On the weak Artinianness and minimax generalized local cohomology modules (Q2872175)

Let \(R\) be a commutative Noetherian ring, \(I\) a proper ideal of \(R\), \(M\) a weakly Laskerian \(R\)-module and \(t\) be a positive integer. Recall that an \(R\)-module \(X\) is said to be weakly Laskerian if each quatient module of \(X\) has finitely many associated prime ideals. Also, an \(R\)-module \(X\) is said to be weakly Artinian if \(E(X)\), its injective envelope, can be written as NEWLINE\[NEWLINEE(X)=\bigoplus_{i=1}^n\mu^0(m_i,X)E(R/m_i),NEWLINE\]NEWLINE where \(m_1,\dots,m_n\) are maximal ideals of \(R\). The elements \(a_1\),\dots, \(a_n\) of \(R\) are said to be weakly filter \(X\)-regular sequence if the \(R\)-module \(0:_{X/(a_1,\dots,a_{i-1})X}a_i\) is weakly Artinian for all \(i=1,\dots,n\). The author proves that \(I\) contains a weakly filter \(M\)-regular sequence of length \(t\) if and only if \(H_I^i(M)\) is weakly Artinian for all \(i<t\). This shows that each maximal weakly filter \(M\)-regular sequence has the same length. The author denotes this length by \(\mathrm{w-f-depth}(I,M)\) and proves that NEWLINE\[NEWLINE\mathrm{w-f-depth}(I+\mathrm{ann} M,N)=\inf\{i\mid H_I^i(M,N) \text{ is not weakly Artinian }\}.NEWLINE\]NEWLINE The author also proves that \(H_I^i(M,N)\) is minimax for all \(i<t\) if and only if \(H_I^i(M,N)_p\) is minimax for all \(i<t\) and all prime ideal \(p\) of \(R\), where \(M\), \(N\) are finitely generated.NEWLINENEWLINE Recall that an \(R\)-module \(X\) is said to be minimax if there is a finitely generate submodule \(Y\) of \(X\) such that \(X/Y\) is Artinian.