Isomorphism conjectures with proper coefficients.

GUILLERMO CORTIÑAS; EUGENIA ELLIS

JOURNAL OF PURE AND APPLIED ALGEBRA

ELSEVIER SCIENCE BV

Lugar: Amsterdam; Año: 2014 vol. 218 p. 1224 - 1263

0022-4049

 Let $G$ be a group and $\cF$ 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  $H^G(-,E(A))$ of $G$-simplicial sets such that $H_*^G(G/H,E(A))=E(A\rtimes H)$. The strong isomorphism conjecture for the quadruple $(G,\cF,E,A)$ asserts that if $X\to Y$ is an equivariant map such that $X^H\to Y^H$ is an equivalence for all $H\in\cF$, then\[H^G(X,E(A))\to H^G(Y,E(A))\]is an equivalence. 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.