  Let $N/K$ be a Galois extension of $p$-adic number fields and let $V$ be a
$p$-adic representation of the absolute Galois group $G_K$ of $K$. The
equivariant local $\epsilon$-constant conjecture $C_{EP}^{na}(N/K, V)$ is
related to the compatibility of the equivariant Tamagawa number conjecture with
the functional equation of Artin $L$-functions and it can be formulated as the
vanishing of a certain element $R_{N/K}$ in $K_0(\mathbb Z_p[G],\mathbb
Q_p^c[G])$. One of the main technical difficulties in the computation of
$R_{N/K}$ arises from the so-called cohomological term $C_{N/K}$, which
requires the construction of a bounded complex $C_{N,T}^\bullet$ of
cohomologically trivial modules which represents $R\Gamma(N,T)$ for a full
$G_K$-stable $\mathbb Z_p$-sublattice $T$ of $V$. In this paper we generalize
the construction of $C_{N,T}^\bullet$ in Thm. 2 of arXiv:1602.07858 to the case
of a higher dimensional $T$.
