Complete cohomological functors on groups (Q1109859)

If \(\Lambda\) is a ring and \(A\) is a \(\Lambda\)-module, then a terminal completion of \(Ext^*_{\Lambda}(A,-)\) is shown to exist if, and only if, \(Ext^ j_{\Lambda}(A,P)=0\) for all projective \(\Lambda\)-modules \(P\) and all sufficiently large \(j\). Such a terminal completion exists for every \(A\) if, and only if, the supremum of the injective lengths of all projective \(\Lambda\)-modules, silp \(\Lambda\), is finite. Analogous results hold for \(Ext^*_{\Lambda}(-,A)\) and involve spli \(\Lambda\), the supremum of the projective lengths of the injective \(\Lambda\)-modules. When \(\Lambda\) is an integral group ring \({\mathbb{Z}}G\), spli \({\mathbb{Z}}G\) is finite implies silp \({\mathbb{Z}}G\) is finite. Also the finiteness of spli is preserved under group extensions. If \(G\) is a countable soluble group, then spli \({\mathbb{Z}}G\) is finite if, and only if, the Hirsch number of \(G\) is finite.