On the first and second problems of Hartshorne on cofiniteness (Q6600713)

Let \(R\) be a commutative noetherian ring, \(\mathfrak{a}\) an ideal of \(R\), and \(M\) an \(R\)-module. For any \(i \geq 0\), the \(i\)th local cohomology module of \(M\) with respect to \(\mathfrak{a}\) is given by \N\[\NH^{i}_{\mathfrak{a}}(M) \cong \underset{n\geq 1}\varinjlim \operatorname{Ext}^{i}_{R}\left(R/ \mathfrak{a}^{n},M\right).\N\]\NHartshorne defines an \(R\)-module \(M\) to be \(\mathfrak{a}\)-cofinite if \(\operatorname{Supp}_{R}(M)\subseteq \operatorname{Var}(\mathfrak{a})\) and \(\operatorname{Ext}^{i}_{R}\left(R/ \mathfrak{a},M\right)\) is a finitely generated \(R\)-module for every \(i\geq 0\). Then he asks the following questions:\N\N\begin{itemize}\N\item For which ideal \(\mathfrak{a}\) of \(R\), is the local cohomology module \(H^{i}_{\mathfrak{a}}(M)\) \(\mathfrak{a}\)-cofinite for every finitely generated \(R\)-module \(M\) and every \(i\geq 0\)?\N\item For which ideal \(\mathfrak{a}\) of \(R\), is the category of \(\mathfrak{a}\)-cofinite \(R\)-modules abelian?\N\end{itemize}\N\NThe author of this paper shows that if \(\mathfrak{a}\) is an ideal of \(R\) which satisfies the condition of the first question, then it also satisfies the condition of the second question. He further provides an example to show that the converse does not hold in general.