Covariant Honda theory

Abstract

Honda's theory gives an explicit description up to strict isomorphism of for- mal groups over perfect fields of characteristic p �= 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 no- tion 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 Dieudonne module is constructed and an equivalence between the categories of formal groups and right modules satisfying certain con- ditions, similar to the classical anti-equivalence between the categories of formal groups, and left modules satisfying certain conditions is established. As an application, the � -isomorphism classes of the deformations of a formal group over and the action of its automorphism group on these classes are studied. 0. Introduction. Let k be a perfect field of characteristic p �= 0a ndO its ring of Witt vectors. Honda's theory (5) gives an explicit description of formal groups over O and k. It attaches to every n-dimensional formal group F a certain n × n-matrix over the non- commutative twisted power series ring E, which is called a type of F. If we restrict our consideration to the p-typical formal groups we will not lose much: every formal group under consideration is strongly isomorphic to a p-typical group. In the present paper, we attach to every p-typical formal group another matrix over E, which we call a 'right type', while Honda's original type will be a 'left type'. The left type describes formal groups up to strict isomorphism and the right type is connected with the reduction of formal groups. In a sense, these notions are dual. Fontaine (4) used Honda's technique to construct an anti-equivalence from the categories of formal groups over k and that over O to the category of left E-modules and the category of the pairs consisting of a left E-module and its O-submodule satisfying certain conditions, respectively. Moreover, the reduction modulo p functor between the former categories corre- sponds to the forgetting of the second component functor between the latter ones. Employing the notion of right type, we obtain an equivalence from the categories of the formal groups over k and that over O to the category of right E-modules and the category of the pairs con- sisting of a right E-module and its O-submodule satisfying certain conditions, respectively. In this construction, the reduction modulo p functor corresponds to the functor of factorization of the first component by the E-linear envelope of the second component.