On the Krull dimension of cofinite local cohomology modules

Let $\mathfrak {a}$ be an ideal of a Noetherian ring $R$ such that the category of $\mathfrak {a}$-cofinite modules is an Abelian subcategory of the category of $R$-modules. Let $M$ be a finitely generated $R$-module such that the $R$-modules $H^i_{\mathfrak {a}}(M)$ are $\mathfrak {a}$-cofinite for all integers $i\in \mathbb{N}_0$. In this paper it is shown that $\dim H^i_{\mathfrak {a}}(M)\leq 1$ for all integers $i\geq 2$.