The Rigid Analytic Period Mapping, Lubin-tate Space, and Stable Homotopy Theory

The geometry of the Lubin-Tate space of deformations of a formal group is studied via an étale, rigid analytic map from the deformation space to projective space. This leads to a simple description of the equivariant canonical bundle of the deformation space which, in turn, yields a formula for the dualizing complex in stable homotopy theory. Introduction Ever since Quillen [22, 1 ] discovered the relationship between formal groups and complex cobordism, stable homotopy theory and the theory of formal groups have been intimately connected. Among other things the height filtration of formal groups has led to the chromatic filtration [21, 20, 23] which offers the best global perspective on stable homotopy theory available. From the point of view of homotopy theory this correspondence has been largely an organizational principle. It has always been easier to make calculations with the algebraic apparatus familiar to topologists. This is due to the fact that the geometry which comes up in studying formal groups is the geometry of affine formal schemes. The point of this paper is to study the Lubin-Tate deformation spaces of 1-dimensional formal groups of finite height using a p-adic analogue of the classical period mapping. This is a rigid analytic, étale morphism, from the LubinTate deformation space to projective space, which is equivariant for the natural group action. With this morphism the global geometry of projective space can be brought to bear on the study of formal groups. When applied to stable homotopy theory, this leads to a formula for the analogue of the Grothendieck-Serre dualizing complex. 1. Formal groups 1.1. The map to projective space. Let k be an algebraically closed field of characteristic p > 0, and let F0 be a formal group of dimension 1 and finite height n over k . Lubin and Täte [18] studied the problem of deforming F0 to a formal group F over R, where R is a complete, local Noetherian ring with residue field k . They defined deformations F and F' to be equivalent if there is an isomorphism 4>: F —> F' over R which reduces to the identity morphism of Fo and showed that the functor which assigns to R the equivalence classes of deformations of F0 to R is representable by a smooth formal scheme X Received by the editors August 22, 1992. 1991 Mathematics Subject Classification. Primary 14L05, 12H25, 55P.