Isomorphism conjectures with proper coefficients

Abstract

Let $G$ be a group and let $E$ be a functor from small $\Z$-linear categories to spectra. Also let $A$ be a ring with a $G$-action. Under mild conditions on $E$ and $A$ one can define an equivariant homology theory of $G$-simplicial sets $H^G(-,E(A))$ with the property that if $H\subset G$ is a subgroup, then \[ H^G_*(G/H,E(A))=E_*(A\rtimes H) \] If now $\cF$ is a nonempty family of subgroups of $G$, closed under conjugation and under subgroups, then there is a model category structure on $G$-simplicial sets such that a map $X\to Y$ is a weak equivalence (resp. a fibration) if and only if $X^H\to Y^H$ is an equivalence (resp. a fibration) for all $H\in\cF$. The strong isomorphism conjecture for the quadruple $(G,\cF,E,A)$ asserts that if $cX\to X$ is the $(G,\cF)$-cofibrant replacement then \[ H^G(cX,E(A))\to H^G(X,E(A)) \] is an equivalence. The isomorphism conjecture says that this holds when $X$ is the one point space, in which case $cX$ is the classifying space $\cE(G,\cF)$. In this paper we introduce an algebraic notion of $(G,\cF)$-properness for $G$-rings, modelled on the analogous notion for $G$-$C^*$-algebras, and show that the strong $(G,\cF,E,P)$ isomorphism conjecture for $(G,\cF)$-proper $P$ is true in several cases of interest in the algebraic $K$-theory context. Thus we give a purely algebraic, discrete counterpart to a result of Guentner, Higson and Trout in the $C^*$-algebraic case. We apply this to show that under rather general hypothesis, the assembly map $H_*^G(\cE(G,\cF),E(A))\to E_*(A\rtimes G)$ can be identified with the boundary map in the long exact sequence of $E$-groups associated to certain exact sequence of rings. Along the way we prove several results on excision in algebraic $K$-theory and cyclic homology which are of independent interest.