Homological characterisation of Lambda-ranks

If G is a pro-p, p-adic, Lie group and if $\mathrm{\Lambda }(G)$ denotes the Iwasawa algebra of G then we present a formula for determining the $\mathrm{\Lambda }(G)$-rank of a finitely generated $\mathrm{\Lambda }(G)$-module. This is given in terms of the G homology groups of the module. We explore some consequences of this for the structure of $\mathrm{\Lambda }(G)$-modules.