2011/08/31 by Guillermo Cortiñas, Guillermo Cortiñas⋆, Eugenia Ellis · 18 citations
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Combinatorics #Conjecture #Crystallography #Equivalence (formal languages) #Equivariant map #Functor #Homology (biology) #Homotopy and Cohomology in Algebraic Topology #Isomorphism (crystallography) #Mathematics #Pure mathematics #math.AT #math.KT #math.OA
paper · pdf · doi:10.1016/j.jpaa.2013.11.016
published in Journal of Pure and Applied Algebra 218(7), 1224-1263 (Elsevier BV) · 55 pages. Minor changes
arxiv created 2011/09/29 · openalex publication_date 2013/11/14 · arxiv updated 2014/03/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 HG(-,E(A)) with the property that if H⊂ G is a subgroup, then HG_*(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→ Y is a weak equivalence (resp. a fibration) if and only if XH→ YH is an equivalence (resp. a fibration) for all H∈\cF. The strong isomorphism conjecture for the quadruple (G,\cF,E,A) asserts that if cX→ X is the (G,\cF)-cofibrant replacement then HG(cX,E(A))→ HG(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))→ 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.