Open questions from the generalized Hartshorne problem: $I$-weak cofiniteness of the generalized local cohomology modules $H_I^i(M,N)$

The 2023 preprint of Dosea, Holanda and Miranda-Neto proves new positive cases of the generalized Hartshorne cofiniteness problem — deciding when the generalized local cohomology modules $H_I^i(M,N)$, with $M$ finite and $N$ weakly Laskerian, are $I$-weakly cofinite — and explicitly poses the questions its methods cannot settle: whether $H^1_I(M,N)$ is $I$-weakly cofinite under the hypotheses of its Theorem 2.10 (Question 2.12); whether, over a local ring, a finite module of finite projective dimension forms an $I$-cohomologically injective pair with every (or some) non-Gorenstein module, and whether this property is independent of the chosen injective resolution (Question 3.5); whether the deviation-based cofiniteness criterion of Theorem 3.16 extends to $\delta\ge 2$ (Question 3.17); and whether the three-way weak-cofiniteness/Ext characterization known for ordinary local cohomology in dimension at most 2 extends to general finite $M$ (Question 3.32). These questions are open in the generality stated in the source.

Background

Throughout, $R$ is a commutative Noetherian ring with identity and $I\subset R$ is a proper ideal. The $I$-torsion functor $\Gamma_I(N)=\{x\in N: I^t x=0 \text{ for } t\gg 0\}$ has right derived functors $H_I^i(N)$, the local cohomology modules of $N$, which are the standard cohomological tool behind duality theory, vanishing theorems and many algebraic-geometric applications. These modules are almost never finitely generated, so one studies weaker finiteness properties. Hartshorne (1970) introduced the $I$-cofinite modules — those $W$ with $\operatorname{Supp}W\subset V(I)=\{\mathfrak p\in\operatorname{Spec}R: I\subset\mathfrak p\}$ such that $\operatorname{Ext}^i_R(R/I,W)$ is finitely generated for every $i$ — and asked when local cohomology modules of finite modules are $I$-cofinite; Huneke later conjectured that over a Noetherian local ring of dimension $d$ the local cohomology modules of finitely generated modules have finitely many associated primes (the case $d\le 3$ is confirmed, while counterexamples exist in general). Both questions generated an extensive literature.

For $R$-modules $M$ and $N$ with $M$ finite (finitely generated), the generalized local cohomology modules are defined, following Herzog, as $H_I^i(M,N)=\varinjlim_n\operatorname{Ext}^i_R(M/I^nM,N)$, the direct system being induced by the natural epimorphisms $M/I^{n+1}M\to M/I^nM$; taking $M=R$ recovers the ordinary theory. Because $N$ need not be finite, one works in the class of weakly Laskerian modules: $W$ is weakly Laskerian if $\operatorname{Ass}_R(W/U)$ is finite for every submodule $U\subseteq W$, a class containing all finite modules and closed under submodules, quotients and extensions. A module $W$ with $\operatorname{Supp}W\subset V(I)$ is $I$-weakly cofinite if $\operatorname{Ext}^i_R(R/I,W)$ is weakly Laskerian for all $i\ge 0$; this is the natural analogue of $I$-cofiniteness in the weakly Laskerian category, and it implies that $\operatorname{Ass}_R W$ is finite. The generalized Hartshorne problem in this setting asks: when is $H_I^i(M,N)$ $I$-weakly cofinite?

The source preprint (Dosea–Holanda–Miranda-Neto, arXiv:2304.11493) proves several new positive cases via spectral sequence techniques. Theorem 2.10: if $M$ is finite, $N$ is weakly Laskerian, $\operatorname{Tor}^R_i(R/I,M)=0$ for all $i>0$ (for instance $M$ flat, or $I$ generated by an $M$-regular sequence), and $\operatorname{depth}_I N>0$ (where $\operatorname{depth}_I N=\inf\{i:\operatorname{Ext}^i_R(R/I,N)\neq 0\}$), then $\operatorname{Ext}^1_R(R/I,H^1_I(M,N))$ is weakly Laskerian. A central technical device is $I$-cohomological injectivity of the pair $M,N$ ($M$ finite): the module $H^0_I(\operatorname{Hom}_R(M,E^i))$ is injective for each term $E^i$ of an injective resolution of $N$; the condition holds, for example, when $M$ is flat or $N$ is a Gorenstein module (a finite module of finite injective dimension over the local ring). Under this condition, Theorem 3.6 shows that if all $H^i_I(M,N)$ with $i\neq s$ are $I$-weakly cofinite, then so is $H^s_I(M,N)$. With the invariants $g(M,N)=\inf_{j\ge 0}\operatorname{depth}_I\operatorname{Ext}^j_R(M,N)$, $c(M,N)=\sup_{j\ge 0}\operatorname{cd}_I\operatorname{Ext}^j_R(M,N)$ (where $\operatorname{cd}_I W=\sup\{i: H^i_I(W)\neq 0\}$ is the cohomological dimension), the deviation $\delta(M,N)=c(M,N)-g(M,N)$ satisfies $H^i_I(M,N)=0$ for $i<g(M,N)$ or $i>c(M,N)$, and Theorem 3.16 settles the $I$-weak cofiniteness of all $H^i_I(M,N)$ when $\delta=0$, and when $\delta=1$ provided that, for each $j\ge 0$, at least one of $H^{g(M,N)}_I(\operatorname{Ext}^j_R(M,N))$ or $H^{c(M,N)}_I(\operatorname{Ext}^j_R(M,N))$ is $I$-weakly cofinite. Finally, Theorem 3.31 shows that for $R$ local and $\dim N\le 2$ the following are equivalent: $H^i_I(N)$ is $I$-weakly cofinite for all $i$; $\operatorname{Ext}^i_R(R/I,N)$ is weakly Laskerian for all $i$; and $\operatorname{Ext}^i_R(R/I,N)$ is weakly Laskerian for $i=0,1$. The paper closes each of these threads with a question that its methods provably cannot settle, and those questions are the present problem.

Problem Statement

The problem is to settle the four questions left open by Dosea–Holanda–Miranda-Neto (arXiv:2304.11493), which together probe one common objective — deciding the $I$-weak cofiniteness of generalized local cohomology modules $H_I^i(M,N)$ with $M$ finite and $N$ weakly Laskerian — from complementary directions (the first module, the technical injectivity hypothesis, the higher-deviation regime, and a low-dimensional equivalence characterization). Each item is a precise statement (item 2 inherits the source's disjunctive scope — every, or at least some, non-Gorenstein $N$ — together with the source's separate open remark on resolution-independence); the problem is completely resolved when each of the four is decided, affirmatively by proof or negatively by explicit counterexample.

1. (Question 2.12.) Let $M$ be finite, $N$ weakly Laskerian, $\operatorname{Tor}^R_i(R/I,M)=0$ for all $i>0$, and $\operatorname{depth}_I N>0$. Is $H^1_I(M,N)$ $I$-weakly cofinite? (Currently only the weak Laskerianess of $\operatorname{Ext}^1_R(R/I,H^1_I(M,N))$ is known.)

2. (Question 3.5.) Let $R$ be local and let $M$ be a finite $R$-module with finite projective dimension. Is the pair $M,N$ $I$-cohomologically injective for every — or at least for some — non-Gorenstein $R$-module $N$? In particular, is the $I$-cohomological injectivity of a pair independent of the choice of injective resolution of $N$ (a separate open remark in the source, Remark 3.3(i), where it is conjectured but not proved)?

3. (Question 3.17.) Let $M$ be finite and $N$ weakly Laskerian, with deviation $\delta(M,N)=\delta\ge 2$. Assume that, for each $j\ge 0$, at least $\delta$ of the modules $H_I^{g(M,N)+k}(\operatorname{Ext}^j_R(M,N))$, $k=0,\dots,\delta$, are $I$-weakly cofinite. Does it follow that $H^i_I(M,N)$ is $I$-weakly cofinite for all $i\ge 0$? (Theorem 3.16 of the source settles $\delta=0$, and $\delta=1$ under the additional hypothesis that for each $j\ge 0$ at least one of $H^{g(M,N)}_I(\operatorname{Ext}^j_R(M,N))$ or $H^{c(M,N)}_I(\operatorname{Ext}^j_R(M,N))$ is $I$-weakly cofinite.)

4. (Question 3.32.) Let $R$ be local and let $M$, $N$ be $R$-modules with $M$ finite and $\dim N\le 2$. Are the following assertions equivalent: (i) $H^i_I(M,N)$ is $I$-weakly cofinite for all $i\ge 0$; (ii) $\operatorname{Ext}^i_R(M/IM,N)$ is weakly Laskerian for all $i\ge 0$; (iii) $\operatorname{Ext}^i_R(M/IM,N)$ is weakly Laskerian for $i=0,1$? (Proved in the source only for $M=R$, where $H^i_I(R,N)=H^i_I(N)$.)

References

1. André Dosea, Rafael Holanda, Cleto B. Miranda-Neto, “Generalized Hartshorne's problem on finiteness properties of local cohomology modules”, preprint, arXiv:2304.11493 (2023). DOI: 10.48550/arXiv.2304.11493, https://arxiv.org/abs/2304.11493
2. Robin Hartshorne, “Affine duality and cofiniteness”, Inventiones Mathematicae 9 (1970), 145–164. DOI: 10.1007/BF01404554, https://doi.org/10.1007/BF01404554
3. Craig Huneke, “Problems on local cohomology”, in: Free resolutions in commutative algebra and algebraic geometry (Sundance, UT, 1990), Research Notes in Mathematics 2, Jones and Bartlett Publishers, Boston, 1992, pp. 93–108. ISBN 978-0-86720-285-4, Zbl 0782.13015, https://zbmath.org/124447
4. André Dosea, Cleto B. Miranda-Neto, “On Huneke's conjecture about associated primes of local cohomology modules”, Journal of Algebra 669 (2025), 143–158. DOI: 10.1016/j.jalgebra.2025.02.008, https://doi.org/10.1016/j.jalgebra.2025.02.008 (preprint version: arXiv:2405.06862, “On the factorial case of Huneke's conjecture for local cohomology modules”, 2024, https://arxiv.org/abs/2405.06862)
5. Rafael Holanda, Cleto B. Miranda-Neto, “An Ext-Tor duality theorem, cohomological dimension, and applications”, Bulletin des Sciences Mathématiques 205 (2025), article 103696. DOI: 10.1016/j.bulsci.2025.103696, https://doi.org/10.1016/j.bulsci.2025.103696 (preprint: arXiv:2312.09725, 2023, https://arxiv.org/abs/2312.09725)
6. André Dosea, Majid Eghbali, Cleto B. Miranda-Neto, “Epimorphisms of local cohomology modules, a general Peskine–Szpiro theorem, and an application to sheaf cohomology vanishing for thickenings”, preprint, arXiv:2604.20150 (2026). DOI: 10.48550/arXiv.2604.20150, https://arxiv.org/abs/2604.20150
7. Alireza Vahidi, Ahmad Khaksari, Mohammad Shirazipour, “Cofiniteness and finiteness of associated prime ideals of generalized local cohomology modules”, preprint, arXiv:2409.05090 (2024). DOI: 10.48550/arXiv.2409.05090, https://arxiv.org/abs/2409.05090
8. Hajar Roshan-Shekalgourabi, Dawood Hassanzadeh-Lelekaami, “Cominimaxness of generalized local cohomology modules”, Bulletin of the Belgian Mathematical Society — Simon Stevin 32, No. 1 (2025), 81–92. DOI: 10.36045/j.bbms.240923, https://doi.org/10.36045/j.bbms.240923
9. Jingwen Shen, Xiaoyan Yang, “Cofiniteness of generalized local cohomology modules”, Journal of Algebra and Its Applications 25, No. 1 (2026), article 2550315. DOI: 10.1142/S0219498825503153, https://doi.org/10.1142/S0219498825503153
10. Nguyen Minh Tri, “Cofiniteness of generalized local cohomology modules over local rings”, Journal of Algebra and Its Applications 25, No. 4 (2026), article 2550368. DOI: 10.1142/S0219498825503682, https://doi.org/10.1142/S0219498825503682
11. Tran Tuan Nam, Nguyen Minh Tri, “Non-vanishing and cofiniteness of generalized local cohomology modules”, Periodica Mathematica Hungarica 88, No. 2 (2024), 461–474. DOI: 10.1007/s10998-023-00567-w, https://doi.org/10.1007/s10998-023-00567-w