The (LC) and (LC*) conjectures: uniform linear bounds for saturating the Frobenius powers of homogeneous ideals

Studying whether tight closure commutes with localization, Huneke (2000) introduced two saturation conditions on the Frobenius powers $I^{[q]}$ ($q=p^{e}$) of a homogeneous ideal $I$ in a positively graded ring of positive characteristic: condition (LC) asks for a constant $N$ with $R_{\geq Nq}\cdot H^{0}_{m}(R/I^{[q]})=0$ for all $q$, and condition (LC*) asks the same with the tight closure $(I^{[q]})^{*}$ in place of $I^{[q]}$. He proved both for homogeneous ideals with $\dim R/I=1$, observed that they are equivalent in the presence of test elements, and wrote that, as far as he knew, all homogeneous ideals satisfy both. The original motivations have since been settled negatively: tight closure provably does not commute with localization (Brenner–Monsky 2010), and the aggregate associated primes of Frobenius powers can be infinite even over $F$-regular unique factorization domains (Katzman 1996; Singh–Swanson 2004). The boundedness questions themselves remain open and now underpin the existence theory of generalized Hilbert–Kunz multiplicities. The problem is to determine, by proof or counterexample, whether every homogeneous ideal satisfies (LC) and (LC*), and to settle the precise logical relationship between the two conditions.

Background

In a commutative Noetherian ring $R$ of positive characteristic $p>0$, the map $r\mapsto r^{p}$ is the Frobenius endomorphism. For an ideal $I$ and a power $q=p^{e}$ of the characteristic, the Frobenius power $I^{[q]}$ is the ideal generated by the $q$-th powers of all elements of $I$. Tight closure, introduced by Hochster and Huneke in 1990, is the closure operation $I\mapsto I^{*}$ defined as follows: $x\in I^{*}$ if there exists an element $c$ of $R$ lying in no minimal prime of $R$ such that $cx^{q}\in I^{[q]}$ for all sufficiently large $q$. Tight closure encodes characteristic-$p$ singularity information — for instance, weak $F$-regularity, the condition that all ideals are tightly closed, is an algebraic analogue of mild singularities — and it supplied uniform proofs of results such as the Briançon–Skoda theorem on integral closures of powers of ideals.

A fundamental structural question from the beginnings of the theory was whether tight closure commutes with localization: for a multiplicative system $S$ and an ideal $I$, is $(S^{-1}I)^{*}=S^{-1}(I^{*})$? The inclusion $S^{-1}(I^{*})\subseteq(S^{-1}I)^{*}$ is elementary; the reverse inclusion resisted all attacks through the 1990s (Aberbach–Hochster–Huneke proved early positive cases). Katzman reformulated the problem in terms of finiteness and boundedness of Frobenius-power data: localization would follow if the set of associated primes (primes that occur as annihilators of elements) of the tight-closure quotients $R/(I^{[q]})^{*}$ had only finitely many maximal elements, together with a uniform linear bound on the degrees needed to kill the zeroth local cohomology of those quotients.

To state the boundedness condition precisely, let $R$ be a Noetherian $\mathbb{N}$-graded ring over a field $R_{0}$ of characteristic $p$, with irrelevant ideal $m=\bigoplus_{i\geq 1}R_{i}$, and let $R_{\geq N}$ denote the ideal generated by all homogeneous elements of degree at least $N$. The saturation of an ideal $I$ with respect to $m$ is $I^{\mathrm{sat}}=\{r\in R:\ m^{n}r\subseteq I\ \text{for some}\ n\}$, and the zeroth local cohomology module $H^{0}_{m}(R/I)$ is the submodule of elements of $R/I$ killed by some power of $m$, so that $H^{0}_{m}(R/I)=I^{\mathrm{sat}}/I$; its Loewy length is the least $f$ with $m^{f}H^{0}_{m}(R/I)=0$. Huneke (2000) introduced two conditions. Condition (LC): there exists a constant $N$ such that $R_{\geq Nq}\cdot H^{0}_{m}(R/I^{[q]})=0$ for all $q=p^{e}$; equivalently $R_{\geq Nq}\cdot(I^{[q]})^{\mathrm{sat}}\subseteq I^{[q]}$ for all $q$. Condition (LC*): the same requirement with the tight closure $(I^{[q]})^{*}$ in place of $I^{[q]}$. Both assert that the Loewy length of the local cohomology of the Frobenius iterates grows at most linearly in $q$ — saturating a Frobenius power costs only linearly many degrees. In local rings one replaces $R_{\geq Nq}$ by $m^{Nq}$.

Huneke proved that (LC) and (LC*) hold, in equidimensional graded rings, for homogeneous ideals with $\dim R/I=1$ (recovering a theorem of Vraciu) and studied how the conditions transfer from an ideal $J$ to ideals of the form $(J,y)$ when the colon ideal $J:y=\{r\in R:\ ry\in J\}$ has large codimension. He observed that the two conditions are equivalent in the presence of test elements — elements $c$ in no minimal prime such that $cx^{q}\in I^{[q]}$ for all $q$ whenever $x\in I^{*}$, which exist for example over isolated singularities (rings whose singular locus is zero-dimensional), where they yield an $m$-primary test ideal uniformly comparable with $I^{[q]}$ — but outside such situations he could establish no implication between them, and he wrote: “As far as we know all homogeneous ideals $I$ satisfy both of these two conditions.”

The subsequent history transformed the surrounding landscape while leaving exactly these questions open. Brenner and Monsky (2010) constructed a three-dimensional normal hypersurface domain in characteristic two, an ideal $I$, an element $f$, and a multiplicative system $S$ with $f\in(S^{-1}I)^{*}$ but $f\notin S^{-1}(I^{*})$: tight closure does not commute with localization in general. On the finiteness side, Katzman (1996) had already produced the first ideal whose Frobenius powers have infinitely many associated primes in aggregate, and Singh–Swanson (2004) strengthened this drastically, exhibiting $F$-regular (that is, every ideal tightly closed — the strongest of the Frobenius-based regularity classes) unique factorization domains — even two-dimensional hypersurface domains over a field — for which $\bigcup_{e}\mathrm{Ass}(R/I^{[p^{e}]})$, and its tight-closure variant, has infinitely many maximal elements. Both of the original motivations for (LC)/(LC*) are therefore settled negatively in general; the boundedness conjectures themselves survive.

They survive with independent weight. Dao and Smirnov attached to an ideal $I$ the generalized Hilbert–Kunz function $gHK(I,q)=\ell(H^{0}_{m}(R/I^{[q]}))$, the finite-length defect between a Frobenius power and its saturation (the classical Hilbert–Kunz function of an $m$-primary ideal $\mathfrak a$ is the analogous $q\mapsto\ell(R/\mathfrak a^{[q]})$), and showed that condition (LC) forces strong regularity properties on it. Vraciu (2016) proved that if every ideal of $R$ satisfies (LC), then the generalized Hilbert–Kunz multiplicity $\lim_{q\to\infty}gHK(I,q)/q^{\dim R}$ exists for every ideal and the function is a $\mathbb{Z}$-linear combination of classical Hilbert–Kunz functions of $m$-primary ideals. Dao–Smirnov call (LC) a “widely open conjecture”; the strongest partial results remain the low-dimensional cases of Vraciu and Huneke and Asgharzadeh's theorem for homogeneous ideals in normal standard-graded Cohen–Macaulay rings (Cohen–Macaulay meaning a standard depth condition on local rings; standard-graded meaning generated by degree-one elements) of dimension below four, in certain cases where a cover of $\mathrm{Proj}(R/I)$ (the projective variety cut out by $I$) is regular, and for ideals of finite projective dimension.

Problem Statement

Let $R$ be a Noetherian $\mathbb{N}$-graded ring over a field $R_{0}$ of positive characteristic $p$, with irrelevant ideal $m=\bigoplus_{i\geq 1}R_{i}$, and let $q$ range over the powers $q=p^{e}$ of the characteristic; write $R_{\geq N}$ for the ideal generated by all homogeneous elements of $R$ of degree at least $N$. For a homogeneous ideal $I$ of $R$, write $I^{[q]}=(x^{q}:x\in I)$ for the Frobenius power, $(I^{[q]})^{\mathrm{sat}}=\{r\in R:\ m^{n}r\subseteq I^{[q]}\ \text{for some}\ n\}$ for its saturation, and $H^{0}_{m}(R/I^{[q]})=(I^{[q]})^{\mathrm{sat}}/I^{[q]}$ for the zeroth local cohomology module, the submodule of $R/I^{[q]}$ consisting of elements killed by some power of $m$. The tight closure of a homogeneous ideal $J$ is $J^{*}=\{x\in R:\ \text{there exists }c\in R\text{ lying in no minimal prime of }R\text{ such that }cx^{Q}\in J^{[Q]}\text{ for all sufficiently large }Q=p^{e}\}$, which is again a homogeneous ideal. Ideal $I$ satisfies condition (LC) if there is a constant $N$ such that

$R_{\geq Nq}\cdot H^{0}_{m}(R/I^{[q]})=0\quad\text{for all }q=p^{e},$

equivalently $R_{\geq Nq}\cdot(I^{[q]})^{\mathrm{sat}}\subseteq I^{[q]}$ for all $q$, and satisfies condition (LC*) if the same requirement holds with $(I^{[q]})^{*}$ in place of $I^{[q]}$. The local-ring analogues replace $R_{\geq Nq}$ by $m^{Nq}$. Determine, with proof or counterexample:

(1) whether every homogeneous ideal $I$ of every such graded ring satisfies condition (LC);

(2) whether every homogeneous ideal $I$ satisfies condition (LC*);

(3) what the precise logical relationship between conditions (LC) and (LC*) is — in particular, whether either condition implies the other for homogeneous ideals without auxiliary hypotheses such as the existence of test elements (elements $c$ of $R$ lying in no minimal prime such that $cx^{q}\in J^{[q]}$ for all $q$ whenever $x\in J^{*}$).

The asserted bound (or implication) holds at the stated generality — every homogeneous ideal in every Noetherian $\mathbb{N}$-graded ring over a field $R_{0}$ of positive characteristic $p$ — or fails, with an explicit graded ring of positive characteristic together with a homogeneous ideal witnessing the failure for infinitely many $q$, so that no constant $N$ can work.

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