On weakly Laskerian and weakly cofinite modules (Q2859256)

Let \(R\) be a Noetherian commutative ring with nonzero identity. Let \(I\) be an ideal of \(R\) and \(M, N\) two \(R\)-modules.NEWLINENEWLINERecall that an \(R\)-module \(L\) is said to be weakly Laskerian if each quotient module of \(L\) has finitely many associated primes. Also, recall that for any nonegative integer \(i,\) the \(i\)th generalized local cohomology module of \(M\) and \(N\) with respect to \(I\) is defined by \(H_I^i(M,N):={\varinjlim}_n \mathrm{Ext}^i_R(M/I^nM,N).\) So, the usual local cohomology corresponds to the case \(M=R\).NEWLINENEWLINEThis paper investigates the question when the Ext modules \(\mathrm{Ext}^j_R(R/I,H_I^i(M,N))\) are weakly Laskerian.NEWLINENEWLINEAssume that \(M\) is fnitely generated and let \(r\) be a natural integer. The authors proved:NEWLINENEWLINEa) If \(N\) and \(H_I^i(M,N)\) are weakly Laskerian for all \(i<r,\) then \(\Hom_R(R/I,H_I^r(M,N))\) is also weakly Laskerian.NEWLINENEWLINEb) If \(H_I^i(N)\) is weakly Laskerian for all \(i\leq r\), then \(\mathrm{Ext}^j_R(R/I,H_I^i(M,N))\) is also weakly Laskerian for all \(j\) and all \(i\leq r\).