Covariant Honda theory

Abstract

Honda's theory gives an explicit description up to strict isomorphism of formal groups over perfect fields of characteristic $p\neq 0$ and over their rings of Witt vectors by means of attaching a certain matrix, which is called its type, to every formal group. A dual notion of right type connected with the reduction of the formal group is introduced while Honda's original type becomes a left type. An analogue of the Dieudonné module is constructed and an equivalence between the categories of formal groups and right modules satisfying certain conditions, similar to the classical anti-equivalence between the categories of formal groups, and left modules satisfying certain conditions is established. As an application, the $\star$-isomorphism classes of the deformations of a formal group over and the action of its automorphism group on these classes are studied.