Finiteness dimension and Bass numbers of generalized local cohomology modules (Q2853972)

In the paper under review, the authors want to generalize Theorem 2.6 of [\textit{K. Bahmanpour} and \textit{R. Naghipour}, J. Algebra 321, No. 7, 1997--2011 (2009; Zbl 1168.13016)] to generalized local cohomology. The main theorem of the present paper is as follows: Let \(R\) be a Noetherian ring, \(\mathfrak a \subset R\) an ideal, \(M\) and \(N\) finitely generated \(R\)-modules and \(t \geq 0\) an integer. If \(\dim \text{Supp} H_{\mathfrak a}^i(M, N) \leq 1\) for all \(i < t\), then there is another ideal \(\mathfrak b \subset R\) such that \(\dim R/\mathfrak b \leq 1\) and \(H_{\mathfrak a}^i(M, N) = H_{\mathfrak b}^i(M, N)\) for \(i < t\). If, in addition, \(M\) is of finite projective dimension, then all the Bass numbers and all the Betti numbers of \(H_{\mathfrak a}^i(M, N)\) are finite for all \(i < t\).