Principal localization of tight closure and the existence of its failure in characteristic $p>2$

Tight closure, a fundamental closure operation on ideals in Noetherian rings of prime characteristic, was expected for two decades to commute with localization, until Brenner and Monsky (2010) and, building on Monsky's Hilbert–Kunz calculations, Borevitz–Nader–Sandstrom–Shapiro–Simpson–Zomback (2022, published 2024) exhibited hypersurfaces over $\overline{\mathbb F_2}$ where tight closure fails to commute with localization at the multiplicative set of nonzero polynomials in a parameter. Two questions explicitly left open by the 2022 work define this problem: (1) does tight closure commute with localization at a singly generated multiplicative set $S=\{1,f,f^2,\dots\}$, which by Epstein's theorem is equivalent to the sheafified tight-closure operation preserving quasi-coherence; and (2) does failure of tight-closure localization occur in any prime characteristic $p>2$, given that all known examples rely on characteristic-two combinatorics and test-element arguments. Both questions remain open as of mid-2026 (Epstein 2024; Smirnov 2023; Zhang 2025; Das–Landsittel–Pham 2026).

Background

Tight closure is a closure operation on ideals in commutative algebra of prime characteristic, introduced by Melvin Hochster and Craig Huneke. Throughout, $R$ is a Noetherian ring of characteristic $p>0$ and $I\subseteq R$ is an ideal. The $e$-th Frobenius power of $I$ is the ideal $I^{[p^e]}$ generated by the $p^e$-th powers of the elements of $I$, where $p^e$ ranges over powers of the characteristic. An element $r\in R$ lies in the tight closure $I^*$ of $I$ if there exists an element $c\in R$, not contained in any minimal prime of $R$, such that $c\,r^{p^e}\in I^{[p^e]}$ for all $e\ge 1$; an element $c$ that witnesses membership uniformly for a class of elements and ideals is called a test element. Tight closure refines integral-closure-type statements (for example the Briançon–Skoda theorem) and underlies the dictionary between singularities in characteristic $p$ and complex birational singularities.

Because localization $R\to S^{-1}R$ at a multiplicative subset $S\subseteq R$ is one of the basic constructions of ring theory, it was expected from the beginning that tight closure commute with it, i.e. that $S^{-1}(I^*)=(S^{-1}I)^*$; this would make tight closure a genuinely local, sheaf-level notion. Partial positive results hold for special classes of rings, for instance rings admitting test exponents (a test exponent for a test element $c$ and a submodule $N\subseteq M$ is a power $q'$ of $p$ such that $u$ lies in the tight closure of $N$ in $M$ whenever $c\,u^{q}\in N^{[q]}$ for some $q\ge q'$) and binomial rings (quotients of polynomial rings by ideals generated by binomials, i.e. differences of monomials). Brenner and Monsky settled the general question negatively: over the algebraic closure $F=\overline{\mathbb F_2}$, in the hypersurface (the quotient of a polynomial ring by one equation) $A=F[x,y,z,t]/(tx^2y^2+z^4+xyz^2+x^3z+y^3z)$ with $I_0=(x^4,y^4,z^4)$ and $S_0=F[t]\setminus\{0\}$, the element $y^3z^3$ lies in $(S_0^{-1}I_0)^*$ but not in $S_0^{-1}(I_0^*)$. Their construction grew out of Hilbert–Kunz theory: the Hilbert–Kunz multiplicity $e_{HK}(J)$ of an ideal $J$ primary to the maximal ideal is the limit $\lim_{e\to\infty}\ell(R/J^{[p^e]})/p^{e\dim R}$ of normalized Frobenius-power lengths, and Monsky's 1998 calculations showed that it varies with the parameter $t$ in two families of quartic hypersurfaces (defining equations of degree four) in the variables $x,y,z$, which he dubbed point-$S_4$ and line-$S_4$ quartics. Building on the line-$S_4$ calculation, the source paper produced the second known counterexample, on the hypersurface $B=F[x,y,z,t]/(tz^4+(x^2+yz)(y^2+xz))$ with the same ideal $I_0$ and the same set $S_0$, via a combinatorial tiling argument with Sierpiński triangles (the self-similar triangle fractal pattern arising in the relevant Frobenius colengths) and a study of the arithmetic dynamics of the maps $t\mapsto t^4+\alpha t^{-4}$ over $F$.

Both known counterexamples localize at $S_0=F[t]\setminus\{0\}$, which is not generated by a single element. Epstein proved that, for tight closure and its main variants, commutation with localization at single elements is equivalent to a geometric property: the sheafified tight-closure operation, applied to a quasi-coherent sheaf of ideals on a scheme, always yields a quasi-coherent sheaf (a quasi-coherent sheaf of ideals is one locally induced by an ideal of functions; quasi-coherence is the minimal condition for a sheaf to be determined by its sections over affine open sets, and coherence additionally requires finite generation). The source paper closes by formulating the two questions that constitute the present problem.

Problem Statement

The problem is to determine the scope of the failure of tight-closure localization left open by the two known characteristic-two counterexamples. It has two linked parts, which jointly delimit how badly tight closure fails to be a local, sheaf-level operation.

(1) Principal localization. For a Noetherian ring $R$ of prime characteristic $p>0$, an ideal $I\subseteq R$, and a single element $f\in R$, let $S_f=\{1,f,f^2,\dots\}$ be the multiplicative set generated by $f$. Is it always the case that $(S_f^{-1}I)^*=S_f^{-1}(I^*)$? Equivalently, by Epstein's theorem, does the sheafified tight-closure operation applied to a quasi-coherent sheaf of ideals on a scheme always produce a quasi-coherent sheaf? A negative answer requires an explicit Noetherian ring of prime characteristic, an ideal, an element $f$, and a witness lying in $(S_f^{-1}I)^*$ but not in $S_f^{-1}(I^*)$; a positive answer requires a proof valid for all Noetherian rings of prime characteristic.

(2) Odd characteristic. Do there exist a Noetherian ring $R$ of prime characteristic $p>2$, an ideal $I\subseteq R$, a multiplicative set $S\subseteq R$, and an element of $(S^{-1}I)^*\setminus S^{-1}(I^*)$? Both known counterexamples live over $\overline{\mathbb F_2}$, and their proofs depend essentially on characteristic-two combinatorics, notably the Brenner–Monsky degree bound that exhibits degree-two homogeneous elements as test elements in the relevant fibers. An answer is either an explicit example in some characteristic $p>2$ or a theorem that no such example exists.

A complete resolution settles both parts; an answer to one part alone leaves the other open. The two parts share one scientific objective: to decide whether the known failure of tight closure to commute with localization is confined to non-principal multiplicative sets in characteristic two, or whether it already manifests on principal open sets and across characteristics.

References

1. Melvin Hochster and Craig Huneke, "Tight Closure, Invariant Theory, and the Briançon–Skoda Theorem", Journal of the American Mathematical Society 3, no. 1 (1990), 31–116. DOI: 10.2307/1990984, https://doi.org/10.2307/1990984
2. Melvin Hochster and Craig Huneke, "Localization and test exponents for tight closure", Michigan Mathematical Journal 48 (2000), 305–329; arXiv:math/0211173. DOI: 10.1307/mmj/1030132721, https://doi.org/10.1307/mmj/1030132721
3. Karen Smith, "Tight closure commutes with localization in binomial rings", Proceedings of the American Mathematical Society 129, no. 3 (2001), 667–669. DOI: 10.1090/s0002-9939-00-05626-4, https://doi.org/10.1090/s0002-9939-00-05626-4
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7. Neil Epstein, "Tight closure, coherence, and localization at single elements", Acta Mathematica Vietnamica 49 (2024), 201–215; arXiv:2206.11949. DOI: 10.1007/s40306-024-00533-9, https://arxiv.org/abs/2206.11949
8. Levi Borevitz, Naima Nader, Theodore J. Sandstrom, Amelia Shapiro, Austyn Simpson, and Jenna Zomback, "On localization of tight closure in line-S4 quartics", Journal of Pure and Applied Algebra 228 (2024), article 107682; arXiv:2211.03220. DOI: 10.1016/j.jpaa.2024.107682, https://arxiv.org/abs/2211.03220
9. Ilya Smirnov, "An invitation to equimultiplicity of F-invariants", arXiv:2309.04322 (2023). https://arxiv.org/abs/2309.04322
10. Wenliang Zhang, "Lectures on the localization problem in tight closure", Contemporary Mathematics 831 (2025), 159–175. DOI: 10.1090/conm/831/16635, https://doi.org/10.1090/conm/831/16635
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