Some properties of generalized local homology and cohomology modules (Q2872190)

Let \(R\) be a commutative Noetherian ring, \(I\) an ideal of \(R\) and \(M,N\) two \(R\)-modules. For each non-negative integer \(i\), \(i\)-th generalized local homology module of \(M\) and \(N\) with respect to \(I\) is defined by NEWLINE\[NEWLINEH^I_i(M,N):={\varprojlim}_n\mathrm{Tor}_i^R(M/I^nM,N).NEWLINE\]NEWLINE This notion is the dual of the notion of generalized local cohomology which has been introduced by J. Herzog.NEWLINENEWLINEIn this paper, the authors study when the generalized local homology modules \(H^I_i(M,N)\) are secondary representable. Assume that \(M\) is finitely generated, \(N\) is Artinian and let \(s\geq 0\) be an integer. The authors show that the following statements are equivalent:NEWLINENEWLINE(i) \(H^I_i(M,N)\) is \(I\)-stable for all \(i<s\);NEWLINENEWLINE(ii) \(H^I_i(M,N)\) is Artinian for all \(i<s\);NEWLINENEWLINE(iii) \(H^I_i(M,N)\) is secondary representable for all \(i<s\).NEWLINENEWLINEAlso, when in addition the projective dimension of \(M\) is finite, they prove that the following statements are equivalent:NEWLINENEWLINE(i) \(H^I_i(M,N)\) is \(I\)-stable for all \(i>\mathrm{pd}_RM+s\);NEWLINENEWLINE(ii) \(H^I_i(M,N)\) is Artinian for all \(i>\mathrm{pd}_RM+s\);NEWLINENEWLINE(iii) \(H^I_i(M,N)\) is secondary representable for all \(i>\mathrm{pd}_RM+s\);NEWLINENEWLINE(iv) \(H^I_i(M,N)=0\) for all \(i>\mathrm{pd}_RM+s\).NEWLINENEWLINERecall that an \(R\)-module \(X\) is said to be \(I\)-stable if for each element \(a\in I\), there is a natural integer \(n\) such that \(a^tX=a^nX\) for all \(t\geq n\).