ON NONABELIAN TENSOR PRODUCT MODULOqOF GROUPS

Modq cohomology and Tate–Vogel cohomology of groups

“…The assertion similar to Theorem 3.1 has been proved in [15] for the classical (co)homology of groups and associative algebras. Moreover, one can obtain a similar result for mod q homology of groups [16].…”

“…Proof (It is essentially [16,Proposition 3.5]). The mapping cone gives rise to an exact homology sequence…”

“…In this section, we shall deÿne a mod q cohomology of groups by using Deÿnition 1.1 and then express it as the Ext * functors in the same way as the mod q homology of groups is expressed as the Tor * functors [16]. The ÿrst and the second mod q cohomology of groups will be described in terms of q-torsors and q-extensions of groups, respectively.…”

“…Then for any C m -module A one getŝH 2n (C m ; A; Z=q) = {(a; a ) | Na + qa = 0; ta = a }=D(A ⊕ A); n∈ Z; H 2n+1 (C m ; A; Z=q) = {(a; a ) | Da + qa = 0; Na = 0}=Ñ (A ⊕ A); n∈ Z with N = 1 + t + · · · + t m−1 , D = t − 1 ∈ Z[G]and where the homomorphismsD : A ⊕ A → A ⊕ A andÑ : A ⊕ A → A ⊕ Aare deÿned byD(a; a ) = (Da + qa ; −Na ) andÑ (a; a ) = (Na + qa ; −Da ).Proof. Follows from Theorem 5.8(ii) and[16, Proposition 3.10].…”

INTERNAL q-HOMOLOGY OF CROSSED MODULES

“…, A) are the q-homology groups introduced in [11]. In the following theorem we show the connection between our internal q-homology of crossed modules and the homology of groups with coefficients in Z/qZ, H q n (G) = H n (G, Z/qZ).…”