Abstract:

Let G be a group and F a nonempty family of subgroups of G, closed under conjugation and under subgroups. Also let E be a functor from small Z-linear categories to spectra, and let A be a ring with a G-action. Under mild conditions on E and A one can define an equivariant homology theory HG (-, E (A)) of G-simplicial sets such that H* G (G / H, E (A)) = E (A ⋊ H). The strong isomorphism conjecture for the quadruple (G, F, E, A) asserts that if X → Y is an equivariant map such that XH → YH is an equivalence for all H ∈ F, thenHG (X, E (A)) → HG (Y, E (A)) is an equivalence. In this paper we introduce an algebraic notion of (G, F)-properness for G-rings, modeled on the analogous notion for G-C*-algebras, and show that the strong (G, F, E, P) isomorphism conjecture for (G, F)-proper P is true in several cases of interest in the algebraic K-theory context. © 2013 Elsevier B.V. All rights reserved.