Tate cohomology for arbitrary groups via satellites

Complete homology and related dimensions of groups

“…It is known that pccd GL j ðKÞ ¼ À1 for all j d 1; see [11], [14]. The following proposition shows that this group also satisfies ichd GL j ðKÞ ¼ À1.…”

“…The following proposition shows that this group also satisfies ichd GL j ðKÞ ¼ À1. Observe that the groups GL j ðO S i ðK i ÞÞ are virtual duality groups of dimension n i ¼ vcdðGL j ðO S i ðK i ÞÞÞ and then H m ðGL j ðO S i ðK i ÞÞ; ZGÞ ¼ 0 for m 0 n i ; see [3], [14]. Thus for any co-induced module C, Recall that a weakly complete resolution for a group G is a (not necessarily acyclic) complex F of projective G-modules such that F coincides with a projective resolution of Z over ZG in su‰ciently high dimension and Hom ZG ðF; QÞ is acyclic for every projective G-module Q; see [11,Definition 2.8].…”

“…The complete cohomology theory for arbitrary groups was defined by Benson and Carlson [1], Mislin [14], and Goichot [9] independently as a generalization of Farrell-Tate cohomology, and their approaches were shown to be isomorphic by Mislin. In [9], Goichot introduced a complete homology theory for arbitrary groups developed by Vogel as follows:…”

“…In 1998, Triulzi investigated another complete homology theory for arbitrary groups in his Ph.D. thesis [20]. Using the satellite approach invented by Mislin [14], he defined a complete homologyĤ H Ã ðG; ÀÞ which is injective complete as follows:…”

Bounding the orders of finite subgroups

“…His proof made use of the complete cohomology introduced by D. Benson, J. Carlson, G. Mislin and F. Vogel [1], [9] as will ours. (Complete cohomology can be viewed as a generalization of Tate cohomology.…”