Single-element localization of tight closure and coherence of sheafified tight closure

Whether tight closure — the central closure operation of prime-characteristic commutative algebra — commutes with localization at single elements, that is whether $(I_f)^*=(I^*)_f$ for every ideal $I$ and every element $f$ of a Noetherian ring of characteristic $p$, is an open problem. It was left untouched by the Brenner–Monsky counterexample (2010) to commutation of tight closure with localization at arbitrary multiplicative sets, and by the second counterexample of Borevitz et al. (2024); both localize at the infinite multiplicative set of nonzero polynomials in a parameter. Epstein (2024) proved the question equivalent to a geometric coherence question: is the sheafified tight closure of every coherent sheaf of ideals on $\operatorname{Spec}R$ coherent? The equivalence extends to the $\mathfrak{b}_\bullet$-tight-closure variants of tight closure, and the companion notion of semi F-regularity — every sheaf of ideals tightly closed — sits between weak F-regularity and F-regularity, coinciding with weak F-regularity for Jacobson rings, while a local coincidence would settle the conjecture that weak F-regularity implies F-regularity. The question remains open as of August 2026.

Background

Tight closure is the central closure operation of commutative algebra in prime characteristic, introduced by Hochster and Huneke around 1990. For a Noetherian ring $R$ of characteristic $p>0$, let $q=p^e$ and let $I^{[q]}$ denote the ideal generated by the $q$-th powers of the elements of an ideal $I$ (the expansion of $I$ under the $e$-fold iterate of the Frobenius map $x\mapsto x^p$). An element $x$ lies in the tight closure $I^*$ of $I$ if there exists an element $c$ of $R$ lying outside every minimal prime of $R$ such that $cx^q\in I^{[q]}$ for all $q=p^e$ (equivalently, for all $q\gg 0$). Tight closure underlies the theory of $F$-singularities: $R$ is weakly F-regular if every ideal is tightly closed, and F-regular if every localization $R_P$ at a prime ideal $P$ is weakly F-regular. For F-finite rings (rings for which $R^{1/p}$ is a finite $R$-module), Hochster and Huneke introduced strong F-regularity, a Frobenius-splitting-type condition that is automatically stable under localization; it coincides with weak F-regularity for Gorenstein rings and for $\mathbb{N}$-graded rings, but whether weak F-regularity implies F-regularity for all Noetherian characteristic-$p$ rings remains a major open conjecture of the subject. From the theory's beginning it was hoped that tight closure, like integral closure, would commute with localization — that $S^{-1}(I^*)=(S^{-1}I)^*$ for every multiplicatively closed subset $S$ — not least because a positive answer would have made tight closure commute with arbitrary regular base change in all cases of geometric interest. Partial positive results were obtained for special classes (for instance modules of finite phantom projective dimension), and Hochster and Huneke reduced the problem to the existence of test exponents, meaning uniform Frobenius bounds on the witnesses $c$ that certify tight-closure membership. Brenner and Monsky settled the general question negatively in 2010: in the normal three-dimensional hypersurface $A=\overline{\mathbb{F}_2}[x,y,z,t]/(tx^2y^2+z^4+xyz^2+x^3z+y^3z)$ over the algebraic closure of the two-element field, for the ideal $I=(x^4,y^4,z^4)$ and the multiplicative set $S=\overline{\mathbb{F}_2}[t]\setminus\{0\}$, the element $y^3z^3$ lies in $(S^{-1}I)^*$ but not in $S^{-1}(I^*)$. A second counterexample of the same shape — the same $S$, the same ideal, and the same witness element in a so-called line–$S_4$ quartic hypersurface — was produced in 2024 by Borevitz, Nader, Sandstrom, Shapiro, Simpson and Zomback. Both counterexamples, however, localize at the infinite multiplicative set of all nonzero polynomials in the parameter $t$, and neither they nor anything else in the literature decides the single-element case $S=\{1,f,f^2,\dots\}$. Epstein (2024) gave this remaining question a geometric reformulation. A quasi-coherent sheaf on a scheme is one that is locally associated to a module, and a coherent sheaf is a quasi-coherent sheaf locally associated to a finitely generated module. Epstein isolates two axioms for a (pre)closure operation on ideals or submodules — open persistence (compatibility with restriction to affine open subschemes) and glueability — under which such an operation extends to a closure-type operation on sheaves of ideals, or on sheaves of submodules of a quasi-coherent sheaf, and he proves that tight closure and its $\mathfrak{b}_\bullet$-tight-closure variants (tight closure relative to a $p$-system of ideals, meaning a suitably compatible sequence $\{\mathfrak{b}_{p^e}\}_{e\ge 1}$ of ideals; $p$-systems simultaneously generalize $p$-families and $F$-graded systems of ideals) satisfy both axioms. For a glueable operation he proves that commutation with localization at single elements is equivalent to the sheafified closure of a quasi-coherent subsheaf remaining quasi-coherent. His Theorem 4.14 specializes this to tight closure: for a Noetherian $\mathbb{F}_p$-algebra $R$ and an ideal $I$, one has $(I_f)^*=(I^*)_f$ for all $f\in R$ if and only if the sheafified tight closure $(\widetilde{I})^*$ of the corresponding sheaf of ideals on $\operatorname{Spec}R$ is coherent. The same paper introduces semi F-regularity — every sheaf of ideals on $\operatorname{Spec} R$ is tightly closed, equivalently $R_f$ is weakly F-regular for every $f\in R$ — a class lying between weak F-regularity and F-regularity that agrees with weak F-regularity for Jacobson rings (rings in which every prime ideal is an intersection of maximal ideals), while a positive answer for local rings would settle the conjecture that weak F-regularity implies F-regularity for all Noetherian $\mathbb{F}_p$-algebras.

Problem Statement

The problem has one central question — posed verbatim in the source paper — together with one companion question that the source poses explicitly. Central question. Does tight closure commute with localization at single elements? That is: for every Noetherian ring $R$ of prime characteristic $p$, every ideal $I\subseteq R$, and every element $f\in R$, is

$$(I_f)^*=(I^*)_f,$$

where $I_f$ and $(I^*)_f$ denote the extensions of $I$ and $I^*$ in the localization $R_f=R[1/f]$? By Epstein's Theorem 4.14 this is equivalent to the geometric coherence question: for every Noetherian $\mathbb{F}_p$-algebra $R$, is the sheafified tight closure $(\widetilde{I})^*$ of the coherent sheaf of ideals attached to $I$ on $\operatorname{Spec} R$ coherent for every ideal $I$ (equivalently quasi-coherent, since $R$ is Noetherian)? The same question arises for submodules — is $(L_f)^*_{M_f}=(L^*_M)_f$ for every inclusion of modules $L\subseteq M$, equivalently, is the sheafified tight closure of every quasi-coherent subsheaf of a quasi-coherent (respectively coherent) sheaf always quasi-coherent — and for the $\mathfrak{b}_\bullet$-tight-closure variants, to which Epstein's equivalence applies equally. Companion question. The source poses this explicitly, immediately before its Proposition 4.20: is semi F-regularity equal to weak F-regularity? The two notions coincide for Jacobson rings; for local rings a positive answer would settle the conjecture that weak F-regularity implies F-regularity for all Noetherian $\mathbb{F}_p$-algebras, so this companion is expected to be at least as hard as that conjecture. A resolution of the central question in either direction — a proof that single-element localization always commutes with tight closure, or an explicit ring, ideal, and element for which it fails — is the decisive target; the companion question then interprets the outcome.

References

1. Neil Epstein, "Tight closure, coherence, and localization at single elements", Acta Mathematica Vietnamica 49(2) (2024), 201–215. DOI: 10.1007/s40306-024-00533-9. https://doi.org/10.1007/s40306-024-00533-9 (preprint: arXiv:2206.11949)
2. Levi Borevitz, Naima Nader, Theodore J. Sandstrom, Amelia Shapiro, Austyn Simpson, Jenna Zomback, "On localization of tight closure in line–S4 quartics", Journal of Pure and Applied Algebra 228(9) (2024), Article 107682. DOI: 10.1016/j.jpaa.2024.107682. https://doi.org/10.1016/j.jpaa.2024.107682 (preprint: arXiv:2211.03220)
3. Holger Brenner, Paul Monsky, "Tight closure does not commute with localization", Annals of Mathematics (2) 171(1) (2010), 571–588. DOI: 10.4007/annals.2010.171.571. https://doi.org/10.4007/annals.2010.171.571
4. Melvin Hochster, Craig Huneke, "Localization and test exponents for tight closure", Michigan Mathematical Journal 48 (2000), 305–329. DOI: 10.1307/mmj/1030132721. https://doi.org/10.1307/mmj/1030132721 (preprint: arXiv:math/0211173)
5. Melvin Hochster, Craig Huneke, "F-regularity, test elements, and smooth base change", Transactions of the American Mathematical Society 346(1) (1994), 1–62. DOI: 10.1090/s0002-9947-1994-1273534-x. https://doi.org/10.1090/s0002-9947-1994-1273534-x
6. Melvin Hochster, Craig Huneke, "Tight closure, invariant theory, and the Briançon–Skoda theorem", Journal of the American Mathematical Society 3(1) (1990), 31–116. DOI: 10.2307/1990984. https://doi.org/10.2307/1990984
7. Melvin Hochster, Craig Huneke, "Tight closure and strong F-regularity", Mémoires de la Société Mathématique de France (N.S.) 38 (1989), 119–133. DOI: 10.24033/msmf.343. https://doi.org/10.24033/msmf.343
8. Gennady Lyubeznik, Karen E. Smith, "Strong and weak F-regularity are equivalent for graded rings", American Journal of Mathematics 121(6) (1999), 1279–1290. DOI: 10.1353/ajm.1999.0042. https://doi.org/10.1353/ajm.1999.0042
9. I. M. Aberbach, Melvin Hochster, Craig Huneke, "Localization of tight closure and modules of finite phantom projective dimension", Journal für die reine und angewandte Mathematik 434 (1993), 67–114. DOI: 10.1515/crll.1993.434.67. https://doi.org/10.1515/crll.1993.434.67
10. Neil Epstein, Peter M. McDonald, Rebecca R.G., Karl Schwede, "Closure operations induced via resolutions of singularities in characteristic zero" (2025), preprint, arXiv:2504.05554. https://arxiv.org/abs/2504.05554