Hochster's question on finiteness of closed-fiber-supported associated primes of local cohomology for flat extensions with regular closed fiber

For a Noetherian local ring $(R,m,K)$ and a flat extension $S$ whose closed fiber $S/mS$ is regular, Mel Hochster asked whether the part of the associated-prime set of each local cohomology module $H^i_I(S)$ supported on the closed fiber — the set $\mathcal{V}(mS)\cap \mathrm{Ass}_S H^i_I(S)$ — is always finite. Núñez-Betancourt (J. Algebra 399, 2014) proved this only when $S$ is a polynomial or power series ring over $R$ with $\dim R/(I\cap R)\le 1$, and, again under $\dim R/(I\cap R)\le 1$, for coefficient-field-aligned extensions with $R$ containing a field, leaving the unrestricted question and the stronger $\Sigma$-finiteness of $H^j_{mS}H^i_I(S)$ open. The general question remains open, with a precise open core: arbitrary flat extensions without coefficient-field hypotheses, and relative dimension at least 2.

Background

Let $R$ be a Noetherian commutative ring, $I\subseteq R$ an ideal and $M$ an $R$-module. The $i$-th local cohomology module $H^i_I(M)$ — computable as the cohomology of the Cech complex obtained by localizing at finitely many generators of $I$ — is a fundamental homological invariant that is almost never finitely generated. A basic structural question is how large the set $\mathrm{Ass}_R H^i_I(M)$ of associated primes, i.e. prime ideals occurring as annihilators $\mathrm{Ann}_R(v)$ of single elements $v$, can be. For regular rings, Huneke-Sharp proved finiteness of $\mathrm{Ass}_R H^i_I(R)$ in characteristic $p>0$ and Lyubeznik in equal characteristic $0$ as well as for unramified regular local rings of mixed characteristic; Bhatt-Blickle-Lyubeznik-Singh-Zhang proved it for smooth $\mathbb{Z}$-algebras. Lyubeznik's conjecture, open for three decades, predicts finiteness over every regular ring. For non-regular rings finiteness fails: Katzman, and Singh-Swanson, constructed ideals whose local cohomology modules have infinitely many associated primes. The frontier of Lyubeznik's conjecture is ramified mixed characteristic: a local ring of mixed characteristic $(0,p)$ has residue field of characteristic $p$ but contains no field of characteristic $p$ (equivalently, $p$ is neither zero nor a unit in the ring), and a regular local ring of mixed characteristic is called unramified when $p\notin m^2$ and ramified when $p\in m^2$; the source paper points to $V[[x,y,z_1,\dots,z_n]]/(\pi-xy)\cong(V[[x,y]]/(\pi-xy))[[z_1,\dots,z_n]]$, with $(V,\pi V,K)$ a discrete valuation ring of mixed characteristic $(0,p)$ (a principal ideal domain whose unique nonzero maximal ideal is $\pi V$, with residue field $K$ of characteristic $p$), as the simplest regular local ring of ramified mixed characteristic where finiteness of $\mathrm{Ass}_R H^i_I(R)$ is unknown. Such rings are formal power series rings over a lower-dimensional local base, which leads to the flat-extension setting of this problem. A flat extension of the local ring $(R,m,K)$ is a Noetherian $R$-algebra $S$ that is flat as an $R$-module (the source paper's global convention is that all rings are commutative Noetherian with unit); its closed fiber is the ring $S/mS$. Writing $\mathcal{V}(mS)$ for the set of prime ideals of $S$ containing $mS$, the primes in $\mathcal{V}(mS)\cap \mathrm{Ass}_S H^i_I(S)$ are exactly the associated primes of the $mS$-torsion submodule $H^0_{mS}(H^i_I(S))$, i.e. those lying over the closed point of $R$.

Mel Hochster raised the question whether this closed-fiber-supported part of the associated primes is always finite; an affirmative answer, combined with the known equal-characteristic and unramified results, would give finiteness of all associated primes of local cohomology for ramified regular local rings of mixed characteristic such as the example above. The main tool introduced in the source paper is the ring $D(S,R)$ of $R$-linear differential operators (the subring of $\mathrm{Hom}_R(S,S)$ generated by iterated derivatives in chosen variables) and its module theory: a $D(S,R)$-module $M$ supported on $\mathcal{V}(mS)$ is called $\Sigma$-finite if it is the directed union of its finite-length $D$-submodules (finite length in the noncommutative sense, via composition series with simple factors), the composition factors of all these submodules form a finite set, and every composition factor lies in Lyubeznik's category $\mathcal{C}(S/mS,R/mR)$ — the smallest subcategory of $D(S/mS,R/mR)$-modules containing the localizations of $S/mS$ and closed under subobjects, extensions and quotients — i.e. essentially lives over the closed fiber (in the source paper this definition is made for $S=R[x_1,\dots,x_n]$ or $R[[x_1,\dots,x_n]]$). A $\Sigma$-finite module has finitely many associated primes, and its local cohomology modules are again $\Sigma$-finite; hence $\Sigma$-finiteness of the iterated local cohomology modules $H^j_{mS}H^i_I(S)$ for all $i,j$ would settle Hochster's question affirmatively; the source paper poses this as its Question 5.1 for $S=R[x_1,\dots,x_n]$ or $R[[x_1,\dots,x_n]]$, asking whether $H^j_{mS}H^i_I(S)$ is a $\Sigma$-finite $D(S,R)$-module for every ideal $I\subseteq S$ and all integers $i,j\ge 0$. Known cases before and after the source paper: Question 1.2 (the polynomial/power series case) for bases of dimension $\le 1$ in any characteristic (Núñez-Betancourt, Illinois J. Math. 2013); finiteness of the full associated-prime set over normal (integrally closed) bases whose singular locus is a single point and which are finitely generated over a characteristic-zero field (Robbins, building on her 2008 thesis); and, after the source paper, the relative-dimension-$\le 1$ cases of the $\Sigma$-finiteness question (Quy 2016) and Murayama's 2025 theorem for excellent locally unramified regular rings. None of these removes the dimension, characteristic, or coefficient-field restrictions.

Problem Statement

Question (raised by Mel Hochster; recorded as Question 1.1 in Núñez-Betancourt, J. Algebra 399 (2014)). Let $(R,m,K)$ be a Noetherian local ring and let $S$ be a Noetherian flat extension of $R$ whose closed fiber $S/mS$ is regular. Is $\mathrm{Ass}_S H^0_{mS}(H^i_I(S))\;=\;\mathcal{V}(mS)\cap \mathrm{Ass}_S H^i_I(S)$ finite for every ideal $I\subseteq S$ and every integer $i\ge 0$? The same question restricted to $S=R[x_1,\dots,x_n]$ or $S=R[[x_1,\dots,x_n]]$ (Question 1.2 of the source) is equivalent to the general case for local extensions whose completion map sends a coefficient field of $\hat{R}$ (a subfield of the completion $\hat{R}$ mapping isomorphically onto the residue field $K$) to a coefficient field of $\hat{S}$ — in particular in equal characteristic zero and over perfect residue fields. The unrestricted question asks about arbitrary flat extensions with regular closed fiber, with no restriction on characteristic, on the relative dimension $\dim R/(I\cap R)$, or on coefficient-field alignment; the restricted cases proved so far answer only special cases.

References

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