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.
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