  Let $\mathfrak{a}$ be an ideal of a commutative noetherian (not necessarily
local) ring $R$. In the case $\cd(\mathfrak{a},R)\leq 1$, we show that the
subcategory of $\mathfrak{a}$-cofinite $R$-modules is abelian. Using this and
the technique of way-out functors, we show that if $\cd(\mathfrak{a},R)\leq 1$,
or $\dim(R/\mathfrak{a}) \leq 1$, or $\dim(R) \leq 2$, then the local
cohomology module $H^{i}_{\mathfrak{a}}(X)$ is $\mathfrak{a}$-cofinite for
every $R$-complex $X$ with finitely generated homology modules and every $i \in
\mathbb{Z}$. We further answer Question 1.3 in the three aforementioned cases,
and reveal a correlation between Questions 1.1, 1.2, and 1.3.
