Conference abstracts

Session S10 - Homological Methods

July 29, 16:30 ~ 17:00

## Isomorphism conjectures with proper coefficients

### Eugenia Ellis

### Universidad de la República, Uruguay - eellis@fing.edu.uy

Let $G$ be a group and $\mathcal{F}$ a nonempty family of subgroups of $G$, closed under conjugation and under subgroups. Also let $E$ be a functor from small $\mathbb{Z}$-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 $(G,\mathcal{F},E,A)$ asserts that if $X\rightarrow Y$ is an equivariant map such that $X^H\rightarrow Y^H$ is an equivalence for all $H\in\mathcal{F}$, then $H^G(X,E(A))\rightarrow H^G(Y,E(A))$ is an equivalence. We introduce an algebraic notion of $(G,\mathcal{F})$-properness for $G$-rings, modelled on the analogous notion for $G$-$C^*$-algebras, and show that the strong $(G,\mathcal{F},E,P)$ isomorphism conjecture for $(G,\mathcal{F})$-proper $P$ is true in several cases of interest in the algebraic $K$-theory context.

Joint work with Guillermo Cortiñas (Universidad de Buenos Aires, Argentina).