Coherence of augmented Iwasawa algebras

The augmented Iwasawa algebra of a p-adic Lie group is a generalisation of the Iwasawa algebra of a compact p-adic Lie group. We prove that a split-semisimple group over a p-adic field has a coherent augmented Iwasawa algebra if and only if its root system is of rank one. We deduce that the general linear group of degree n has a coherent augmented Iwasawa algebra precisely when n is at most two. We also characterise when certain solvable p-adic Lie groups have a coherent augmented Iwasawa algebra.
69 pages. Section 2 substantially expanded to develop basic results for augmented Iwasawa algebras. Section 7 expanded to prove a module-theoretic criterion. Section 9 expanded to provide a new characterisation for general linear groups. Minor corrections throughout. Manuscript accepted in Advances in Mathematics