Descent of associated-prime and Bass-number finiteness of local cohomology along cyclically pure ring extensions

For a Noetherian ring $R$, ideal $I$, and integer $i$, the local cohomology module $H^i_I(R)$ need not have finitely many associated primes, but it does when $R$ is regular in equal characteristic, an unramified regular local ring of mixed characteristic, or a smooth $\mathbb{Z}$-algebra, and Lyubeznik conjectured this finiteness for all regular rings. Bhattacharyya (2015) studied when such finiteness — for local cohomology and, more generally, for Lyubeznik functors — descends from an extension ring to a base ring along pure and cyclically pure extensions, proving descent for pure extensions and for cyclically pure extensions whose local base is analytically unramified, with Bass-number analogs under Cohen–Macaulay and module-finite hypotheses on the extension. The remaining open case is whether the finiteness of associated primes (and of Bass numbers) of Lyubeznik functors descends along an arbitrary cyclically pure extension without the analytic-unramifiedness hypothesis on the base, and without the Cohen–Macaulay and module-finite hypotheses on the extension. The adjacent case — Lyubeznik's finiteness question for ramified regular local rings at primes containing $p$ — was refuted in 2026 by Linquan Ma, who constructed ramified regular local rings of mixed characteristic whose local cohomology modules have infinitely many associated primes (all containing $p$) and infinite Bass numbers; the conditional descent question below survives and is unaffected.

Background

For a commutative Noetherian ring $R$, an ideal $I\subset R$, and an integer $i\ge 0$, the $i$-th local cohomology module $H^i_I(R)$ is the $i$-th right derived functor of the torsion functor sending an $R$-module to the submodule of elements annihilated by some power of $I$; it is the algebraic counterpart of cohomology with support in the closed subset $V(I)$ of $\operatorname{Spec}R$. An associated prime of an $R$-module $M$ is a prime ideal of the form $\operatorname{Ann}(x)$ for some $x\in M$; finiteness of $\operatorname{Ass}_RM$ is a basic measure of module-theoretic tameness. Although $H^i_I(R)$ is almost never finitely generated as an $R$-module, Huneke asked in 1992 whether its set of associated primes is always finite, and Katzman (2002) gave examples over hypersurface rings where it is infinite. For regular rings the expectation stayed positive: Lyubeznik conjectured in 1993 that for every regular ring $R$ and ideal $I$, each $H^i_I(R)$ has only finitely many associated primes, and asked the analogous question for Bass numbers, the invariants $\mu_j(\mathfrak{p},M)=\dim_{\kappa(\mathfrak{p})}\operatorname{Ext}^j_{R_\mathfrak{p}}(\kappa(\mathfrak{p}),M_\mathfrak{p})$ that measure the injective resolutions of a module. The conjecture was proved for regular rings of prime characteristic (Huneke–Sharp), for regular local and affine rings of characteristic zero via the theory of $D$-modules, i.e. rings of differential operators (Lyubeznik), for unramified regular local rings of mixed characteristic (Lyubeznik) — a regular local ring of mixed characteristic $(0,p)$ being called unramified if $p\notin\mathfrak{m}^2$ and ramified otherwise — and for smooth $\mathbb{Z}$-algebras (Bhatt–Blickle–Lyubeznik–Singh–Zhang). Núñez-Betancourt proved that in mixed-characteristic regular local rings the associated primes of $H^i_I(R)$ that do not contain $p$ are finite, so the open heart of the question in mixed characteristic sat at the primes containing $p$, i.e. exactly in the ramified case. The conjecture extends naturally from single local cohomology modules to Lyubeznik functors: the smallest class of functors on $R$-modules containing the functors $H^i_Z(-)$ of local cohomology with support in a locally closed subset $Z\subset\operatorname{Spec}R$ and closed under kernels, images, and cokernels of the natural transformations arising from the long exact sequences relating them.

A separate strand concerns how these finiteness properties behave under ring extensions. An injective homomorphism $R\to S$ is pure if $M\to M\otimes_RS$ is injective for every $R$-module $M$, and cyclically pure if $R/I\to S/IS$ is injective for every ideal $I$ of $R$, equivalently $IS\cap R=I$; faithfully flat maps and split inclusions are pure, and purity implies cyclic purity, so cyclic purity is the weakest of these conditions. Hochster proved in 1977 that cyclic purity implies purity when the base ring is approximately Gorenstein, meaning its ideals are cofinal with irreducible ideals; every complete Noetherian local reduced ring is approximately Gorenstein. A Noetherian local ring is called analytically unramified when its completion is reduced. In this setting Núñez-Betancourt (2012) proved that finiteness of associated primes, and of Bass numbers, of Lyubeznik functors descends from $S$ to $R$ along split inclusions, and Bhattacharyya (2015) extended this: descent holds along arbitrary pure extensions, and along cyclically pure extensions whose local base is analytically unramified (Theorem 4.3, via the observation that a cyclically pure extension stays cyclically pure after completing the base, at which point the reduced complete base is approximately Gorenstein, so cyclic purity upgrades to purity and the completed extension splits); for Bass numbers the cyclically pure case required the extension to be Cohen–Macaulay and finitely generated as an $R$-module (Theorem 5.3). Consequently every cyclically pure subring of an equicharacteristic or unramified regular local ring inherits finiteness. The general case outside these hypotheses remained open.

The landscape shifted in April 2026: Linquan Ma constructed complete ramified regular local rings of mixed characteristic possessing local cohomology modules with infinitely many associated primes and infinite Bass numbers, refuting Lyubeznik's question and Huneke's conjectures in the ramified case. Since Ma's modules are annihilated by $p$, their infinitely many associated primes all contain $p$, precisely the zone left open by Núñez-Betancourt's theorem; and since the ambient ramified regular local ring itself fails finiteness, Bhattacharyya's corollary on cyclically pure subrings of regular local rings can never be extended to ramified ambient rings. The descent question below is conditional on the extension ring having the finiteness property, so it survives as a well-posed question — but the refutation shows that this hypothesis is now a genuine restriction on the extension, not a formality.

Problem Statement

Let $R\to S$ be an injective cyclically pure homomorphism of commutative Noetherian rings with identity, that is, $IS\cap R=I$ for every ideal $I\subset R$, and let $(R,\mathfrak{m})$ be local. Let $T$ be a Lyubeznik functor.

(1) Associated primes. Suppose $\operatorname{Ass}_ST(S)$ is a finite set. Known results give $\operatorname{Ass}_RT(R)$ finite when the extension is pure, and when $R$ is analytically unramified. If $R$ is not analytically unramified, that is, the $\mathfrak{m}$-adic completion $\hat{R}$ is not reduced, and the extension is merely cyclically pure: is $\operatorname{Ass}_RT(R)$ necessarily finite?

(2) Bass numbers. Suppose all Bass numbers of $T(S)$ are finite. Known results give the same for $T(R)$ when the extension splits, when it is pure with $S$ Cohen–Macaulay and finitely generated as an $R$-module, and when $R$ is analytically unramified with $S$ Cohen–Macaulay and finitely generated as an $R$-module. Without the analytic-unramifiedness hypothesis on $R$ and without the Cohen–Macaulay and module-finite hypotheses on $S$: are all Bass numbers of $T(R)$ finite?

Both facets are asked in the stated generality: descent for arbitrary cyclically pure extensions of Noetherian rings with local base, with no analytic-unramifiedness, Cohen–Macaulay, or module-finiteness hypotheses; the known positive results recalled above cover only the cases excluded there. Thus the alternative to a positive answer to facet (1) is a cyclically pure extension $R\to S$ with $(R,\mathfrak{m})$ local that is not pure and has $R$ not analytically unramified, with $\operatorname{Ass}_ST(S)$ finite but $\operatorname{Ass}_RT(R)$ infinite, and the alternative to a positive answer to facet (2) is a cyclically pure extension $R\to S$ with $(R,\mathfrak{m})$ local that is not split, is not pure with $S$ Cohen–Macaulay and finitely generated as an $R$-module, and is not cyclically pure with $R$ analytically unramified and $S$ Cohen–Macaulay and finitely generated as an $R$-module, with all Bass numbers of $T(S)$ finite but some Bass number of $T(R)$ infinite.

References

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