The remaining cases of Huneke's associated-primes conjecture in dimensions five and six: non-principal height-one ideals in factorial local rings and the second local cohomology module over six-dimensional regular rings

Huneke's 1990 conjecture predicts that over a Noetherian local ring $R$, every local cohomology module $H^i_I(M)$ of a finitely generated $R$-module has only finitely many associated primes. The conjecture is false in general but survives in restricted classes of rings. Dosea and Miranda-Neto proved it for four-dimensional factorial local rings, for five-dimensional factorial local rings when the support ideal is not a non-principal ideal of height one (or the module has dimension at most four), and — over six-dimensional regular local rings containing a field, for height-unmixed ideals satisfying two annihilator conditions — for every cohomological degree except $i = 2$. The two questions their paper (J. Algebra 669, 2025) leaves open are: (i) does the five-dimensional factorial statement survive, as full $I$-weak cofiniteness, for non-principal height-one ideals, and (ii) is $\operatorname{Ass}_R H^2_I(M)$ finite under the hypotheses of the six-dimensional theorem? Both questions are open as of August 2026: all known counterexamples (Katzman 2002; Singh–Swanson 2004; Ma 2026 for ramified mixed-characteristic regular rings) violate the hypotheses at issue, and no later work resolves either question.

Background

For a commutative Noetherian ring $R$, an ideal $I \subset R$, and a finitely generated $R$-module $M$, the $i$-th local cohomology module of $M$ with support in $I$ is $H^i_I(M) = \varinjlim_n \operatorname{Ext}^i_R(R/I^n, M)$, where $\operatorname{Ext}$ denotes the derived functors of $\operatorname{Hom}$. Introduced by Grothendieck, these modules control cohomology with support conditions in algebraic geometry, but they are almost never finitely generated, so one studies weaker finiteness invariants. A basic one is the set of associated primes $\operatorname{Ass}_R N$ of a module $N$ — the prime ideals occurring as annihilators of elements of $N$; its finiteness forces the support of $N$ to be Zariski-closed. In his 1990 problem list, Craig Huneke conjectured that over a Noetherian local ring $R$ the set $\operatorname{Ass}_R H^i_I(M)$ is finite for every ideal $I$, every finitely generated module $M$, and every $i \ge 0$.

The conjecture is false in general, and the surviving territory is charted by dimension and by hypotheses on the ring. The height $\operatorname{ht}\mathfrak{p}$ of a prime $\mathfrak{p}$ is $\dim R_{\mathfrak{p}}$, and $\operatorname{ht} I$ is the least height of a prime containing $I$; a minimal prime divisor of $I$ is a prime minimal among those containing $I$. Marley (2001) established finiteness for local rings of dimension at most three. Katzman (2002) constructed the first counterexample: over a specific five-dimensional local hypersurface domain $T$ over a field — necessarily not a unique factorization domain, since the offending ideal $(u,v)$ is a height-one prime that is not principal — the module $H^2_{(u,v)}(T)$ has infinitely many associated primes. Singh and Swanson (2004) gave further counterexamples: five-dimensional local domains whose offending ideals are again height-one and non-principal, and unique factorization domains of dimension 8 (hypersurfaces with rational singularities — a mild-singularity condition from birational geometry — in characteristic zero, and $F$-regular — a strong condition from tight closure theory — in prime characteristic) with infinitely many associated primes in degree three. On the positive side, Lyubeznik (1993) proved that if $R$ is a regular local ring containing a field (i.e., after completion, a formal power series ring over a field), then $H^i_I(R)$ has finitely many associated primes for every $I$ and $i$ — but only for the module $R$ itself, not for arbitrary finitely generated $M$.

For arbitrary finite modules, Dosea and Miranda-Neto (2025) pushed the frontier forward. Their central notion is weak cofiniteness: a module $H$ is weakly Laskerian if $\operatorname{Ass}_R(H/U)$ is finite for every submodule $U \subseteq H$, and $I$-weakly cofinite if moreover $\operatorname{Supp}_R H \subseteq V(I)$, the set of primes containing $I$, and every $\operatorname{Ext}^i_R(R/I, H)$ is weakly Laskerian; $I$-weak cofiniteness implies that $\operatorname{Ass}_R H$ is finite. They prove: (a) if $R$ is a four-dimensional factorial local ring (a local unique factorization domain), then $H^i_I(M)$ is $I$-weakly cofinite for every ideal $I$, every finite $M$, and every $i$; (b) if $R$ is a five-dimensional factorial local ring and $\operatorname{ann}_R M$ is contained in no minimal prime divisor of $I$ (equivalently, for radical $I$, $\operatorname{ann}_R M$ contains an $R/I$-regular element), the same conclusion holds provided $\operatorname{ht} I \ne 1$ or $I$ is principal, and — in a companion theorem — the weaker conclusion that $\operatorname{Ass}_R H^i_I(M)$ is finite for all $i$ holds for height-one non-principal $I$ whenever $\dim M \le 4$; and (c) if $(R, \mathfrak{m})$ is a six-dimensional regular local ring containing a field, $I$ is height-unmixed (all associated prime divisors of $I$, i.e. the primes in $\operatorname{Ass}_R R/I$, have the same height), $M$ is finite with $\operatorname{Ann}_R M$ contained in no minimal prime divisor of $I$ and $\operatorname{Ann}_R M \not\subset \mathfrak{m}^2$, then $\operatorname{Ass}_R H^i_I(M)$ is finite for all $i \ne 2$. Result (c) rests on the Hartshorne–Lichtenbaum vanishing theorem, which states that over a $d$-dimensional complete local domain the top local cohomology module $H^d_I(R)$ vanishes whenever $\dim R/I \ge 1$; the authors state that (c) is the first result toward the six-dimensional case of Huneke's conjecture.

Both remaining gaps are sharp rather than cosmetic. In a factorial domain every height-one prime is principal, yet a height-one ideal need not be (e.g., $(x^2, xy)$ in $k[[x,y]]$); the height-one non-principal case in (b) matches the ideal-theoretic configuration of the Katzman and Singh–Swanson five-dimensional counterexamples, but those examples also fail (b)'s annihilator hypothesis, since their module is the ring itself and hence $\operatorname{Ann}_R M = 0$ is contained in every minimal prime divisor of $I$. In (c), the single remaining degree $i = 2$ is genuinely delicate: viewed over its ambient six-dimensional regular ring (using independence of local cohomology from the base ring), Katzman's example shows that $\operatorname{Ass}_R H^2_I(M)$ can be infinite in dimension six over a regular ring containing a field, but with $\operatorname{Ann}_R M$ contained both in a minimal prime divisor of $I$ and in $\mathfrak{m}^2$ — precisely the two hypotheses that the six-dimensional theorem excludes, so the open question is whether finiteness returns once they are imposed.

Problem Statement

Dosea and Miranda-Neto leave two linked questions open (Questions 4.1 and 5.3 of their paper).

Question 4.1. Let $R$ be a five-dimensional factorial local ring (a Noetherian local unique factorization domain), let $I \subset R$ be a non-principal ideal of height one, and let $M$ be a finitely generated $R$-module whose annihilator is not contained in any minimal prime divisor of $I$. Is $H^i_I(M)$ $I$-weakly cofinite for every $i \ge 0$ — in particular, is $\operatorname{Ass}_R H^i_I(M)$ finite for every $i \ge 0$?

Question 5.3. Let $(R, \mathfrak{m})$ be a six-dimensional regular local ring containing a field, let $I \subset R$ be a height-unmixed ideal, and let $M$ be a finitely generated $R$-module such that $\operatorname{Ann}_R M$ is contained in no minimal prime divisor of $I$ and $\operatorname{Ann}_R M \not\subset \mathfrak{m}^2$. Is $\operatorname{Ass}_R H^2_I(M)$ finite?

References

1. André Dosea, Cleto B. Miranda-Neto, 'On the factorial case of Huneke's conjecture for local cohomology modules', preprint arXiv:2405.06862 (2024). URL: https://arxiv.org/abs/2405.06862
2. 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. URL: https://doi.org/10.1016/j.jalgebra.2025.02.008
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, Boston, 1992, pp. 93–108. DOI (Taylor & Francis reprint): 10.1201/9781003420187-10. URL: https://doi.org/10.1201/9781003420187-10
4. Thomas Marley, 'The associated primes of local cohomology modules over rings of small dimension', Manuscripta Mathematica 104 (2001), 519–525. DOI: 10.1007/s002290170024. URL: https://doi.org/10.1007/s002290170024
5. Gennady Lyubeznik, 'Finiteness properties of local cohomology modules (an application of D-modules to commutative algebra)', Inventiones Mathematicae 113 (1993), 41–55. DOI: 10.1007/BF01244301. URL: https://doi.org/10.1007/BF01244301
6. Mordechai Katzman, 'An example of an infinite set of associated primes of a local cohomology module', Journal of Algebra 252 (2002), no. 1, 161–166. DOI: 10.1016/S0021-8693(02)00032-7; arXiv:math/0209353. URL: https://arxiv.org/abs/math/0209353
7. Anurag K. Singh, Irena Swanson, 'Associated primes of local cohomology modules and of Frobenius powers', International Mathematics Research Notices 2004, no. 33, 1703–1733. DOI: 10.1155/S1073792804133424; arXiv:math/0406357. URL: https://arxiv.org/abs/math/0406357
8. Linquan Ma, 'Infinitely many associated primes of local cohomology modules of ramified regular local rings', preprint arXiv:2604.10964 (2026). URL: https://arxiv.org/abs/2604.10964
9. Takumi Murayama, 'Finiteness of associated primes for local cohomology modules of excellent locally unramified regular rings of finite Krull dimension', preprint arXiv:2506.17875 (2025). URL: https://arxiv.org/abs/2506.17875
10. Ryotaro Hanyu, 'Finiteness of the set of associated primes for local cohomology modules of ideals via properties of almost factorial rings', preprint arXiv:2506.17586 (2025; v3 2026). URL: https://arxiv.org/abs/2506.17586
11. Tony J. Puthenpurakal, 'Associate primes of local cohomology modules over certain quotients of regular rings', preprint arXiv:2503.20539 (2025). URL: https://arxiv.org/abs/2503.20539
12. Maryam Jahangiri, 'Koszul cohomology of Čech cohomology modules', preprint arXiv:2608.20171 (2026). URL: https://arxiv.org/abs/2608.20171
13. Alireza Vahidi, Ahmad Khaksari, Mohammad Shirazipour, 'Cofiniteness and finiteness of associated prime ideals of generalized local cohomology modules', preprint arXiv:2409.05090 (2024). URL: https://arxiv.org/abs/2409.05090
14. Alberto F. Boix, Majid Eghbali, 'On Hellus–Lyubeznik–Yildirim's conjecture of local cohomology modules', preprint arXiv:2606.07518 (2026). URL: https://arxiv.org/abs/2606.07518
15. 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). URL: https://arxiv.org/abs/2604.20150