Z[G]
. . . −→ Z[Gi+1]ϕi+1
−→ Z[Gi]ϕi
−→ Z[Gi−1]−→ . . . −→ 0
ϕi:Z[Gi]→Z[Gi−1]
ϕi(g1, . . . , gi) =
i
X
j=1
(−1)i(g1,..., ˆgj, . . . , gi)
(g1,..., ˆgj, . . . , gi)i(g1, . . . , gi)gj
ϕi◦ϕi+1 =
0
(g1, . . . , gi+1)∈Gi+1
ϕi◦ϕi+1(g1, . . . , gi+1) =
i+1
X
j=0
(−1)j(
j−1
X
k=1
(−1)k(g1,..., ˆgk,..., ˆgj,...gi+1) +
i
X
k=j
(−1)k(g1,..., ˆgj,..., ˆgk+1, . . . , gi+1))
(g1,..., ˆgk,..., ˆgj,...gi+1)
j ϕi+1 k ϕi
(−1)j(−1)kk ϕi+1 j ϕi
(−1)k(−1)j−1
C
0−→ Ai
−→ Bπ
−→ C−→ 0
i π Z[G]A B C
0−→ Z[Gj]⊗Z[G]A−→ Z[Gj]⊗Z[G]B−→ Z[Gj]⊗Z[G]C−→ 0