  Let $n$ be a nonnegative integer. For each composition $\alpha$ of $n$, Berg
$\textit{et al.}$ introduced a cyclic indecomposable $H_n(0)$-module
$\mathcal{V}_\alpha$ with a dual immaculate quasisymmetric function as the
image of the quasisymmetric characteristic. In this paper, we study
$\mathcal{V}_\alpha$'s from the homological viewpoint. To be precise, we
construct a minimal projective presentation of $\mathcal{V}_\alpha$ and a
minimal injective presentation of $\mathcal{V}_\alpha$ as well. Using them, we
compute ${\rm Ext}^1_{H_n(0)}(\mathcal{V}_\alpha, {\bf F}_\beta)$ and ${\rm
Ext}^1_{H_n(0)}( {\bf F}_\beta, \mathcal{V}_\alpha)$, where ${\bf F}_\beta$ is
the simple $H_n(0)$-module attached to a composition $\beta$ of $n$. We also
compute ${\rm Ext}_{H_n(0)}^i(\mathcal{V}_\alpha,\mathcal{V}_{\beta})$ when
$i=0,1$ and $\beta \le_l \alpha$, where $\le_l$ represents the lexicographic
order on compositions.
