  Rational discrete cohomology and homology for a totally disconnected locally
compact group $G$ is introduced and studied. The $\mathrm{Hom}$-$\otimes$
identities associated to the rational discrete bimodule $\mathrm{Bi}(G)$ allow
to introduce the notion of rational duality groups in analogy to the discrete
case. It is shown that semi-simple groups defined over a non-discrete,
non-archimedean local field are rational t.d.l.c. duality groups, and the same
is true for certain topological Kac-Moody groups. However, Y. Neretin's group
of spheromorphisms of a locally finite regular tree is not even of finite
rational discrete cohomological dimension. For a unimodular t.d.l.c. group $G$
of type $\mathrm{FP}$ it is possible to define an Euler-Poincar\'e
characteristic $\chi(G)$ which is a rational multiple of a Haar measure. This
value is calculated explicitly for Chevalley groups defined over a
non-discrete, non-archimedean local field $K$ and some other examples.
