Singular coefficients in the \(K\)-theoretic Farrell-Jones conjecture (Q258834)

Let \(G\) be a group and consider \(\mathcal{F}\) a family of subgroups of \(G\), by this we mean a collection of subgroups of \(G\) closed under conjugation by elements of \(G\) and by taking subgroups of elements of \(\mathcal{F}\). Let \(X\) and \(Y\) be \(G\) spaces, a map \(f:X\to Y\) is called an \(\mathcal{F}\)-equivalence if the restriction to fixed points \(f:X^H\to Y^H\) is a weak equivalence for every \(H\in\mathcal{F}\). A \(G\)-complex \(X\) is called a \((G,\mathcal{F})\)-space if the stabilizer of every simplex of \(X\) is in \(\mathcal{F}\). Given a functor \(E\) from the category of small \(\mathbb{Z}\)-linear categories to the category of spectra, there is a well defined equivariant homology theory for \(G\)-spaces: \(H_*^G(X;E(R))\). Let \(R\) be a ring with unit, the \textit{strong isomorphism conjecture} for the quadruple \((G,\mathcal{F},E,R)\) asserts that given an \(\mathcal{F}\)-equivalence \(f:X\to Y\), then the induced map \(H_*^G(f,E(R))\) is an isomorphism. The \textit{isomorphism conjecture} for the the quadruple \((G,\mathcal{F},E,R)\) asserts that if the projection map \(\mathcal{E}(G,\mathcal{F})\to pt\) is a cofibrant replacement then the induced map \(H_*^G(\mathcal{E}(G,\mathcal{F});E(R))\to E_*(R[G])\) is an isomorphism. Let \(\mathcal{VC}\) be the family of virtually cyclic subgroups of \(G\) and \(K\) the \(K\)-theory spectrum of the ring \(R\). The \textit{Farrell-Jones isomorphism conjecture} is the isomorphism conjecture for the quadruple \((G,\mathcal{VC},K,R)\).NEWLINENEWLINEThe main theorem of this paper is the following:NEWLINENEWLINE Theorem 1.1. Let \(\mathcal{F}\) be a family of subgroups of \(G\) that contains all the cyclic subgroups of \(G\). Let \(k\) be a field of characteristic zero and \(f:X\to Y\) be a map of \(G\) spaces that is a \((G,\mathcal{F})\)-equivalence. Assume that \(H^G(f;K(R))\) is a weak equivalence for every commutative smooth \(k\)-algebra \(R\). Then \(H^G(f;K(R))\) is a weak equivalence for every commutative \(k\)-algebra \(R\). As a corollary one has that: If a group \(G\) satifies the \(K\)-theoretic Farrell-Jones conjecture with coefficients in every smooth \(\mathbb{Q}\)-algebra \(R\), then \(G\) also satisfies the Farrell-Jones conjecture with coefficients in any commutative \(\mathbb{Q}\)-algebra.