Rational discrete cohomology for totally disconnected locally compact groups

Rational discrete cohomology and homology for a totally disconnected locally compact group G are introduced and studied. The Hom-⊗ identity associated to the rational discrete bimodule Bi(. G) allows to introduce the notion of rational duality group in analogy to the discrete case. It is shown that a semi-simple algebraic group G(. K) defined over a non-discrete, non-archimedean local field K is a rational t.d.l.c. duality group, and the same is true for certain topological Kac-Moody groups. Indeed, for these groups the Tits (or Davis) realization of the associated building is a finite-dimensional model of the classifying space E_C(G(K)) one may define for any t.d.l.c. group. In contrast, Y. Neretin's group of spheromorphisms of a locally finite regular tree is not even of finite rational discrete cohomological dimension.