Finiteness of associated primes of local cohomology in dimension four: the Robbins hypersurface family and the non-local three-dimensional maximal-ideal question

For a Noetherian ring $R$, ideal $I$, and finitely generated module $M$, the local cohomology modules $H^i_I(M)$ need not be finitely generated, and Craig Huneke asked, in a problem list from the 1990 Sundance conference published in 1992, whether their sets of associated primes are at least always finite when $R$ is local. The answer is negative in general, and the exact boundary is a recognized open problem: finiteness holds for local rings of dimension at most three (Marley), while the smallest known local counterexamples have dimension five (Singh-Swanson), leaving dimension four as the decisive undecided local case. Robbins (2012) proved non-local finiteness criteria via ideal transforms and $S_2$-ification and reduced the four-dimensional local case to an explicit family of hypersurface local rings $S=V[[X_1,X_2,X_3,Y]]/(f)$ with $V$ a complete discrete valuation ring, and $S=V[[X_1,X_2,X_3,X_4,Y]]/(f)$ with $V$ a field, where $f$ is monic in $Y$ with constant term divisible by $X_1$: finiteness of $\operatorname{Ass}_S H^2_{(X_1,Y)}(G)$ for all faithful finite $S$-modules $G$ of pure dimension four is equivalent to the four-dimensional local case of Huneke's question. This problem asks for a decision on that family, by proof or explicit counterexample, together with the companion question the same paper left open: whether, for three-dimensional rings that are not local and whose quotients admit $S_2$-ifications, the set $\operatorname{Ass}_R H^2_I(M)$ can contain infinitely many distinct maximal ideals. The factorial subcase of the four-dimensional question was settled positively in 2025; the general case and the non-local three-dimensional question remain open.

Background

Let $R$ be a commutative Noetherian ring, $I$ an ideal, and $M$ a finitely generated $R$-module. The local cohomology modules $H^i_I(M)$ are the right derived functors of the functor sending a module to its submodule of elements killed by some power of $I$; concretely $H^i_I(M)=\varinjlim_n \operatorname{Ext}^i_R(R/I^n,M)$. They connect the algebra of $R$ with the geometry of the complement of the closed set $V(I)$ in $\operatorname{Spec}R$, and although they are generally not finitely generated, their finiteness invariants control vanishing, connectedness, and cofiniteness phenomena. The most basic invariant is the set of associated primes: a prime $\mathfrak p$ is associated to a module $N$ when it is the annihilator of some element of $N$, and $\operatorname{Ass}_R N$ records the primes at which $N$ has elements. In his 1992 problem list, Craig Huneke asked whether $\operatorname{Ass}_R H^i_I(M)$ is finite for all $i$ when $R$ is Noetherian local.

The answer is now known to be negative in general, but the exact boundary is unresolved. On the positive side, Huneke and Sharp proved finiteness for regular rings (rings locally isomorphic to power series rings over a field or discrete valuation ring) of positive characteristic, Lyubeznik extended it to regular rings containing a field in any characteristic and to unramified regular local rings of mixed characteristic (mixed characteristic meaning that the residue field and the fraction field have different characteristics, as for $p$-adic rings; unramified meaning $p$ generates the maximal ideal minimally), and Marley proved that a Noetherian local ring of dimension at most three always has $\operatorname{Ass}_R H^i_I(M)$ finite; Marley also settled local dimension four under Serre's condition $(R_2)$ (regularity in codimension at most two) with open regular locus, and dimension five for torsion-free modules. On the negative side, Singh constructed the first example with infinitely many associated primes (a six-dimensional non-local hypersurface, that is, a quotient of a polynomial or power series ring by a single equation, over $\mathbb Z$), Katzman gave another, and Singh and Swanson produced normal hypersurface domains, including unique factorization domains of characteristic zero with rational singularities (a mild birational-geometric singularity condition) and $F$-regular unique factorization domains of positive characteristic ($F$-regularity being the strongest regularity condition of tight closure theory), with a local cohomology module having infinitely many associated primes. Their constructions exploit the Frobenius map $r\mapsto r^p$ of positive characteristic together with ideals of height one. Two of their facts delimit the boundary: localizing their example at the origin gives a five-dimensional local counterexample, while their four-dimensional example is non-local, and they state explicitly that no example is known in local dimension four. Dimension four is thus the precise undecided local case. (For regular rings, Lyubeznik's finiteness question remained open only in ramified mixed characteristic until Ma's 2026 twelve-dimensional ramified regular local counterexamples, which do not affect dimension four.)

Robbins's 2012 paper attacks the non-local and low-dimensional cases with an embedding technique. Its engine is the ideal transform: for elements $x,y$ that are non-zerodivisors on $M$, the transform $D_{(x,y)}(M)$ is the submodule of the localization $M_{xy}$ consisting of classes $u$ for which $x^Nu$ and $y^Nu$ lie in $M$ for all large $N$; such a transform has depth at least two on the ideal $(x,y)$, so mapping $M$ into modules of $I$-depth at least two with kernel of dimension at most one can force finiteness of $\operatorname{Ass}_R H^2_I(M)$. The needed transforms are finite exactly when the ring has enough integral structure: a module satisfies Serre's condition $S_2$ when its depth at each prime of its support is at least the minimum of $2$ and that prime's height, and an $S_2$-ification of a ring $A$ is a finite birational extension of $A$ (an overring inside the total quotient ring agreeing with $A$ at the minimal primes) satisfying $S_2$; when the quotients $R/\mathfrak p$ admit such extensions, the ideal transforms $D_{(x,y)}(R)$ are finitely generated. The paper proves: if $D_{(x,y)}(R)$ is finitely generated for every height-two pair of non-zerodivisors $x,y$, and $\operatorname{ht}(I(R/\mathfrak p))\ge 2$ for every $\mathfrak p\in\operatorname{Ass}_R M$ with $\operatorname{Ass}_R M\subseteq\operatorname{Ass}_R R$, then $\operatorname{Ass}_R H^2_I(M)$ is finite, with no locality assumption on $R$; and the same hypotheses with $\dim R=3$ give $\operatorname{Ass}_R H^i_I(M)$ finite for every $i$.

The paper also contains a reduction of the four-dimensional local case. Using that formal power series rings over a field or a complete discrete valuation ring are unique factorization domains, together with Weierstrass preparation (a series that is monic in a variable $Y$ equals a unit times a monic polynomial in $Y$), it funnels the whole question to an explicit family: the finiteness of $\operatorname{Ass}_R H^i_I(M)$ for all four-dimensional local rings, all ideals, all modules, and all degrees is equivalent to the finiteness of $\operatorname{Ass}_S H^2_{(X_1,Y)}(G)$ for all faithful finite modules $G$ of pure dimension four over the hypersurfaces $S=V[[X_1,X_2,X_3,Y]]/(f)$ with $V$ a complete discrete valuation ring (the completion of a principal ideal domain with exactly one nonzero prime ideal, such as the $p$-adic integers $\mathbb Z_p$ or a formal power series ring $k[[t]]$) and $S=V[[X_1,X_2,X_3,X_4,Y]]/(f)$ with $V$ a field, where $f$ is monic in $Y$ with constant term divisible by $X_1$. No general proof of finiteness for this family is known, and the paper leaves a second gap in the non-local three-dimensional setting: even when the quotients $R/\mathfrak p$ admit $S_2$-ifications, the possibility that infinitely many distinct maximal ideals appear in $\operatorname{Ass}_R H^2_I(M)$ after localization is not ruled out. Later work has mapped only parts of this territory: the Faltings-type local-global theorems control associated primes of positive dimension but not the maximal-ideal tail; Hochster and Núñez-Betancourt obtain finiteness for rings of finite $F$-representation type in positive characteristic (rings with only finitely many indecomposable summands of iterates of the Frobenius); Quinlan-Gallego's 2024 survey still cites Marley and Robbins as the state of the art in small dimensions; and Dosea and Miranda-Neto settled the four-dimensional case in 2025 only for factorial local rings, while Robbins's family contains non-factorial hypersurfaces.

Problem Statement

The common objective of the two components below is to close the remaining low-dimensional boundary of the finiteness of associated primes of local cohomology that the ideal-transform and $S_2$-ification method left open, between Marley's positive results and the known counterexamples.

Component (A). Decide whether the following statement is true: for every coefficient ring $V$ that is either a complete discrete valuation ring or a field, every power series $f\in V[[X_1,\dots,X_r,Y]]$ (where $r=3$ if $V$ is a complete discrete valuation ring and $r=4$ if $V$ is a field) that is monic in $Y$ and whose constant term is divisible by $X_1$, and every finitely generated module $G$ over $S=V[[X_1,\dots,X_r,Y]]/(f)$ that is faithful ($\operatorname{Ann}_S G=0$) and of pure dimension four (every associated prime of $G$ is minimal of height zero in $S$), the set $\operatorname{Ass}_S H^2_{(X_1,Y)}(G)$ is finite. By the reduction established in the source this is equivalent to the four-dimensional local case of Huneke's finiteness question, namely that $\operatorname{Ass}_R H^i_I(M)$ is finite for every Noetherian local ring $R$ of Krull dimension four, every ideal $I$, every finitely generated $R$-module $M$, and every $i\ge 0$; since every member of the family is itself a four-dimensional local ring, the two formulations decide each other. An answer of 'false' must be given as an explicit counterexample: a specific $V$, $f$, and $G$ as above with $\operatorname{Ass}_S H^2_{(X_1,Y)}(G)$ infinite, which would be the first counterexample in local dimension four.

Component (B). Decide whether infinitely many distinct maximal ideals can occur among the associated primes of the second local cohomology module in the non-local three-dimensional setting of the same method: either prove that, for every three-dimensional Noetherian ring $R$ that is not local and in which the quotients $R/\mathfrak p$ over the relevant associated primes admit $S_2$-ifications (in the sense of the source, under which the ideal transforms $D_{(x,y)}(R)$ required by the method are finitely generated), every ideal $I$ of $R$, and every finitely generated $R$-module $M$ with $\operatorname{ht}(I(R/\mathfrak p))\ge 2$ for every $\mathfrak p\in\operatorname{Ass}_R M$, the set $\operatorname{Ass}_R H^2_I(M)$ is finite; or construct an explicit such $R$, $I$, and $M$ for which $\operatorname{Ass}_R H^2_I(M)$ contains infinitely many distinct maximal ideals.

References

1. Craig Huneke, 'Problems on Local Cohomology', in Free Resolutions in Commutative Algebra and Algebraic Geometry (Sundance, 1990), Research Notes in Mathematics 2, Jones and Bartlett, Boston, 1992, pp. 93-108; chapter DOI: 10.1201/9781003420187-10, https://doi.org/10.1201/9781003420187-10
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12. Eamon Quinlan-Gallego, 'A Formalism of F-modules for Rings with Complete Local Finite F-Representation Type', International Mathematics Research Notices 2024(12) (2024), 9562-9587. DOI: 10.1093/imrn/rnae054, arXiv: 2303.02705, https://doi.org/10.1093/imrn/rnae054
13. André Dosea and 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 (preprint arXiv:2405.06862, posted under the earlier title 'On the factorial case of Huneke's conjecture for local cohomology modules'), https://doi.org/10.1016/j.jalgebra.2025.02.008
14. Takumi Murayama, 'Finiteness of associated primes for local cohomology modules of excellent locally unramified regular rings of finite Krull dimension', preprint (2025). arXiv: 2506.17875, https://arxiv.org/abs/2506.17875
15. Linquan Ma, 'Infinitely many associated primes of local cohomology modules of ramified regular local rings', preprint (2026). arXiv: 2604.10964, https://arxiv.org/abs/2604.10964