chapter gives an example of a ring spectrum whose unit map is of effective descent and whose algebraic properties are amenable to algebro–geometric study. Following these ideas to their conclusion leads us in turn to the study of the moduli of formal groups, and we dedicate several sections to the description of the geometry of this moduli stack. We produce reflections in the stable homotopy category of our main algebraic results, emphasizing especially the periodicity theorems of Devinatz, Hopkins, and Smith, which simultaneously give shape to the structure of the “prime ideals” of the stable homotopy category as well as organize the stable stems into identifiable families.