Formal groups and stable homotopy of commutative rings

Abstract

We explain a new relationship between formal group laws and ring spectra in stable homotopy theory. We study a ring spectrum denoted $DB$ which depends on a commutative ring $B$ and is closely related to the topological André–Quillen homology of $B$. We present an explicit construction which to every 1–dimensional and commutative formal group law $F$ over $B$ associates a morphism of ring spectra $F_{\ast }:H\mathbb{Z}\to DB$ from the Eilenberg–MacLane ring spectrum of the integers. We show that formal group laws account for all such ring spectrum maps, and we identify the space of ring spectrum maps between $H\mathbb{Z}$ and $DB$. That description involves formal group law data and the homotopy units of the ring spectrum $DB$.