On the associated primes and the support of generalized local cohomology modules (Q1002458)

This paper contains finiteness and vanishing properties of generalized local cohomology modules \(H^i_I(M,N)\); some of these results were known before in the special case of non-generalized local cohomology modules \(H^i_I(N)\). Let \(I\) be an ideal of a noetherian, local ring \(R\), let \(M\) and \(N\) be finite \(R\)-modules. Recall that the \(i\)-th generalized local cohomology module is defined as \[ H^i_I(M,N)=\text{dirlim}_n\text{Ext}^i_R(M/I^n,N). \] For the case \(\dim N\leq 3\) it is shown that all \(H^i_I(M,N)\) have only finitely many associated prime ideals. In the general case, if \(N\) has finite injective dimension, there are the following results (\(d:=\dim R\)): \(\bullet \;H^i_I(M,N)=0\) for all \(i>d-s\), where \(s\) is the stable value of \(\text{depth} (M/I^nM)\) for \(n\gg 0\). \(\bullet \;H^d_I(M,N)\) is Artinian. \(\bullet \;\text{Supp}^{d-1}_I(M,N))\) is finite.