On complete resolutions

Complete homology and related dimensions of groups

“…This module has played an important role in complete cohomology. For more details, see [7]. We also recall that the flat dimension of a left G-module M is defined by…”

“…Note that any two weakly complete resolution for G are chain homotopy equivalent (cf. [7,Lemma 2.4]). Thus the weakly complete cohomology of G is well defined [11,Definition 2.9].…”

“…In a similar way as for a complete projective resolution (see [7], [8], and [15]), we can define the notion of a complete injective resolution. A complete injective resolution of a G-module M is an acyclic complex of injective G-modules which agrees with an ordinary injective resolution in a su‰ciently high dimension k. The number k A N is called the coincidence index of the complete injective resolution; see [16].…”

Finiteness conditions in the stable module category

“…It follows from Mis-lin's approach to complete cohomology that the functors Ext * R (M, ) may be computed, in the case where M has a complete resolution P * , as the cohomology groups of the complex Hom R (P * , ), (cf. [7], Theorem 1.2). In particular, if M is Gorenstein projective, then the complete cohomology group Ext 0 R (M, N ) may be naturally identified for any left R-module N with the quotient Hom R (M, N ) of the abelian group Hom R (M, N ) by the subgroup consisting of all R-linear maps f : M −→ N that factor through a projective left R-module.…”

“…The first and third authors of the present work introduced a theory in a series of papers [6,8,7,16] the last one of which builds on the ideas of Benson. In view of the original Bieri-Eckmann theory, it seems very natural to consider what happens if one requires some or many cohomology functors to commute with filtered colimits. The behaviour of groups for which this kind of continuity of functors begins after some steps was investigated by the third author and then further in a series of papers by Hamilton [17,11,13,12].…”