Does finiteness of associated primes of $H^{i-1}_I(R/J)$ imply finiteness of associated primes of $H^i_I(J)$ for complete-intersection ideals $J$ in LC-finite regular rings?

For a commutative Noetherian ring $R$ , an ideal $J$ generated by an $R$ -regular sequence (a complete-intersection ideal) can be viewed as an $R$ -module, and its local cohomology modules $H^i_I(J)$ with respect to a larger ideal $I \supseteq J$ can be compared with the local cohomology $H^{i-1}_I(R/J)$ of the quotient ring. Hellus-type isomorphisms and the work of Hochster–Núñez-Betancourt, Lewis, and Hanyu show that, under structural hypotheses on $R/J$ - essentially that some quotient of $R/J$ by an $R/J$ -regular sequence from $I$ is locally almost factorial normal - finiteness of $\operatorname{Ass} H^{i-1}_I(R/J)$ is equivalent to finiteness of $\operatorname{Ass} H^i_I(J)$ . The open question, posed as Question 3 by Lewis (2022) and reposed as Question 1.2/1.3 by Hanyu (2025), asks whether this implication holds in full generality for LC-finite regular rings, without the almost-factorial hypotheses, in the critical case $\operatorname{depth}_I(R) = i-1$ . No counterexample is known: every known construction producing infinitely many associated primes on the $J$ -side produces them on the $R/J$ -side simultaneously.

Background

Let $R$ be a commutative Noetherian ring and $I$ an ideal. The $I$ -torsion functor $\Gamma_I(M) = \{m \in M : I^n m = 0 \text{ for some } n\}$ has right derived functors $H^i_I(M)$ , the local cohomology modules, which measure the failure of $M$ to be supported on the Zariski-closed subset defined by $I$ . An associated prime of an $R$ -module $M$ is a prime ideal of the form $\operatorname{Ann}_R(m)$ for some element $m \in M$ ; the set $\operatorname{Ass} M$ records the generic points of support components that are detected by single elements. Local cohomology modules are almost never finitely generated, so finiteness of $\operatorname{Ass} H^i_I(M)$ is a fundamental and delicate structural finiteness property: Huneke–Sharp (1993, prime characteristic, via finiteness of Bass numbers, the multiplicities occurring in a minimal injective resolution of a module) and Lyubeznik (1993, characteristic zero, via $D$ -module and $F$ -module techniques - a $D$ -module being a module over the ring of differential operators on $R$ ) proved that over a regular ring containing a field, $\operatorname{Ass} H^i_I(R)$ is finite for every ideal $I$ and every $i$ , and Bhatt–Blickle–Lyubeznik–Singh–Zhang (2013) established the same for smooth $\mathbb{Z}$ -algebras, covering unramified mixed characteristic. A regular ring is called LC-finite if $\operatorname{Ass} H^i_I(R)$ is finite for every ideal $I$ and all $i$ . Regularity is essential: Katzman (2002) exhibited a hypersurface over which a local cohomology module has infinitely many associated primes.

A regular sequence (or $R$ -sequence) in $R$ is a sequence $x_1,\dots,x_c$ such that each $x_j$ is a non-zerodivisor on $R/(x_1,\dots,x_{j-1})$ ; an ideal generated by such a sequence is a complete-intersection ideal. For such an ideal $J$ , viewed as an $R$ -module, one may compare the local cohomology of $J$ with that of the quotient ring $R/J$ . Hellus (2001) initiated this comparison by proving, in a principal-parameter context, an isomorphism of the form $H^{d+1}_I(R)/(x) \cong H^d_I(R/(x))$ relating associated primes across a mod-one-parameter reduction. The program gained geometric motivation from Hochster–Núñez-Betancourt (2017): over hypersurfaces and rings with finite $F$ -representation type (FFRT - a prime-characteristic condition meaning every finitely generated module appears as a direct summand of a finite direct sum of Frobenius iterates of the ring), finiteness of $\operatorname{Ass} H^{i+1}_I(J)$ forces the support of $H^i_I(R/J)$ to be Zariski-closed, so finiteness on the $J$ -side carries geometric information about the $R/J$ -side.

Lewis (2022) posed and partially answered the comparison question. Her Lemma 4.1 shows that when $\operatorname{depth}_I(R) > i-1$ , finiteness of $\operatorname{Ass} H^i_I(J)$ is equivalent to finiteness of $\operatorname{Ass} H^{i-1}_I(R/J)$ , and her generalized Hellus isomorphism (Theorem 4.1/Corollary 4.3) reduces the cases $\operatorname{depth}_I(R) < i-1$ to the range $\operatorname{depth}_I(R) \geq i-1$ , so the critical case is exactly $\operatorname{depth}_I(R) = i-1$ . Her Theorem 2.1 proves $\operatorname{Ass} H^2_I(J)$ finite when $R$ itself is locally almost factorial normal - a normal Krull domain in which, locally, some positive power of every height-one prime ideal is principal, equivalently a normal domain whose local divisor class groups (the groups of codimension-one Weil divisors modulo principal divisors) are torsion; since every regular ring is locally almost factorial and $\operatorname{Ass} H^1_I(R/J)$ is always finite, the instances $i \leq 2$ of the comparison question are settled outright and the open cases effectively begin at $i = 3$ . Her Theorem 4.2 treats $i = 3$ (when $\operatorname{Spec}(R/J)$ has disjoint irreducible components) and $i = 4$ (when $R/J$ is normal locally almost factorial). Her Section 3 answers negatively the separate question whether $\operatorname{Ass} H^i_I(J)$ is always finite: over a 7-dimensional polynomial ring she constructs $J$ generated by a length-2 $R$ -sequence and a 4-generated ideal $I$ with $R/J$ isomorphic to Katzman's hypersurface, so $\operatorname{Ass} H^2_I(R/J)$ is infinite - but her example has infinitely many associated primes on the quotient side too, so the implication question survives. She states it (her Question 3) as remaining open.

Hanyu (2025, arXiv preprint, revised 2026) extended the positive results: for the critical top instance, under the hypotheses that $\operatorname{Ass} H^{d+1}_I(R)$ is finite and that there exists an $R/J$ -regular sequence $x_1,\dots,x_{e-1} \in I$ , where $e = \operatorname{depth}_I(R/J) \geq 2$ , with $R/(J, x_1,\dots,x_{e-1})$ locally almost factorial normal, he proved that $\operatorname{Ass} H^{d+1}_I(J)$ is finite if and only if $\operatorname{Ass} H^d_I(R/J)$ is finite (Theorem 1.4/4.3), with companion results for $\operatorname{depth}_I(R/J) \leq 1$ (Theorem 3.3), unconditional equivalence for parameter-generated ideals in LC-finite regular local rings (Theorem 5.2), unramified mixed-characteristic cases (Proposition 5.4), and low dimensions via Marley's small-dimension finiteness results (Propositions 5.5–5.6, dimensions at most 6, and 7 with residue-field characteristic conditions). Neither Lewis nor Hanyu removes the almost-factorial-type structural hypotheses, and the general comparison question - Lewis's Question 3, reposed as Hanyu's Question 1.2/1.3 - remains open. One clarification: in the setting of Question 1.2 itself $R$ is assumed LC-finite, so finiteness of $\operatorname{Ass} H^{d+1}_I(R)$ is automatic; that hypothesis in Hanyu's Theorem 1.4 is load-bearing only because he states the theorem for general Noetherian rings. The genuinely open structural hypothesis to remove is the locally-almost-factorial one.

Problem Statement

Let $R$ be an LC-finite regular ring, let $J \subseteq I$ be ideals of $R$ such that $J$ is generated by an $R$ -regular sequence (a complete-intersection ideal, viewed here as an $R$ -module), and fix an integer $i \geq 1$ . Suppose $\operatorname{Ass} H^{i-1}_I(R/J)$ is finite. Must $\operatorname{Ass} H^i_I(J)$ be finite? By Lewis's depth lemma (Lemma 4.1 of her 2022 paper) the cases $\operatorname{depth}_I(R) > i-1$ are settled (the two finiteness conditions are equivalent there), and her generalized Hellus isomorphism (Theorem 4.1/Corollary 4.3) reduces the cases $\operatorname{depth}_I(R) < i-1$ to the range $\operatorname{depth}_I(R) \geq i-1$ , so the question is the critical case $\operatorname{depth}_I(R) = i-1$ (the depth-reduced form, Hanyu's Question 1.3, additionally assumes $J$ is generated by an $R$ -sequence of length at least $2$ ): does the implication hold without any structural hypothesis on the quotient, in particular without the existence of an $R/J$ -regular sequence $x_1,\dots,x_{e-1} \in I$ such that $R/(J,x_1,\dots,x_{e-1})$ is locally almost factorial normal, and without the lower-depth hypotheses (Cohen-Macaulay locally-a-domain or locally almost factorial $R/J$ ) used in the known partial results? A complete answer either proves the implication - or, as in all known positive results, the equivalence of the two finiteness conditions - for all such $R$ , $J$ , $I$ , $i$ , or exhibits a counterexample: an LC-finite regular ring $R$ , a complete-intersection ideal $J \subseteq I$ , and an index $i \geq 1$ with $\operatorname{Ass} H^{i-1}_I(R/J)$ finite but $\operatorname{Ass} H^i_I(J)$ infinite.

References

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