Does tight closure commute with localization at a single element?

Tight closure, the closure operation on ideals introduced by Hochster and Huneke in the late 1980s, is a central tool of commutative algebra in prime characteristic. Whether it commutes with localization was a famous open problem for two decades, until Brenner and Monsky (Annals of Mathematics, 2010) constructed a three-dimensional normal hypersurface in characteristic two and an ideal for which tight closure fails to commute with localization at the multiplicative set of all nonzero polynomials in one parameter. Their counterexample leaves open the principal case, which Brenner and Monsky explicitly single out: does tight closure commute with localization at a single element, i.e., is $(I_f)^* = (I^*)_f$ for every ideal $I$ and every element $f$ of every noetherian ring of prime characteristic (the inclusion $(I^*)_f \subseteq (I_f)^*$ being trivial)? Later literature confirms the question is still open and has deepened it: Epstein (2024) proved it is equivalent to the tight closure of a coherent sheaf of ideals on a noetherian $\mathbb{F}_p$-scheme remaining quasi-coherent, and Borevitz–Nader–Sandstrom–Shapiro–Simpson–Zomback (2024), exhibiting only the second known counterexample to localization (again in characteristic two), state explicitly that the single-element question is unresolved. The problem asks for either a proof of commutation in full generality or an explicit counterexample.

Background

Tight closure theory works over noetherian rings of prime characteristic $p > 0$ (rings containing the finite field $\mathbb{F}_p$), where the Frobenius map $r \mapsto r^p$ is a ring homomorphism. For $q = p^e$ a power of $p$ and an ideal $I$, the Frobenius power $I^{[q]}$ is the ideal generated by the $q$-th powers of all elements of $I$. The tight closure $I^*$ of $I$ is the set of elements $u$ for which there exists a multiplier $c$ not in any minimal prime of $R$ such that $cu^q \in I^{[q]}$ for all $q = p^e$. Tight closure, introduced by Hochster and Huneke, always contains $I$, behaves like integral closure but is much smaller, and gave unified proofs of many results such as the Briançon–Skoda theorem (a uniform containment of the integral closures of powers of an ideal in a fixed higher power) and the existence of big Cohen–Macaulay algebras (algebras over a local ring in which some system of parameters is a regular sequence).

Since localization $R \to S^{-1}R$ at a multiplicative set $S$ is a fundamental operation, it was natural to ask whether tight closure commutes with it: is $(S^{-1}I)^* = S^{-1}(I^*)$ for all ideals $I$? One inclusion, $S^{-1}(I^*) \subseteq (S^{-1}I)^*$, is immediate from persistence of tight closure (under any ring homomorphism, the image of an element of $I^*$ lies in the tight closure of the extension of $I$); the reverse inclusion was the celebrated localization problem. Huneke's 1996 CBMS monograph devoted a chapter to what he called the most frustrating problem in the theory. Positive results accumulated: commutation holds in binomial rings (quotients of polynomial rings by ideals generated by binomials; K. E. Smith, 2001), in rings of dimension at most two and in two-dimensional local rings (Divaani-Aazar–Tousi, 2005), and the theory of test elements and test exponents (Hochster–Huneke, 2000) gave criteria equivalent to commutation in several classes of rings. A test element is an element $t$ not in any minimal prime such that $u \in I^*$ if and only if $tu^q \in I^{[q]}$ for all $q$, which reduces tight closure membership to a single multiplier. One attack proceeds through plus closure: the plus closure $I^+$ of an ideal $I$ in a domain $R$ consists of the elements of $R$ that lie in the extension of $I$ in some finite integral extension ring of $R$; since plus closure commutes with localization, the equality $I^* = I^+$ would imply commutation, and it was known for parameter ideals in excellent domains (K. E. Smith) and for graded $R_+$-primary ideals in two-dimensional standard-graded domains over finite fields (Brenner, 2006).

Brenner and Monsky settled the localization problem negatively. They study the family of hypersurfaces $g_\alpha = z^4 + xyz^2 + x^3z + y^3z + \alpha x^2y^2$ in characteristic two, whose Hilbert–Kunz multiplicities (a Frobenius-based multiplicity introduced by Monsky that measures the asymptotic behavior of the lengths of $R/I^{[q]}$) jump between transcendental and algebraic values of $\alpha$. Taking $A = L[x,y,z,t]/(g_t)$ over the algebraic closure $L$ of $\mathbb{F}_2$, the ideal $I = (x^4, y^4, z^4)$, the element $f = y^3z^3$, and the multiplicative set $S = L[t] \setminus \{0\}$ (localization at the generic point of the parameter line), they prove $f \in (S^{-1}I)^*$ but $f \notin S^{-1}(I^*)$. The argument combines a geometric deformation principle for tight closure, a test-element theorem for normal homogeneous hypersurfaces, and Monsky's explicit Hilbert–Kunz computations. The example also answers negatively Hochster's 'tantalizing question' whether tight closure always equals plus closure, already in dimension two. Crucially, the multiplicative set $L[t] \setminus \{0\}$ is not generated by a single element, so the paper states that it is still possible that tight closure always commutes with localization at a single element, namely $(I_f)^* = (I^*)_f$, and that $I^* = I^+$ always holds for algebras of finite type over a finite field.

Later work treats the single-element question as the surviving core of the localization problem. Borevitz–Nader–Sandstrom–Shapiro–Simpson–Zomback (2024) constructed only the second known example of non-commutation (in the quartic hypersurface $\overline{\mathbb{F}_2}[x,y,z,t]/(tz^4 + (x^2+yz)(y^2+xz))$ (the line-$S_4$ quartic family), with the same ideal and element), again using a multiplicative set that is not singly generated, and state that the single-element question remains open. Epstein (2024) recast it geometrically: tight closure is compatible with restriction to affine open subsets and with gluing, so it extends to a closure operation on sheaves of ideals on noetherian $\mathbb{F}_p$-schemes, and commutation with localization at single elements is equivalent to this operation sending coherent sheaves of ideals to quasi-coherent ones. He also introduces semi F-regularity — every sheaf of ideals is tightly closed — a singularity class sitting between weak F-regularity (every ideal of the ring is tightly closed) and F-regularity (the same after every localization), and shows that its equivalence with weak F-regularity for local rings would settle the long-standing conjecture that weakly F-regular rings are F-regular. A 2025 lecture series by Zhang is dedicated to the localization problem, confirming its status as an active open topic.

Problem Statement

Let $R$ be a noetherian ring of prime characteristic $p > 0$, let $I \subseteq R$ be an ideal, and let $f \in R$ be an element. Writing $I_f$ for the extension of $I$ in the localization $R_f$ and $I^*$ for the tight closure, the inclusion $(I^*)_f \subseteq (I_f)^*$ always holds, since tight closure is persistent under ring homomorphisms (the image of an element of $I^*$ lies in the tight closure of the extension of $I$), in particular under $R \to R_f$. The question left open by Brenner and Monsky's counterexample to general localization, and still open in the literature as of 2024-2025, is the reverse inclusion.

Question: does tight closure commute with localization at a single element in full generality — that is, is it true that for every noetherian ring $R$ of prime characteristic, every ideal $I \subseteq R$, and every element $f \in R$, one has $(I_f)^* = (I^*)_f$? Equivalently: whenever an element of $R_f$ lies in the tight closure of $I_f$, must some $f$-multiple of it already lie in the tight closure of $I$ in $R$?

Settling the question requires either a proof that commutation holds at this level of generality or an explicit counterexample consisting of a noetherian ring $R$ of prime characteristic, an ideal $I$, an element $f$, and an element $u \in (I_f)^* \setminus (I^*)_f$. By Epstein's theorem, an answer is equivalently an answer to the geometric form: does the tight closure of a coherent sheaf of ideals on a noetherian $\mathbb{F}_p$-scheme remain quasi-coherent? The companion questions raised by Brenner and Monsky — whether $I^* = I^+$ for all algebras of finite type over a finite field, whether non-commutation can occur in characteristic $p > 2$, and whether the deformation behavior of cohomological dimension (the least number of affine open subsets needed to cover a scheme) observed by Brenner and Monsky can occur in characteristic zero — are related open directions but are logically independent of this question and do not belong to its completion boundary.

References

1. Holger Brenner and Paul Monsky, "Tight closure does not commute with localization", Annals of Mathematics (2) 171 (2010), no. 1, 571-588. DOI: 10.4007/annals.2010.171.571; arXiv:0710.2913. https://doi.org/10.4007/annals.2010.171.571
2. Neil Epstein, "Tight closure, coherence, and localization at single elements", Acta Mathematica Vietnamica 49 (2024), no. 2, 201-215. DOI: 10.1007/s40306-024-00533-9; arXiv:2206.11949. https://doi.org/10.1007/s40306-024-00533-9
3. 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), no. 9, Article 107682. DOI: 10.1016/j.jpaa.2024.107682; arXiv:2211.03220. https://doi.org/10.1016/j.jpaa.2024.107682
4. Wenliang Zhang, "Lectures on the localization problem in tight closure", in Lectures on Tight Closure and Its Applications, Contemporary Mathematics 831, American Mathematical Society, Providence, RI, 2025, pp. 159-175. DOI: 10.1090/conm/831/16635. https://doi.org/10.1090/conm/831/16635
5. Melvin Hochster, "Tight closure and strongly F-regular rings", Research in the Mathematical Sciences 9 (2022), no. 3, Article 56. DOI: 10.1007/s40687-022-00353-z. https://doi.org/10.1007/s40687-022-00353-z
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