The Landweber Exact Functor Theorem, Stacks, and the Presheaf of Elliptic Homology Theories

The goal of this chapter is the construction of the presheaf of elliptic homology theories on the moduli stack of elliptic curves Mell . This sets the stage for many of the later chapters where the objective will be to turn this presheaf into to a sheaf of E∞-ring spectra (using obstruction theory). Even though we use the language of stacks, much of this chapter is closely related to the classical story of elliptic cohomology. Constructing a presheaf of homology theories on Mell means to associate to every elliptic curve C (satisfying a certain flatness condition) a homology theory EllC . The construction of EllC is based on Landweber’s exact functor theorem, and we verify the assumptions of Landweber’s theorem using the argument given by Landweber, Stong, Ravenel, and Franke, see [LRS], [Fr]. However, in the approach we describe (due to Hopkins and Miller) the use of Landweber’s theorem is elegantly hidden in the statement that the morphism