Homological properties of 0-Hecke modules for dual immaculate quasisymmetric functions

Let $n$ be a nonnegative integer. For each composition $\alpha$ of $n$, Berg $\mathit{\text{et}}\mathit{\text{al.}}$ 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 ${\mathrm{E}\mathrm{x}\mathrm{t}}_{H_n(0)}^1({\mathcal{V}}_{\alpha },{\mathbf{F}}_{\beta })$ 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 ${\mathrm{E}\mathrm{x}\mathrm{t}}_{H_n(0)}^i({\mathcal{V}}_{\alpha },{\mathcal{V}}_{\beta })$ when $i=0,1$ and $\beta {\le }_l\alpha$, where ${\le }_l$ represents the lexicographic order on compositions.