The structure of the Hopf algebroid associated with the elliptic homology theory

Abstract

In representation theory of groups, there is an equivalence between the category of affine group and the category of commutative Hopf algebras. This fact can be generalized to affine groupoid schemes, that is, a groupoid scheme is an internal groupoid in the category of and there is an equivalence between the category of affine groupoid and the category of Hopf algebroids. On the other hand, by a work of J.F. Adams on generalized homology theory ([l], [2].) a commutative ring spectrum E such that E*E is flat over E* = E*(S°) gives a Hopf algebroid (£*, E*E). One of the most important examples of such E is the complex cobordism spectrum MU. A theorem of Quillen enable us to identify the affine groupoid scheme represented by the Hopf algebroid defined from the complex cobordism theory with an affine groupoid scheme of formal groups and strict isomorphisms between them. Many homology theories related with complex cobordism (for example, fiP-theory, Morava .fί-theory, elliptic homology) give closed (or locally closed) subgroupoid schemes of above groupoid scheme. This viewpoint is originally due to Jack Morava ([lθ]). We give basic definitions and constructions on internal groupoids in sections 1 and 2. In section 3, we translate the language of previous sections in terms of Hopf algebroids. We give a general description of Hopf algebroids associated with complex oriented cohomology theories satisfying certain condition in Section 4. Finally, we determine the structure of the Hopf algebroid associated with the elliptic homology theory given by Landweber ([8]).