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Let $R$ be a commutative Noetherian ring, and let $\mathfrak a$ be a properideal of $R$. Let $M$ be a non-zero finitely generated $R$-module with thefinite projective dimension $p$. Also, let $N$ be a non-zero finitely generated$R$-module with $N\neq\mathfrak{a} N$, and assume that $c$ is the greatestnon-negative integer with the property that $\operatorname{H}^i_{\mathfraka}(N)$, the $i$-th local cohomology module of $N$ with respect to $\mathfraka$, is non-zero. It is known that $\operatorname{H}^i_{\mathfrak a}(M, N)$, the$i$-th generalized local cohomology module of $M$ and $N$ with respect to$\mathfrak a$, is zero for all $i>p+c$. In this paper, we obtain thecoassociated prime ideals of $\operatorname{H}^{p+c}_{\mathfrak a}(M, N)$.Using this, in the case when $R$ is a local ring and $c$ is equal to thedimension of $N$, we give a necessary and sufficient condition for thevanishing of $\operatorname{H}^{p+c}_{\mathfrak a}(M, N)$ which extends theLichtenbaum-Hartshorne vanishing theorem for generalized local cohomologymodules.

Lichtenbaum-Hartshorne vanishing theorem for generalized local cohomology modules