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Hartshorne’s question on cofiniteness of complexes

Part of: Commutative algebra: Homological methods

Published online by Cambridge University Press:  07 October 2025

Xiaoyan Yang
Xiaoyan Yang*
Affiliation:
School of Science, Zhejiang University of Science and Technology , China
*
e-mail: yangxy@zust.edu.cn
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Abstract

Let d be a positive integer, and let $\mathfrak {a}$ be an ideal of a commutative Noetherian ring R. We answer Hartshorne’s question on cofiniteness of complexes posed in Hartshorne (1970, Invent. Math. 9, 145–164) in the cases $\mathrm {dim}R=d$ or $\mathrm {dim}R/\mathfrak {a}=d-1$ or $\mathrm {ara}(\mathfrak {a})=d-1$, show that if $d\leqslant 2$, then a complex $X\in \mathrm {D}_\sqsubset (R)$ is $\mathfrak {a}$-cofinite if and only if each homology module $\mathrm {H}_i(X)$ is $\mathfrak {a}$-cofinite; if R is regular local, $\mathfrak {a}$ is perfect and $d\leqslant 2$, then $X\in \mathrm {D}(R)$ is $\mathfrak {a}$-cofinite if and only if every $\mathrm {H}_i(X)$ is $\mathfrak {a}$-cofinite; if $d\geqslant 3$, then $X\in \mathrm {D}_\sqsubset (R)$ is $\mathfrak {a}$-cofinite and $\mathrm {Ext}^j_R(R/\mathfrak {a},\mathrm {H}_i(X))$ is finitely generated for $j\leqslant d-2$ and $i\in \mathbb {Z}$ if and only if every $\mathrm {H}_{i}(X)$ is $\mathfrak {a}$-cofinite.

Keywords

Cofinite complexcofinite modulelocal cohomology

MSC classification

Primary: 13D45: Local cohomology 13D09: Derived categories

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Type
Article
Information
Canadian Mathematical Bulletin , First View , pp. 1 - 14
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

This research was supported by the National Natural Science Foundation of China (Grant No. 12571035).

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