Given two equivalent locally compact Hausdorff groupoids, We prove that the Bost conjecture with Banach algebra coefficients is true for one if and only if it is true for the other. This also holds for the Bost conjecture with C*- coefficients. To show these results, the functoriality of Lafforgue's KK-theory for Banach algebras and groupoids with respect to generalised morphisms of groupoids is established. It is also shown that equivalent groupoids have Morita equivalent L1-algebras (with Banach algebra coefficients).

We prove that the Bost conjecture with C*-algebra coefficients for locally compact Hausdorff groups passes to open subgroups. We also prove that if a locally compact Hausdorff group acts on a tree, then the Bost conjecture with C*-coefficients is true for the group if and only if it is true for the stabilisers of the vertices.

In his fundamental paper on group cohomology [20] R.G. Swan defined a homomorphism for any finite group G which, in this restricted context, has since been used extensively both in the classification of projective modules and the algebraic homotopy theory of finite complexes ([3], [18], [21]). We extend the definition so that, for suitable modules J over reasonably general rings Λ, it takes the form here is the quotient of the category of Λ-homomorphisms obtained by setting ‘projective = 0’. We then employ it to give an exact classification of homotopy classes of extensions 0 → J → Fn → … → F0 → F0 → M → 0 where each Fr is finitely generated free.

We consider the manifold Fln(ℍ) = Sp(n)/Sp(1)n of all complete flags in ℍn, where ℍ is the skew-field of quaternions. We study its equivariant complex K-theory rings with respect to the action of two groups: Sp(1)n and a certain canonical subgroup T = (S1)n (a maximal torus). For the first group action we obtain a Goresky-Kottwitz-MacPherson type description. For the second one, we describe the ring KT(Fln(ℍ)) as a subring of KT(Sp(n)/T). This ring is well known, since Sp(n)/T is a complex flag variety.

For a tame Deligne-Mumford stack X with the resolution property, we show that the Cartan-Eilenberg resolutions of unbounded complexes of quasicoherent sheaves are K-injective resolutions. This allows us to realize the derived category of quasi-coherent sheaves on X as a reflexive full subcategory of the derived category of X-modules.

We then use the results of Neeman and recent results of Kresch to establish the localization theorem of Thomason-Trobaugh for the K-theory of perfect complexes on stacks of above type which have coarse moduli schemes. As a byproduct, we get a generalization of Krause's result about the stable derived categories of schemes to such stacks.

We prove Thomason's classification of thick triangulated tensor subcategories of D(perf / X). As the final application of our localization theorem, we show that the spectrum of D(perf / X) as defined by Balmer, is naturally isomorphic to the coarse moduli scheme of X, answering a question of Balmer for the tensor triangulated categories arising from Deligne-Mumford stacks.