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Here $F_{\\cdot}^R$ denote a flat resolution of $M$. In this article we will prove the vanishing and non-vanishing properties of $L_i\\Lambda^I (N,M)$. We denote $H^{i}_{I}(N,M)$ (resp. $U^I_i(N,M)$) by the generalized local cohomology modules (resp. the generalized local homology modules). 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