On the second problem of Hartshorne on cofiniteness

Let \({\mathfrak {a}}\) $\mathfrak{a}$ be an ideal of a commutative Noetherian ring R with \(\dim H^i_{{\mathfrak {a}}}(R)\le 1\) $dim{H}_{\mathfrak{a}}^i(R)\le 1$ for all integers \(i\ge 2\) $i\ge 2$ . In this paper, it is shown that the category of \({\mathfrak {a}}\) $\mathfrak{a}$ -cofinite modules forms an Abelian subcategory of the category of all R-modules. This assertion provides a partially affirmative answer to a question raised by R. Hartshorne in [Affine duality and cofiniteness, Invent. Math. 9(1970), 145-164]. Moreover, some applications of this result will be included.