Cofiniteness and associated-prime finiteness of generalized local cohomology modules: the open boundary beyond dimension two

For a commutative Noetherian ring $R$ , an ideal $\mathfrak a$ , and finitely generated modules $M,N$ , the generalized local cohomology modules are $\mathrm{H}^i_{\mathfrak a}(M,N)=\varinjlim_n\operatorname{Ext}^i_R(M/\mathfrak a^nM,N)$ , recovering ordinary local cohomology at $M=R$ . The source preprint (arXiv:2409.05090) proves that $\mathrm{H}^i_{\mathfrak a}(M,N)$ is $(\mathrm{FD}_{<n},\mathfrak a)$ -cofinite with $\operatorname{Ass}_{\ge n}$ finite whenever $\dim_R M\le n+2$ or $\dim_R N\le n+2$ , that $\operatorname{Ass}_R(\mathrm{H}^i_{\mathfrak a}(M,N))$ is finite over semi-local $R$ when $\dim_R M\le 3$ or $\dim_R N\le 3$ , and Hom-finiteness/cofiniteness equivalence theorems with parity propagation for $\dim R/\mathfrak a\le 2$ . The unrestricted questions (cofiniteness and Ass-finiteness for arbitrary $M,N$ ) are already false via $M=R$ (Hartshorne's and Singh's counterexamples), so the open problem is the boundary itself: extend or optimally stop the dimension and ring hypotheses — associated-prime finiteness at the dimension-3 boundary without semi-locality, the $n+3$ rung of the $(\mathrm{FD}_{<n},\mathfrak a)$ -ladder, and the extension of the equivalence and parity theorems to $\dim R/\mathfrak a\ge 3$ . These questions remain open.

Background

Let $R$ be a commutative Noetherian ring with identity, $\mathfrak a$ an ideal, and $X$ an $R$ -module. The $i$ -th local cohomology module of $X$ with support in $V(\mathfrak a)$ is $\mathrm{H}^i_{\mathfrak a}(X)=\varinjlim_n\operatorname{Ext}^i_R(R/\mathfrak a^n,X)$ , the right-derived functor of $\mathfrak a$ -torsion and a fundamental tool of commutative algebra and algebraic geometry. Generalized local cohomology, studied since the 1970s, replaces the first argument: for $R$ -modules $M$ and $N$ , set $\mathrm{H}^i_{\mathfrak a}(M,N)=\varinjlim_n\operatorname{Ext}^i_R(M/\mathfrak a^nM,N)$ , which recovers ordinary local cohomology when $M=R$ ; typically $M$ and $N$ are finitely generated, while several results also allow an arbitrary module in the second slot.

A finiteness conjecture from the 1960s asked whether $\operatorname{Hom}_R(R/\mathfrak a,\mathrm{H}^i_{\mathfrak a}(N))$ is always finitely generated; Hartshorne disproved it and in the same 1970 paper introduced the central notion: an $\mathfrak a$ -torsion module $X$ (every element is killed by a power of $\mathfrak a$ ) is $\mathfrak a$ -cofinite when $\operatorname{Ext}^i_R(R/\mathfrak a,X)$ is finitely generated for all $i$ . Cofinite modules have finite sets of associated primes — the prime ideals occurring as annihilators $\operatorname{Ann}_R(x)$ of elements — so cofiniteness refines the classical problem, popularized in the early 1990s, of whether $\operatorname{Ass}_R(\mathrm{H}^i_{\mathfrak a}(N))$ is finite. Both hopes fail without hypotheses. Hartshorne's example, over the local ring $R_P=k[[u,v]][x,y]_P$ with $\mathfrak a=(x,y)R_P$ and $N=R_P/(ux+vy)R_P$ (so $\dim N=3$ and $\dim R/\mathfrak a=2$ ), has $\mathrm{H}^2_{\mathfrak a}(N)$ with infinite-dimensional socle $\operatorname{Hom}_R(R/\mathfrak m,-)$ , so it is not $\mathfrak a$ -cofinite; Singh's example, over the six-dimensional hypersurface $\mathbb Z[U,V,W,X,Y,Z]/(UX+VY+WZ)$ with $\mathfrak a=(x,y,z)$ (here $\dim R/\mathfrak a=4$ ), has $\mathrm{H}^3_{\mathfrak a}(R)$ containing $p$ -torsion for every prime integer $p$ , hence infinitely many associated primes. Positive results delineate a boundary: $\mathrm{H}^i_{\mathfrak a}(N)$ is $\mathfrak a$ -cofinite for all $i$ when $\dim R\le 2$ or $\dim R/\mathfrak a\le 1$ (Melkersson), and Cuong–Goto–Hoang proved the generalized version: $\mathrm{H}^i_{\mathfrak a}(M,N)$ is $\mathfrak a$ -cofinite for all $i$ whenever $\dim_R M\le 2$ or $\dim_R N\le 2$ .

The source preprint (Vahidi–Khaksari–Shirazipour, arXiv:2409.05090) pushes this boundary within a module-class framework. An $R$ -module is $\mathrm{FD}_{<n}$ if it has a finitely generated submodule whose quotient has Krull dimension (maximal length of chains of prime ideals) less than $n$ ; $X$ is $(\mathrm{FD}_{<n},\mathfrak a)$ -cofinite when it is $\mathfrak a$ -torsion and every $\operatorname{Ext}^i_R(R/\mathfrak a,X)$ is $\mathrm{FD}_{<n}$ . The case $n=0$ is ordinary $\mathfrak a$ -cofiniteness, and $(\mathrm{FD}_{<n},\mathfrak a)$ -cofiniteness forces the truncation $\operatorname{Ass}_{\ge n}=\{\mathfrak p\in\operatorname{Ass}_R(X):\dim R/\mathfrak p\ge n\}$ to be finite. The paper proves: (i) $\mathrm{H}^i_{\mathfrak a}(M,N)$ is $(\mathrm{FD}_{<n},\mathfrak a)$ -cofinite and $\operatorname{Ass}_R(\mathrm{H}^i_{\mathfrak a}(M,N))_{\ge n}$ is finite for all $i$ whenever $\dim_R M\le n+2$ or $\dim_R N\le n+2$ ; (ii) over a semi-local ring $R$ (a ring with only finitely many maximal ideals), $\operatorname{Ass}_R(\mathrm{H}^i_{\mathfrak a}(M,N))$ is finite for all $i$ when $\dim_R M\le 3$ or $\dim_R N\le 3$ ; and (iii) for $\dim R/\mathfrak a\le 2$ , under finiteness side conditions on $\operatorname{Ext}^i_R(M/\mathfrak a M,X)$ and on the maximal ideals in the support, finiteness of $\operatorname{Hom}_R(R/\mathfrak a,\mathrm{H}^j_{\mathfrak a}(M,X))$ in low degrees is equivalent to cofiniteness in low degrees, and cofiniteness propagates by parity (all even, or all odd, degrees cofinite implies all degrees cofinite).

What remains open is the boundary itself. The paper's Questions 1.5–1.8, taken literally for arbitrary $M,N$ , are already false via $M=R$ and the counterexamples above; the genuine problem is to determine how far these dimension and ring hypotheses can be weakened and where Hartshorne–Singh-type failure begins.

Problem Statement

Throughout, $R$ is a commutative Noetherian ring, $\mathfrak a$ an ideal, and $M,N$ finitely generated $R$ -modules; $\mathrm{H}^i_{\mathfrak a}(M,N)=\varinjlim_n\operatorname{Ext}^i_R(M/\mathfrak a^nM,N)$ . Because the unrestricted questions 'is $\mathrm{H}^i_{\mathfrak a}(M,N)$ $\mathfrak a$ -cofinite for all $i$ ' and 'is $\operatorname{Ass}_R(\mathrm{H}^i_{\mathfrak a}(M,N))$ finite for all $i$ ' already fail for $M=R$ (Hartshorne 1970; Singh 2000), the open problem is to determine the boundary between the cofinite/finite-Ass regimes and the counterexample regimes for generalized local cohomology, along the following concrete axes:

(1) Associated primes at the dimension-3 boundary. Is $\operatorname{Ass}_R(\mathrm{H}^i_{\mathfrak a}(M,N))$ finite for all $i\ge 0$ whenever $\dim_R M\le 3$ or $\dim_R N\le 3$ , with no semi-locality hypothesis on $R$ (the known results assume $R$ local or semi-local)? (For $\dim_R M,\dim_R N\ge 4$ , the case $M=R$ already contains the classical associated-primes problem for ordinary local cohomology.)

(2) The $(\mathrm{FD}_{<n},\mathfrak a)$ -ladder. For each $n\ge 1$ , is $\mathrm{H}^i_{\mathfrak a}(M,N)$ $(\mathrm{FD}_{<n},\mathfrak a)$ -cofinite, with $\operatorname{Ass}_R(\mathrm{H}^i_{\mathfrak a}(M,N))_{\ge n}$ finite, for all $i$ whenever $\dim_R M\le n+3$ or $\dim_R N\le n+3$ (the source proves $n+2$ ; the $n=0$ rung at dimension 3 is refuted by Hartshorne's example with $M=R$ , while for $n\ge 1$ — e.g. $(\mathrm{FD}_{<1},\mathfrak a)$ -cofiniteness at $\dim\le 4$ over arbitrary Noetherian rings — no answer is known)?

(3) Beyond $\dim R/\mathfrak a\le 2$ . Do the equivalence theorems (finiteness of $\operatorname{Hom}_R(R/\mathfrak a,-)$ in low degrees is equivalent to cofiniteness in low degrees, under the analogous $\operatorname{Ext}$ -finiteness and maximal-ideal-finiteness side conditions) and the parity-propagation theorems of the source admit analogues for ideals with $\dim R/\mathfrak a\ge 3$ , or does cofiniteness break in that range even under those side conditions?

These three items are facets of one objective: locate the exact boundary, in module dimension, ideal dimension, and ring hypotheses, between the regimes where generalized local cohomology modules are cofinite or have finite associated primes and the regimes where Hartshorne–Singh-type failure occurs.

References

Alireza Vahidi, Ahmad Khaksari, and Mohammad Shirazipour, 'Cofiniteness and finiteness of associated prime ideals of generalized local cohomology modules' (2024), arXiv:2409.05090, https://arxiv.org/abs/2409.05090.
Robin Hartshorne, 'Affine duality and cofiniteness', Inventiones Mathematicae 9 (1970), 145–164, DOI 10.1007/BF01404554, https://doi.org/10.1007/BF01404554.
Anurag K. Singh, 'p-torsion elements in local cohomology modules', Mathematical Research Letters 7 (2000), no. 2–3, 165–176, DOI 10.4310/MRL.2000.v7.n2.a3, arXiv:math/0406354, https://arxiv.org/abs/math/0406354.
Leif Melkersson, 'Modules cofinite with respect to an ideal', Journal of Algebra 285 (2005), no. 2, 649–668, DOI 10.1016/j.jalgebra.2004.08.037, https://doi.org/10.1016/j.jalgebra.2004.08.037.
Nguyen Tu Cuong, Shiro Goto, and Nguyen Van Hoang, 'On the cofiniteness of generalized local cohomology modules', Kyoto Journal of Mathematics 55 (2015), no. 1, 169–185, DOI 10.1215/21562261-2848151, https://doi.org/10.1215/21562261-2848151.
Kamal Bahmanpour, Reza Naghipour, and Monireh Sedghi, 'Cofiniteness with respect to ideals of small dimensions', Algebras and Representation Theory 18 (2015), no. 2, 369–379, DOI 10.1007/s10468-014-9498-3, https://doi.org/10.1007/s10468-014-9498-3.
Alireza Vahidi, Ahmad Khaksari, and Mohammad Shirazipour, 'Cofinite modules and cofiniteness of local cohomology modules', Revista de la Unión Matemática Argentina 67 (2024), no. 1, 317–325, DOI 10.33044/revuma.3535, https://doi.org/10.33044/revuma.3535.
Kamal Bahmanpour, 'Cofiniteness over Noetherian complete local rings' (2019), arXiv:1901.06668, https://arxiv.org/abs/1901.06668.
Kamal Bahmanpour, 'On the second problem of Hartshorne on cofiniteness', Ricerche di Matematica 74 (2025), 1541–1561, DOI 10.1007/s11587-024-00908-y, https://doi.org/10.1007/s11587-024-00908-y.
Tran Tuan Nam and Nguyen Minh Tri, 'Non-vanishing and cofiniteness of generalized local cohomology modules', Periodica Mathematica Hungarica 88 (2024), 461–474, DOI 10.1007/s10998-023-00567-w, https://doi.org/10.1007/s10998-023-00567-w.
Jingwen Shen and Xiaoyan Yang, 'Cofiniteness of generalized local cohomology modules', Journal of Algebra and Its Applications 25 (2026), no. 1, Article 2550315, DOI 10.1142/S0219498825503153, https://doi.org/10.1142/S0219498825503153.
Nguyen Minh Tri, 'Cofiniteness of generalized local cohomology modules over local rings', Journal of Algebra and Its Applications 25 (2026), no. 4, Article 2550368, DOI 10.1142/S0219498825503682, https://doi.org/10.1142/S0219498825503682.