Equality of the finitistic and full tight-closure annihilators in $F$-pure local rings

Let $(R,\mathfrak m)$ be an $F$-pure local ring of prime characteristic $p$, let $E$ be the injective envelope of its residue field, and let $S$ be a multiplicatively closed subset of $R$. In Sharp's theory of tight closure with respect to $S$, two annihilator invariants are naturally attached to $(R,S)$: the finitistic $S$-test ideal $\tau^{\mathrm{fg},S}(R)=(0:_R 0^{*,\mathrm{fg},S}_E)$, built from the $S$-tight closures of zero in the finitely generated submodules of $E$, and the full annihilator $(0:_R 0^{*,S}_E)$, which equals the $S$-test ideal $\tau^S(R)$ whenever $S$ is the complement of finitely many prime ideals. Both are members of the finite set $\mathcal I(\Phi(E))$ of $\Phi(E)$-special ideals, and the containment $(0:_R 0^{*,S}_E)\subseteq(0:_R 0^{*,\mathrm{fg},S}_E)$ always holds. In the classical case $S=R^\circ$ these invariants are the finitistic test ideal $\tau(R)$ and the big test ideal $\widetilde{\tau}(R)$, and the problem posed here is the conjecture recorded in the source for that case: whether the two annihilators coincide for every $F$-pure local ring. This is exactly the Hochster--Huneke conjecture that the finitistic test ideal equals the big test ideal, open since 1990 and established only in special classes (excellent Gorenstein rings, graded localizations, rings Gorenstein or Cohen--Macaulay on the punctured spectrum, isolated singularities, numerically $\mathbb Q$-Gorenstein pairs, certain four-dimensional rings, and weakly $F$-regular rings with Noetherian anti-canonical algebra on the punctured spectrum); an affirmative answer would settle whether weakly $F$-regular local rings are strongly $F$-regular. The analogous equality for a general multiplicatively closed subset $S$ is a natural strictly stronger extension that Sharp's framework makes precise, but the source does not pose it and it too remains unresolved; it is not part of the problem posed here.

Background

Tight closure, introduced by Hochster and Huneke in 1990, is a closure operation for ideals and submodules over commutative Noetherian rings of prime characteristic $p>0$. For an ideal $\mathfrak a\subseteq R$, the $n$-th Frobenius power $\mathfrak a^{[p^n]}$ is the ideal generated by the $p^n$-th powers of the elements of $\mathfrak a$, and the tight closure $\mathfrak a^*$ consists of all $r\in R$ for which some multiplier $c\in R^\circ$ (an element contained in no minimal prime of $R$) satisfies $cr^{p^n}\in\mathfrak a^{[p^n]}$ for all $n\gg 0$. The construction extends to submodules $N\subseteq M$ by the same Frobenius-power condition, giving the tight closure $N^*_M$; for the zero submodule, $0^*_M$ records the elements of $M$ killed by a multiplier after sufficiently many Frobenius twists.

Fix a local ring $(R,\mathfrak m)$ with residue field $k=R/\mathfrak m$. The ring is $F$-pure when the Frobenius endomorphism $F:R\to R$, $r\mapsto r^p$, is a pure homomorphism, meaning that tensoring $F$ with any $R$-module remains injective; this is the characteristic-$p$ analogue of a log canonical singularity. Write $E=E_R(k)$ for the injective envelope of $k$, the minimal injective $R$-module containing $k$. Two test ideals measure how far tight closures are from being trivial: the (finitistic) test ideal $\tau(R)=\bigcap(N:_R N^*_M)$, the intersection over submodules $N$ of finitely generated modules $M$, and the big test ideal $\widetilde{\tau}(R)$, the same intersection over all modules. Hochster and Huneke proved $\tau(R)=(0:_R 0^{*\mathrm{fg}}_E)$, where $0^{*\mathrm{fg}}_E=\bigcup_N 0^*_N$ is the finitistic tight closure of zero in $E$, the union over finitely generated submodules $N\subseteq E$; when $R$ has a big test element — always the case for excellent $F$-pure local rings, excellence being a standard finiteness hypothesis satisfied by rings essentially of finite type over a field — one also has $\widetilde{\tau}(R)=(0:_R 0^*_E)$, where $0^*_E$ is the full tight closure of zero in $E$.

Whether the finitistic and full tight closures of zero in $E$ have the same annihilator, equivalently $\tau(R)=\widetilde{\tau}(R)$, is the finitistic test ideal conjecture of Hochster and Huneke, open since 1990. It is known when $R$ is an excellent Gorenstein local ring (Gorenstein means finite injective dimension as a module over itself), by K. E. Smith; when $R$ is the localization of a finitely generated $\mathbb N$-graded algebra over an $F$-finite field (a ring finitely generated as a module over its own Frobenius image) at its homogeneous maximal ideal, by Lyubeznik and Smith; and when $R$ is Cohen--Macaulay (admitting a system of parameters generating a free submodule) and Gorenstein on the punctured spectrum (the open set $\operatorname{Spec}R\setminus\{\mathfrak m\}$), or is an isolated singularity, again by Lyubeznik and Smith. The conjecture carries the celebrated weak-implies-strong question of $F$-regularity: $R$ is weakly $F$-regular when every ideal is tightly closed, equivalently $\tau(R)=R$; and, in the Lyubeznik--Smith definition that extends to non-$F$-finite rings, strongly $F$-regular when the zero submodule of $E$ is tightly closed, equivalently $\widetilde{\tau}(R)=R$. Equality of the two test ideals would therefore make weak and strong $F$-regularity coincide.

Sharp's paper embeds this question in a family parametrized by multiplicatively closed subsets. For a multiplicatively closed subset $S\subseteq R$ with $1\in S$, the $S$-tight closure of an ideal $\mathfrak a$ is $\mathfrak a^{*,S}=\{r\in R:\text{ some }s\in S\text{ has }sr^{p^n}\in\mathfrak a^{[p^n]}\text{ for all }n\gg 0\}$, and the submodule version $N^{*,S}_M$ is defined analogously with the multiplier class $R^\circ$ replaced by $S$; in Hochster--Huneke's axiomatics this is tight closure with respect to the family of principal ideals $sR$, $s\in S$, directed by reverse inclusion. The choice $S=R^\circ$ recovers ordinary tight closure. An $S$-test element is an element of $S$ usable as the multiplier in every $S$-tight closure computation, the ideal $\tau^S(R)$ generated by all $S$-test elements is the $S$-test ideal, and Sharp proves that every $F$-pure local ring possesses $S$-test elements. The bookkeeping device is the Frobenius skew polynomial ring $R[x,f]$, whose elements are finite sums $\sum r_i x^i$ with multiplication twisted by $xr=r^p x$; each $R$-module $M$ yields a left $R[x,f]$-module $\Phi(M)=\bigoplus_{n\ge 0}Rx^n\otimes_R M$, and the $R$-annihilators of its $R[x,f]$-submodules form a finite set $\mathcal I(\Phi(E))$ of radical ideals of $R$, the $\Phi(E)$-special ideals, which contains $\tau(R)$ and $\widetilde{\tau}(R)$ and is closed under taking primary components. Sharp's classification theorem shows that an ideal of $R$ belongs to $\mathcal I(\Phi(E))$ if and only if it is the $S$-test ideal $\tau^S(R)$ for some multiplicatively closed subset $S$; when $S$ is the complement of finitely many prime ideals, $\tau^S(R)$ is the smallest member of $\mathcal I(\Phi(E))$ that meets $S$.

Exactly as in the classical case one defines the finitistic $S$-tight closure $0^{*,\mathrm{fg},S}_E=\bigcup_N 0^{*,S}_N$, the union over the finitely generated submodules $N\subseteq E$, and the finitistic $S$-test ideal $\tau^{\mathrm{fg},S}(R)=(0:_R 0^{*,\mathrm{fg},S}_E)$. Sharp proves that $\tau^{\mathrm{fg},S}(R)$ and $(0:_R 0^{*,S}_E)$ are both members of $\mathcal I(\Phi(E))$, that $(0:_R 0^{*,S}_E)$ is the smallest member meeting $S$, and that the containment $(0:_R 0^{*,S}_E)\subseteq(0:_R 0^{*,\mathrm{fg},S}_E)$ always holds. The source records, for the classical choice $S=R^\circ$, the standing conjecture that the two annihilators coincide; that recorded conjecture is the problem posed here. The source does not pose the analogous equality for a general multiplicatively closed subset $S$, which remains open as a strictly stronger extension of the problem.

Problem Statement

Let $(R,\mathfrak m)$ be a commutative Noetherian local ring of prime characteristic $p>0$ that is $F$-pure (its Frobenius endomorphism is pure), and let $E=E_R(R/\mathfrak m)$ be the injective envelope of the residue field. Following Sharp — whose $S$-tight closure with respect to $S=R^\circ$ is ordinary tight closure — the tight closure of zero in an $R$-module $M$ is

$0^*_M=\{\,m\in M:\text{ there exists }c\in R^\circ\text{ with }cx^n\otimes m=0\text{ in }Rx^n\otimes_R M\text{ for all }n\gg 0\,\},$

where $R^\circ$ denotes the elements of $R$ contained in no minimal prime, $x$ is a Frobenius variable satisfying $xr=r^p x$ for $r\in R$, and $n\gg 0$ means that the condition is required only for all sufficiently large $n$; and the finitistic tight closure of zero in $E$ is

$0^{*\mathrm{fg}}_E=\bigcup_N 0^*_N,$

the union over all finitely generated $R$-submodules $N\subseteq E$.

Problem. Does

$(0:_R 0^{*\mathrm{fg}}_E)=(0:_R 0^*_E)$

hold for every $F$-pure local ring $R$? Equivalently (when $R$ has a big test element, as is always the case for excellent $F$-pure local rings): is the finitistic test ideal $\tau(R)=(0:_R 0^{*\mathrm{fg}}_E)$ equal to the big test ideal $\widetilde{\tau}(R)$? In Sharp's $S$-relative language this is the classical case $S=R^\circ$: $\tau^{\mathrm{fg},R^\circ}(R)=(0:_R 0^{*,\mathrm{fg},R^\circ}_E)$, while $(0:_R 0^{*,R^\circ}_E)$ is what Sharp identifies with the smallest $\Phi(E)$-special ideal meeting $R^\circ$ (the $\Phi(E)$-special ideals being the annihilators of the $R[x,f]$-submodules of the Frobenius skew module $\Phi(E)=\bigoplus_{n\ge 0}Rx^n\otimes_R E$) and, in this case, with the big test ideal generated by the big test elements, that is, the elements of $R^\circ$ usable as the multiplier in every tight-closure computation.

The containment $(0:_R 0^*_E)\subseteq(0:_R 0^{*\mathrm{fg}}_E)$ always holds, since $0^{*\mathrm{fg}}_E\subseteq 0^*_E$. This is exactly the classical Hochster–Huneke finitistic test ideal conjecture $\tau(R)=\widetilde{\tau}(R)$, recorded as a standing conjecture in the source. The equality is settled either by a proof for all $F$-pure local rings or by an explicit $F$-pure local ring $R$ for which the two annihilators differ.

References

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