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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).

References

Bahmanpour, K. and Naghipour, R., On the cofiniteness of local cohomology modules . Proc. Amer. Math. Soc. 136(2008), 2359–2363.Google Scholar
Bahmanpour, K. and Naghipour, R., Cofiniteness of local cohomology modules for ideals of small dimension . J. Algebra. 321(2009), 1997–2011.Google Scholar
Bahmanpour, K., Naghipour, R., and Sedghi, M., On the category of cofinite modules which is abelian . Proc. Amer. Math. Soc. 142(2014), 1101–1107.Google Scholar
Bahmanpour, K., Naghipour, R., and Sedghi, M., Cofiniteness with respect to ideals of small dimensions . Algebra Represent. Theor. 18(2015), 369–379.Google Scholar
Brodmann, M. P. and Faghani, A. L., A finiteness result for associated primes of local cohomology modules . Proc. Amer. Math. Soc. 128(2000), 2851–2853.Google Scholar
Brodmann, M. P. and Sharp, R. Y., Local cohomology: an algebraic introduction with geometric applications, Cambridge University Press, United Kingdom, 1998.Google Scholar
Bruns, W. and Herzog, J., Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, 39, Cambridge University Press, United Kingdom, 1993.Google Scholar
Christensen, L. W., Foxby, H.-B., and Holm, H., Derived category methods in commutative algebra, Springer Monographs in Mathematics, Switzerland, 2024.Google Scholar
Delfino, D. and Marley, T., Cofinite modules and local cohomology . J. Pure Appl. Algebra. 121(1997), 45–52.Google Scholar
Divaani-Aazar, K., Faridian, H., and Tousi, M., A new outlook on cofiniteness . Kyoto J. Math. 60(2020), 1033–1045.Google Scholar
Eto, K. and Kawasaki, K., A characterization of cofinite complexes over complete Gorenstein domains . J. Comm. Algebra. 3(2011), 537–550.Google Scholar
Faridian, H., Gorenstein homology and finiteness properties of local (co)homology. Ph. D. thesis, Shahid Beheshti University, 2020. arXiv:2010.03013v1.Google Scholar
Greenlees, J. P. C. and May, J. P., Derived functors of $I$ -adic completion and local homology. J. Algebra 149(1992), 438–453.Google Scholar
Grothendieck, A., Cohomologie locale des faisceaux cohèrents et thèorèmes de Lefschetz locaux et globaux, (SGA 2), North-Holland, Amsterdam, 1968.Google Scholar
Hartshorne, R., Affine duality and cofiniteness . Invent. Math. 9(1970), 145–164.Google Scholar
Huneke, C., Problems on local cohomology . In: Eisenbud, D. and Huneke, C. (eds.), Free resolutions in commutative algebra and algebraic geometry, Sundance, Boston, 90, 93–108.Google Scholar
Lipman, J., Lectures on local cohomology and duality . In: G. Lyubeznik (ed.), Local cohomology and its applications (Guanajuato, Mexico), Lecture Notes in Pure and Applied Mathematics, 226, Marcel Dekker, New York, 2002, 39–89.Google Scholar
Melkersson, L., Modules cofinite with respect to an ideal . J. Algebra. 285(2005), 649–668.Google Scholar
Melkersson, L., Cofiniteness with respect to ideals of dimension one . J. Algebra. 372(2012), 459–462.Google Scholar
Nazari, M. and Sazeedeh, R., Cofiniteness with respect to two ideals and local cohomology . Algebra Represent. Theor. 22(2019), 375–385.Google Scholar
Rotman, J. J., An introduction to homological algebra. 2nd ed., Springer, Berlin, 2009.Google Scholar