Rational discrete cohomology for totally disconnected locally compact groups

TL;DR

This paper introduces rational discrete cohomology for totally disconnected locally compact groups, explores duality properties, and calculates explicit invariants for certain algebraic groups, advancing understanding of their topological and algebraic structures.

Contribution

It develops the theory of rational discrete cohomology for t.d.l.c. groups, introduces rational duality groups, and computes Euler-Poincaré characteristics for specific classes of groups.

Findings

Semi-simple groups over non-archimedean local fields are rational duality groups.

Y. Neretin's group has infinite rational discrete cohomological dimension.

Explicit Euler-Poincaré characteristic formulas are provided for Chevalley groups over local fields.

Abstract

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.