Reformulation of the stable Adams conjecture

We revisit methods of proof of the Adams Conjecture in order to correct and supplement earlier efforts to prove an analogous conjecture in the stable homotopy category. Our proof relies heavily on Segal Gamma spaces and completion results of Bousfield and Kan. Our input involves simplicial schemes over an algebraically closed field of positive characteristic and a rigid version of Artin-Mazur etale homotopy theory. The central challenge in extending proofs of the original Adams Conjecture to the stable homotopy category is the lack of suitable functor from relevant categories of schemes to the category of topological spaces which commutes with products; Segal Gamma spaces enable us to utilize an ``etale functor" which commutes up to homotopy with products of simplicial schemes occurring in our arguments

General linear and functor cohomology over finite fields

In recent years, there has been considerable success in computing Ext-groups of modular representations associated to the general linear group by relating this problem to one of computing Ext-groups in functor categories. In this paper, we extend our ability to make such Ext-group calculations by establishing several fundamental results. Throughout this paper, we work over fields of positive characteristic p.Comment: 66 pages, published version, abstract added in migratio