The (LC) conjecture: uniform linear annihilation of the local cohomology of Frobenius powers, and the relation between (LC) and (LC*)

In a Noetherian ring of prime characteristic $p>0$, Huneke's condition (LC) asserts that the zeroth local cohomology of the Frobenius powers of an ideal is annihilated by a power of the maximal ideal that is linear in the Frobenius exponent: there exists a constant $N$ with $\mathfrak m^{Nq}H^0_{\mathfrak m}(R/I^{[q]})=0$ for all $q=p^e$. The condition, its tight-closure analogue (LC$^*$), and the module version of Dao and Smirnov (via the Peskine–Szpiro functor) control the existence and finiteness of generalized Hilbert–Kunz invariants and feed the problem of whether weakly F-regular rings are F-regular. The conjecture is open in general: the settled cases include dimension at most two, normal Cohen–Macaulay standard graded rings of dimension less than four, ideals of finite projective dimension, ideals locally generated by height-many elements, and rings of finite F-representation type. The precise logical relation between (LC) and (LC$^*$) is likewise unknown outside isolated-singularity and test-element situations. This problem asks for a proof or a counterexample in the general case, for the analogous module-level statement, and for the determination of the (LC)–(LC$^*$) relation.

Background

This problem lies in commutative algebra in positive prime characteristic. In a Noetherian ring $R$ of characteristic $p>0$, the Frobenius map sends $r\mapsto r^p$; for each power $q=p^e$, the Frobenius power of an ideal $I$ is $I^{[q]}=(x^q:x\in I)$. For an ideal $\mathfrak m$, the zeroth local cohomology $H^0_{\mathfrak m}(M)$ is the submodule of a module $M$ consisting of the elements killed by some power of $\mathfrak m$; it measures the finite-length defect of $M$ at $\mathfrak m$, and $H^0_{\mathfrak m}(R/J)\cong J^{\mathrm{sat}}/J$, where the saturation $J^{\mathrm{sat}}=\{r:\mathfrak m^n r\subseteq J\text{ for some }n\}$. Equivalently, $H^0_{\mathfrak m}(R/J)$ is nonzero exactly when saturating $J$ enlarges it, and the least power of $\mathfrak m$ annihilating it is a Loewy length (the length of the shortest chain of annihilator ideals) of this finite-length piece.

For a fixed ideal $J$ some power of $\mathfrak m$ always kills $H^0_{\mathfrak m}(R/J)$, but when $J=I^{[q]}$ runs over the infinitely many Frobenius powers, the required exponent could in principle grow faster than linearly in $q$. Studying whether tight closure commutes with localization, Huneke isolated the linear-growth condition (LC): for a standard graded ring $R$ over a field of characteristic $p$ with irrelevant ideal $\mathfrak m=\bigoplus_{i\geq 1}R_i$, writing $R_{\geq n}$ for the ideal generated by the forms of degree at least $n$, a homogeneous ideal $I$ satisfies (LC) if there is a constant $N$ such that $R_{\geq Nq}\cdot H^0_{\mathfrak m}(R/I^{[q]})=0$ for all $q=p^e$; equivalently $R_{\geq Nq}(I^{[q]})^{\mathrm{sat}}\subseteq I^{[q]}$ for all $q$. The same condition for a local ring $(R,\mathfrak m)$ replaces $R_{\geq Nq}$ by $\mathfrak m^{Nq}$. Thus (LC) says exactly that saturating the Frobenius powers of $I$ only ever adds elements of bounded degree, linearly controlled by $q$.

Here tight closure is the closure operation, introduced by Hochster and Huneke, in which $x\in I^*$ when there is an element $c$ outside all minimal primes with $cx^q\in I^{[q]}$ for all large $q$; it underlies the singularity classes of F-regular rings. A ring is weakly F-regular if all of its ideals are tightly closed, and F-regular if this remains true after every localization; whether weak implies strong F-regularity is a long-standing open question, and the cyclic-module instances of (LC) in dimension one imply it. The companion condition (LC$^*$) is (LC) with $I^{[q]}$ replaced by its tight closure $(I^{[q]})^*$. Katzman had shown that localization of tight closure would follow from finiteness of the set of associated primes of the Frobenius iterates together with conditions of this type. If $R$ has an isolated singularity (the non-regular locus of $\operatorname{Spec}R$ is just the closed point), the theory of test elements — elements that test tight closure for every ideal — shows that (LC) holds for $I$ if and only if (LC$^*$) does; without such hypotheses, no implication between the two conditions is known in either direction. Huneke's 2000 paper proved (LC) and (LC$^*$) in several special situations (for example ideals of dimension at most one in equidimensional graded rings, and ideals locally generated by height-many elements over Cohen–Macaulay standard graded rings) and stated that, as far as was known, all homogeneous ideals satisfy both conditions; Vraciu the same year established (LC) for two-dimensional graded affine algebras.

Dao and Smirnov later gave the condition its module-theoretic form and its Hilbert–Kunz meaning. For an $R$-module $M$, the $n$-th Peskine–Szpiro functor is the base change $F^n(M)=M\otimes_R R$ of $M$ along the iterated Frobenius, so that $F^n(R/I)\cong R/I^{[p^n]}$; a finitely generated module $M$ over a local ring $(R,\mathfrak m)$ satisfies (LC) if there is an integer $l$ with $\mathfrak m^{lp^n}H^0_{\mathfrak m}(F^n(M))=0$ for all $n$. The generalized Hilbert–Kunz function of $M$ is $n\mapsto \ell(H^0_{\mathfrak m}(F^n(M)))$ normalized by $p^{n\dim R}$, and its limit, when it exists, is the generalized Hilbert–Kunz multiplicity $e_{gHK}(M)$, generalizing the classical Hilbert–Kunz multiplicity of $\mathfrak m$-primary ideals; the limsup of the normalized function is the upper generalized Hilbert–Kunz multiplicity $e^+_{gHK}(M)$, so $e^+_{gHK}(M)$ is finite exactly when the normalized function stays bounded. Dao and Smirnov proved that if all finitely generated modules over $R$ satisfy (LC) (and $R$ admits countable prime avoidance, i.e. a countable set of primes can be avoided by an element of a given ideal, which holds for complete rings or uncountable residue fields), then the upper generalized Hilbert–Kunz multiplicity $e^+_{gHK}(M)$ is finite for every finitely generated $M$; they describe (LC) as resting on a widely open conjecture. Asgharzadeh refers to the statement as the (LC) conjecture of Hochster and Huneke and verified it for standard graded Cohen–Macaulay rings that are normal of dimension less than four, for ideals with $\operatorname{Proj}(R/I)$ regular in codimension two, for ideals of finite projective dimension, and for rings of finite F-representation type (rings whose indecomposable Frobenius summands are finite in number).

One piece of the original motivation has since been settled negatively: Brenner and Monsky constructed a ring in which tight closure does not commute with localization, so the localization problem has a negative answer in general. This does not resolve (LC), which survives as an independent uniform-bounds problem; its consequences now run through the existence of generalized Hilbert–Kunz limits (Vraciu proved these exist for a subclass of rings satisfying (LC); Hernández and Jeffries obtained existence results for more general $p$-families of ideals — families $I_{p^n}$ with $I_{p^{m+n}}\subseteq I_{p^m}^{[p^n]}$ — under stronger (LC)-type hypotheses), through adjacent linear-growth questions for primary decompositions of Frobenius powers, and through the weak-implies-strong F-regularity program, for which Aberbach and Polstra recently obtained the four-dimensional case via local cohomology bounds. What is missing is any result, positive or negative, for arbitrary ideals in arbitrary Noetherian rings of prime characteristic, and any theorem comparing (LC) with (LC$^*$) in that generality.

Problem Statement

Let $R$ be a Noetherian ring of prime characteristic $p>0$, either standard graded over a field $R_0$ with irrelevant ideal $\mathfrak m$, or local with maximal ideal $\mathfrak m$; write $q=p^e$ and $I^{[q]}=(x^q:x\in I)$ for the Frobenius power of an ideal $I$.

1. (The (LC) conjecture, ideal form.) Prove, or disprove by an explicit counterexample, that for every homogeneous ideal $I$ of $R$ (in the local case: every ideal $I$) there exists a constant $N$, independent of $q$, such that $\mathfrak m^{Nq}\,H^0_{\mathfrak m}(R/I^{[q]})=0$ for all $q=p^e$; equivalently, in the graded case, $R_{\geq Nq}(I^{[q]})^{\mathrm{sat}}\subseteq I^{[q]}$ for all $q$.

2. (Module form, Dao–Smirnov.) For $(R,\mathfrak m)$ local of prime characteristic and the Peskine–Szpiro functor $F^n(-)$ (base change along the $n$-fold iterated Frobenius, with $F^n(R/I)\cong R/I^{[p^n]}$), prove or disprove that for every finitely generated $R$-module $M$ there exists an integer $l$ such that $\mathfrak m^{lp^n}H^0_{\mathfrak m}(F^n(M))=0$ for all $n$. A positive answer implies $e^+_{gHK}(M)$ is finite for every finitely generated $M$, provided $R$ satisfies countable prime avoidance as defined in the background.

3. (Relation between the saturation conditions.) Determine the precise logical relation between (LC) and (LC$^*$), where (LC$^*$) is the same condition with $I^{[q]}$ replaced by its tight closure $(I^{[q]})^*$: does one imply the other for every ideal of every such ring, are they equivalent, or is there a ring and ideal satisfying exactly one of the two?

The three are not logically independent. Because $F^n(R/I)\cong R/I^{[p^n]}$, the module form of part 2 restricted to the cyclic modules $R/I$ is exactly the local case of part 1; hence a counterexample to part 1 over a local ring refutes parts 1 and 2 at once, a counterexample to part 1 in the graded case refutes part 1 only, and a counterexample to part 2 for a general (non-cyclic) module refutes part 2 only. Part 3 stands apart from parts 1 and 2 and is settled either by proving that one of (LC), (LC$^*$) implies the other for every ideal of every such ring (in particular, that they are equivalent) or by exhibiting a ring and ideal for which exactly one of the two conditions holds.

The problem is posed in this full generality: no hypothesis of isolated singularity, existence of test elements, bounded dimension, finite projective dimension, finite F-representation type, or special singularity class may be assumed, since precisely those cases are what the existing literature has already settled.

References

1. Craig Huneke, The saturation of Frobenius powers of ideals, Communications in Algebra 28 (2000), 5563–5572, DOI 10.1080/00927870008827175, https://doi.org/10.1080/00927870008827175 (arXiv:1408.7100, https://arxiv.org/abs/1408.7100)
2. Hailong Dao and Ilya Smirnov, On the generalized Hilbert-Kunz function and multiplicity, Israel Journal of Mathematics 237 (2020), 155–184, DOI 10.1007/s11856-020-2003-2, https://doi.org/10.1007/s11856-020-2003-2 (arXiv:1305.1833, https://arxiv.org/abs/1305.1833)
3. Mohsen Asgharzadeh, On the (LC) conjecture, arXiv:1512.02518 (2015, revised v5 2016), https://arxiv.org/abs/1512.02518
4. Adela Vraciu, Local cohomology of Frobenius images over graded affine algebras, Journal of Algebra 228 (2000), 347–356, DOI 10.1006/jabr.1999.8266, https://doi.org/10.1006/jabr.1999.8266
5. Adela Vraciu, An observation on generalized Hilbert-Kunz functions, Proceedings of the American Mathematical Society 144 (2016), 3221–3229, DOI 10.1090/proc/13000, https://doi.org/10.1090/proc/13000 (arXiv:1510.00668, https://arxiv.org/abs/1510.00668)
6. Holger Brenner and Paul Monsky, Tight closure does not commute with localization, Annals of Mathematics (2) 171 (2010), 571–588, DOI 10.4007/annals.2010.171.571, https://doi.org/10.4007/annals.2010.171.571 (arXiv:0710.2913, https://arxiv.org/abs/0710.2913)
7. Daniel J. Hernández and Jack Jeffries, Local Okounkov bodies and limits in prime characteristic, Mathematische Annalen 372 (2018), 139–178, DOI 10.1007/s00208-018-1651-6, https://doi.org/10.1007/s00208-018-1651-6 (arXiv:1701.02575, https://arxiv.org/abs/1701.02575)
8. Mordechai Katzman, Finiteness of $\bigcup_e \operatorname{Ass} F^e(M)$ and its connections to tight closure, Illinois Journal of Mathematics 40 (1996), no. 2, 330–337, DOI 10.1215/ijm/1255986108, https://doi.org/10.1215/ijm/1255986108
9. Ian M. Aberbach and Thomas Polstra, Local cohomology bounds and the weak implies strong conjecture in dimension 4, Journal of Algebra 605 (2022), 37–57, DOI 10.1016/j.jalgebra.2022.04.020, https://doi.org/10.1016/j.jalgebra.2022.04.020
10. Wenliang Zhang, Lectures on the localization problem in tight closure, in: Lectures on Tight Closure and Its Applications, Contemporary Mathematics 831, American Mathematical Society (2025), 159–175, DOI 10.1090/conm/831/16635, https://doi.org/10.1090/conm/831/16635
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12. Sudipta Das, Stephen Landsittel, and Vinh Anh Pham, Transcendental Hilbert-Kunz multiplicities, arXiv:2607.28205 (2026), https://arxiv.org/abs/2607.28205