Formal groups over non-commutative rings

We develop an extension of the usual theory of formal group laws where the base ring is not required to be commutative and where the formal variables need neither be central nor have to commute with each other. We show that this is the natural kind of formal group law for the needs of algebraic topology in the sense that a (possibly non-commutative) complex oriented ring spectrum is canonically equipped with just such a formal group law. The universal formal group law is carried by the Baker-Richter spectrum M which plays a role analogous to MU in this non-commutative context. As suggested by previous work of Morava the Hopf algebra B of "formal diffeomorphisms of the non-commutative line" of Brouder, Frabetti and Krattenthaler is central to the theory developed here. In particular, we verify Morava's conjecture that there is a representation of the Drinfeld quantum-double D (B) through cohomology operations in M.