  Let $R$ be a commutative noetherian ring and $I$ an ideal of $R$. Assume that
for all integers $i$ the local cohomology module $H_I^i(R)$ is $I$-cofinite.
Suppose that $R_\mathfrak{p}$ is a regular local ring for all prime ideals
$\mathfrak{p}$ that do not contain $I$. In this paper, we prove that if the
$I$-cofinite modules forms an abelian category, then for all finitely generated
$R$-modules $M$ and all integers $i$, the local cohomology module $H_I^i(M)$ is
$I$-cofinite.
