| Summary: | We study the notion of Cartan-Eilenberg Gorenstein-injective $m$-complexes.
We show that a $m$-complex $G$ is Cartan-Eilenberg Gorenstein-injective
if and only if $G_n$, $\mathrm{Z}_n^{t}(G)$, $\mathrm{B}_n^{t}(G)$
and $\mathrm{H}_n^{t}(G)$ are Gorenstein-injective modules
for each $n\in\mathbb{Z}$ and $t=1,2,\ldots,m$.
As an application, we show that an iteration of the procedure
used to define the Cartan-Eilenberg Gorenstein-injective $m$-complexes
yields exactly the Cartan-Eilenberg Gorenstein-injective $m$-complexes.
Specifically, given a Cartan-Eilenberg exact sequence of Cartan-Eilenberg Gorenstein-injective $m$-complexes
$$\mathbb{G}=\cdots\rightarrow G^{-1}\rightarrow
G^0\rightarrow G^1\rightarrow \cdots$$ such that
the functor $\mathrm{Hom}_{\mathcal{C}_m({R})}(H,-)$
leave $\mathbb{G}$ exact
for each Cartan-Eilenberg Gorenstein-injective
$m$-complex $H$, then $\mathrm{Ker}(G^0\rightarrow G^1)$
is a Cartan-Eilenberg Gorenstein-injective
$m$-complex.
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