Complete cohomological functors on groups

Abstract

If Λ is a ring and A is a Λ-module, then a terminal completion of Ext ∗ Λ ( A, ) is shown to exist if, and only if, Ext j Λ( A, P)=0 for all projective Λ-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 Λ-modules, silp Λ, is finite. Analogous results hold for Ext ∗ Λ (, A) and involve spli Λ, the supremum of the projective lengths of the injective Λ-modules. When Λ is an integral group ring Z G, spli Z G is finite implies silp Z G is finite. Also the finiteness of spli is preserved under group extensions. If G is a countable soluble group, the spli Z G is finite if, and only if, the Hirsch number of G is finite.