United Elliptic Homology

Abstract

We study the categories of modules over real K-theory and TMF. Inspired by work of Bousfield, we consider TMF-modules at the prime 3 which are relatively free with respect to TMF(2). We show that a large class of these can be iteratively built from TMF by coning off torsion elements and killing generators. This is based on a detailed study of vector bundles on the moduli stack of elliptic curves. Furthermore, we consider examples of TMF-modules and show that the categories of TMF-modules and quasi-coherent sheaves on the derived moduli stack of elliptic curves are equivalent (at primes bigger than 2).