A new outlook on cofiniteness

摘要

Let a be an ideal of a commutative Noetherian (not necessarily local) ring R. In the case cd(a,R) ≤ 1, we show that the subcategory of a-cofinite R-modules is Abelian. Using this fact and the technique of way-out functors, we show that, if cd(a,R) ≤ 1, or if dim(R/a) ≤ 1, or if dim(R) ≤ 2, then the local cohomology module H_a~i(X) is a-cofinite for every R-complex X with finitely generated homology modules and every i ∈ Z. We further answer Hartshorne's Question 1.3 in the three aforementioned cases, and we reveal a correlation between his Questions 1.1, 1.2, and 1.3.