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Let $M$ be an $R$-module over a Noetherian ring $R$ and $\mathfrak{a}$ be anideal of $R$ with $c={\rm cd}(\mathfrak{a},M)$. First, we prove that $M$ isfinite $\mathfrak{a}$-relative Cohen-Macaulay if and only if ${\rmH}_i(\Lambda_{\mathfrak{a}}({\rm H}_{\mathfrak{a}}^c(M)))=0$ for all $i\neq c$and ${\rm H}_c(\Lambda_{\mathfrak{a}}({\rm H}_{\mathfrak{a}}^c(M))) \cong\widehat{M}^{\mathfrak{a}}$. Next, over an $\mathfrak{a}$-relativeCohen-Macaulay local ring $(R,\mathfrak{m})$, we provide a characterization of$\mathfrak{a}$-relative sequentially Cohen-Macaulay modules $M$ in terms of$\mathfrak{a}$-relative Cohen-Macaulayness of the $R$-modules ${\rmExt}^{d-i}_{R}(M,{\rm D}_{\mathfrak{a}})$ for all $i\geq 0$, where ${\rmD}_{\mathfrak{a}} = {\rm Hom}_R({\rm H}^d_{\mathfrak{a}}(R),{\rmE}(R/\mathfrak{m}))$ and $d={\rm cd}(\mathfrak{a},R)$. Finally, we provideanother characterization of $\mathfrak{a}$-relative sequentially Cohen-Macaulaymodules $M$ in terms of vanishing of the local homology modules ${\rmH}_j(\Lambda_{\mathfrak{a}}({\rm H}_{\mathfrak{a}}^i(M)))=0$ for all $0\leqi\leq c$ and for all $j\neq i$.

Some Characterizations of Relative Sequentially Cohen-Macaulay and Relative Cohen-Macaulay Modules