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