The normalized homogeneous bar complex

For the homogeneous complex, let $D_n(G)$ be the $\mathbb{Z}[G]$-submodule of $B_n(G)$ generated by tuples $(g_0,\dots ,g_n)$ with $g_i=g_{i+1}$ for some $i$. The normalized homogeneous bar complex is the augmented quotient ${\bar{B}}_{\bullet }(G)=B_{\bullet }(G)\mathrm{/}{D}_{\bullet }(G)⟶\mathbb{Z},$ The quotient differential is well defined: if $g_i=g_{i+1}$, deleting either of these two entries gives the same tuple with opposite signs in $d_n$, while every other deletion leaves an adjacent equal pair. Thus $d_n{D}_n\subseteq D_{n-1}$ for $n\ge 1$. Here $D_0=0$, and for $n=1$ the two faces cancel exactly, so the augmentation descends as well. Degeneracy is invariant under diagonal translation, making this a quotient of $\mathbb{Z}[G]$-complexes.

Each ${\bar{B}}_n(G)$ is free over $\mathbb{Z}[G]$ on the orbits of nondegenerate tuples: every orbit has the unique representative $(1,g_0^{-1}{g}_1,\dots ,g_0^{-1}{g}_n)$, and the degenerate tuples span a disjoint union of the other free orbits. The next contractibility and homotopy-equivalence results prove that this augmented free complex is a resolution. Under the coordinates $x_i=g_{i-1}^{-1}{g}_i$, an adjacent equal pair is exactly $x_i=1$. Thus its cochains are the inhomogeneous cochains that vanish whenever one argument is $1$.