Dual of Faltings' Theorems on Finiteness of Local Cohomology

Let $R$ be a commutative Noetherian ring and $\text{fa}$ an ideal of $R$. We intend to establish the dual of two Faltings’ Theorems for local homology modules of an Artinian module. As a consequence of this, we show that, if $A$ is an Artinian module over semi-local complete ring $R$ and $j$ is an integer such that $H_i^{\text{fa}}(A)$ is Artinian for all $i<j$, then the set ${\text{Coass}}_R(H_j^{\text{fa}}(A))$ is finite. We also introduce the notion of the $n$th Artinianness dimension $g_n^\fa(A)=\inf{g^{\fa R_{\fp}}(^\fp A): \fp\in\Spec® \ \ $\text{and}$ \ \ \dim R/\fp\geq n}$,forall$n\in$\mathbb{N}$_{0}$andprovethat$g_1^\fa(A)=\inf{i\in$\mathbb{N}$_0: H_i^\fa(A) \ \ $\text{is not minimax}$}$,whenever$R$ is a semi-local complete ring. Moreover, in this situation we show that ${\text{Coass}}_R(H_{g_1^{\text{fa}}(A)}^{\text{fa}}(A))$ is a finite set.