  In this paper we investigate finiteness properties of totally disconnected
locally compact groups for general commutative rings $R$, in particular for $R
= \mathbb{Z}$ and $R= \mathbb{Q}$. We show these properties satisfy many
analogous results to the case of discrete groups, and we provide analogues of
the famous Bieri's and Brown's criteria for finiteness properties and deduce
that both $FP_n$-properties and $F_n$-properties are quasi-isometric invariant.
Moreover, we introduce graph-wreath products in the category of totally
disconnected locally compact groups and discuss their finiteness properties.
