Class Groups and General Linear Group Cohomology for a Ring of Algebraic Integers

Suppose that F is a number field, with ring of integers OF . Let ` denote an odd prime and let R = OF [1/`]. In [3], the author and W.Dwyer gave an explicit conjectural computation of the mod ` cohomology of the infinite general linear group GLR. Here is the quickest and simplest statement of the conjecture (all homology and cohomology groups have Z/` coefficients): let U denote the infinite unitary group. Let J a denote the mod ` cohomology of the homotopy-fibre of the ` -th power map SU→SU . Thus as an algebra, J a is the tensor product of a certain polynomial algebra P ∗ a and a companion exterior algebra E a . An explicit description is given in Section 1; here we just remark that the Hopf algebra Pa dual to P ∗ a is H∗BU/(` -th powers). Let P denote the algebra of Steenrod `-th power operations.