Lichtenbaum-Hartshorne vanishing theorem for generalized local cohomology modules

Let $R$ be a commutative Noetherian ring, and let $\mathfrak{a}$ be a proper ideal of $R$. Let $M$ be a non-zero finitely generated $R$-module with the finite projective dimension $p$. Also, let $N$ be a non-zero finitely generated $R$-module with $N\mathrm{\ne }\mathfrak{a}N$, and assume that $c$ is the greatest non-negative integer with the property that ${\mathrm{H}}_{\mathfrak{a}}^i(N)$, the $i$-th local cohomology module of $N$ with respect to $\mathfrak{a}$, is non-zero. It is known that ${\mathrm{H}}_{\mathfrak{a}}^i(M,N)$, 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 the coassociated prime ideals of ${\mathrm{H}}_{\mathfrak{a}}^{p+c}(M,N)$. Using this, in the case when $R$ is a local ring and $c$ is equal to the dimension of $N$, we give a necessary and sufficient condition for the vanishing of ${\mathrm{H}}_{\mathfrak{a}}^{p+c}(M,N)$ which extends the Lichtenbaum-Hartshorne vanishing theorem for generalized local cohomology modules.