Non-vanishing and cofiniteness of generalized local cohomology modules

In this paper, we show some results on the non-vanishing of the generalized local cohomology modules \(H^i_I(M,N)\) $H_I^i(M,N)$ . In a Cohen–Macaulay local ring \((R,\mathop {\mathfrak {m}})\) $(R,\mathfrak{m})$ , we prove, by using induction on \(\dim N\) $dimN$ , that if M, N are two finitely generated R-modules with \({\text {id}}\,M<\infty \) $\text{id}M<\infty$ and \({\text {Gid}}\,N<\infty \) $\text{Gid}N<\infty$ , then \(H^{\dim R-grade _R({\text {Ann}}_RN,M)}_{\mathop {\mathfrak {m}}}(M,N)\ne 0\) $H_{\mathfrak{m}}^{dimR-grad{e}_R({\text{Ann}}_{R}N,M)}(M,N)\ne 0$ . We also study the I-cofiniteness of the generalized local cohomology module \(H^i_{\mathop {\mathfrak {m}}}(M,N)\) $H_{\mathfrak{m}}^i(M,N)$ .