On the first and second problems of Hartshorne on cofiniteness

Abstract

Let $\mathfrak{a}$ be an ideal of a given commutative Noetherian ring R which satisfies the condition of the first problem of R. Hartshorne in [Affine duality and cofiniteness, Invent. Math. 9 (1970), 145–164]. In this paper, we prove that $\mathfrak{a}$ also satisfies the condition of his second problem in the same article. We also provide an example to show that the converse statement does not hold in general.

Keywords:

• Abelian category
• cofinite module
• local cohomology
• local homology

2020 Mathematics Subject Classification:

• Primary: 13D45
• Secondary: 14B15
• 13E05

Acknowledgments

The author appreciates the valuable and insightful comments of the anonymous referee.