---
abstract: 'Let $I$ denote an $R_+$-primary homogeneous ideal in a normal standard-graded Cohen-Macaulay domain over a field of positive characteristic $p$. We give a linear degree bound for the Frobenius powers $I^{[q]}$ of $I$, $q=p^{e}$, in terms of the minimal slope of the top-dimensional syzygy bundle on the projective variety ${\operatorname{Proj}}R$. This provides an inclusion bound for tight closure. In the same manner we give a linear bound for the Castelnuovo-Mumford regularity of the Frobenius powers $I^{[q]}$.'
address: 'Department of Pure Mathematics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH, United Kingdom'
author:
- Holger Brenner
bibliography:
- 'bibliothek.bib'
title: A linear bound for Frobenius powers and an inclusion bound for tight closure
---

Mathematical Subject Classification (2000): 13A35; 13D02; 14J60

Introduction {#introduction .unnumbered}
============

Let $R$ denote a noetherian ring, let ${\mathfrak{m}}$ denote a maximal ideal in $R$ and let $I$ denote an ${\mathfrak{m}}$-primary ideal. This means by definition that ${\mathfrak{m}}$ is the radical of $I$. Then there exists a (minimal) number $k$ such that ${\mathfrak{m}}^k \subseteq I \subseteq {\mathfrak{m}}$ holds. If $R$ contains a field of positive characteristic $p$, then the Frobenius powers of the ideal $I$, that is $$I^{[q]} = \{ f^q: f \in I\} \,  , \, \, \, q = p^{e} ,$$ are also ${\mathfrak{m}}$-primary and hence there exists a minimal number $k(q)$ such that ${\mathfrak{m}}^{k(q)} \subseteq I^{[q]}$ holds. In this paper we deal with the question how $k(q)$ behaves as a function of $q$, in particular we look for linear bounds for $k(q)$ from above. If ${\mathfrak{m}}^k \subseteq I$ and if $l$ denotes the number of generators for ${\mathfrak{m}}^k$, then we get the trivial linear inclusion $({\mathfrak{m}}^k)^{lq} \subseteq ({\mathfrak{m}}^k)^{[q]} \subseteq I^{[q]}$.

The main motivation for this question comes from the theory of tight closure. Recall that the tight closure of an ideal $I$ in a domain $R$ containing a field of positive characteristic $p$ is the ideal $$I^* = \{f \in R: \exists  0 \neq c \in R \mbox{ such that }
cf^q \in I^{[q]} \mbox{ for all } q=p^{e} \} \, .$$ A linear inclusion relation $ {\mathfrak{m}}^{\lambda q + \gamma} \subseteq I^{[q]}$ for all $q=p^{e}$ implies the inclusion ${\mathfrak{m}}^\lambda \subseteq
I^*$, since then we can take any element $0 \neq c \in {\mathfrak{m}}^\gamma$ to show for $f \in {\mathfrak{m}}^\lambda$ that $cf^q \in {\mathfrak{m}}^{\lambda q
+\gamma} \subseteq I^{[q]}$, hence $f \in I^*$. The trivial bound mentioned above yields ${\mathfrak{m}}^{kl} \subseteq I^*$, but in fact we have already $ {\mathfrak{m}}^{kl} \subseteq {\mathfrak{m}}^k\subseteq I$, so this does not yield anything interesting.

We restrict in this paper to the case of a normal standard-graded domain $R$ over an algebraically closed field $K=R_0$ of positive characteristic $p$ and a homogeneous $R_+$-primary ideal $I$. The question is then to find the minimal degree $k(q)$ such that $R_{\geq k(q)} \subseteq I^{[q]}$ or at least a good linear bound $k(q) \leq \lambda q+\gamma$. In this setting we work mainly over the normal projective variety $Y=  {\operatorname{Proj}}R$, endowed with the very ample invertible sheaf ${\mathcal{O}}_Y(1)$. If $I=(f_1 { , \ldots , }f_n)$ is given by homogeneous ideal generators $f_i$ of degree $d_i = \deg (f_i)$, then we get on $Y$ the following short exact sequences of locally free sheaves, $$0 {\longrightarrow}{\operatorname{Syz}}(f_1^q { , \ldots , }f_n^q)(m) {\longrightarrow}\bigoplus_{i=1}^n {\mathcal{O}}_Y(m -qd_i)
\stackrel{f_1^q { , \ldots , }f_n^q}{{\longrightarrow}} {\mathcal{O}}_Y(m) {\longrightarrow}0 \, .$$ Another homogeneous element $h \in R$ of degree $m$ yields a cohomology class $\delta(h) \in H^1(Y,{\operatorname{Syz}}(f_1^q { , \ldots , }f_n^q)(m))$, and therefore the question whether $h \in (f_1^q { , \ldots , }f_n^q)=I^{[q]}$ is equivalent to the question whether $\delta(h)=0$. Since ${\operatorname{Syz}}(f_1^q { , \ldots , }f_n^q)(0) =F^{*e} ({\operatorname{Syz}}(f_1 { , \ldots , }f_n)(0))$ is the pull-back under the $e$-th absolute Frobenius morphism $F^{e}:Y {\rightarrow}Y$, our question is an instance of the following more general question: given a locally free sheaf ${\mathcal{S}}$ on a normal projective variety $(Y,{\mathcal{O}}_Y(1))$, find an (affine-linear) bound $\ell (q)$ such that for $m \geq \ell (q)$ we have $H^1(Y, {\mathcal{S}}^q (m))=0$, where we set ${\mathcal{S}}^{q}=F^{e*}({\mathcal{S}})$. Using a resolution ${\mathcal{G}}_\bullet {\rightarrow}{\mathcal{S}}{\rightarrow}0$ where ${\mathcal{G}}_j=
\bigoplus_{(k,j)} {\mathcal{O}}_Y( -\alpha_{k,j})$, we can shift the problem (at least if $Y= {\operatorname{Proj}}R$ with $R$ Cohen-Macaulay, so that $H^{i}(Y,{\mathcal{O}}_Y(m))=0$ for $0< i < \dim (Y)$) to the problem of finding a bound such that $H^t(Y, {\mathcal{S}}_t^q(m)) =0$, where ${\mathcal{S}}_t =
{\operatorname{kern}}({\mathcal{G}}_t {\rightarrow}{\mathcal{G}}_{t-1})$ and $t = \dim (Y)$. By Serre duality this translates to ${\operatorname{Hom}}( {\mathcal{S}}_t^q(m), \omega_Y)=0$. Now the existence of such mappings is controlled by the minimal slope of ${\mathcal{S}}_t^q(m)$. Let $\bar{\mu}_{\min} ({\mathcal{S}}_t)= \lim \inf_{q=p^{e}}
\mu_{\min}({\mathcal{S}}_t^q)/q$ and set $\nu =- \bar{\mu}_{\min} ({\mathcal{S}}_t)/
\deg (Y)$. With these notations applied to ${\mathcal{S}}={\operatorname{Syz}}(f_1 { , \ldots , }f_n)(0)$ our main results are the following theorems (Theorems \[theoreminclusion\] and \[tightinclusion\]).

\[theoreminclusionintro\] Let $R$ denote a standard-graded normal Cohen-Macaulay domain over an algebraically closed field $K$ of characteristic $p >0$. Suppose that the dualizing sheaf $\omega_Y$ of $Y=  {\operatorname{Proj}}R$ is invertible. Let $I$ denote a homogeneous $R_+$-primary ideal. Then $R_{ > q \nu + \frac{\deg (\omega_Y)}{\deg(Y)} } \subseteq I^{[q]}$.

From this linear bound for the Frobenius powers we get the following inclusion bound for tight closure.

\[tightinclusionintro\] Under the assumptions of Theorem \[theoreminclusionintro\] we have the inclusion $R_{\geq  \nu}
\subseteq I^*$, where $I^*$ denotes the tight closure of $I$.

This theorem generalizes [@brennerslope Theorem 6.4] from dimension two to higher dimensions. We also obtain an inclusion bound for the Frobenius closure (Corollary \[frobenius\]) and a linear bound for the Castelnuovo-Mumford regularity of the Frobenius powers $I^{[q]}$ (Theorem \[regularitybound\]), which improves a recent result of M. Chardin [@chardinregularitypowers].

I thank M. Blickle for useful remarks.

Some projective preliminaries
=============================

Let $K$ denote an algebraically closed field and let $Y$ denote a normal projective variety over $K$ of dimension $t$ together with a fixed ample Cartier divisor $H$ with corresponding ample invertible sheaf ${\mathcal{O}}_Y(1)$. The degree of a coherent torsion-free sheaf ${\mathcal{S}}$ (with respect to $H$) is defined by the intersection number $\deg({\mathcal{S}})= \deg (c_1({\mathcal{S}}) )= c_1({\mathcal{S}}) . H^{t-1}$, see [@maruyamagrauertmuelich Preliminaries] for background of this notion. The degree is additive on short exact sequences [@maruyamagrauertmuelich Lemma 1.5(2)].

The slope of ${\mathcal{S}}$ (with respect to $H$), written $\mu({\mathcal{S}})$, is defined by dividing the degree through the rank. The slope fulfills the property that $\mu ({\mathcal{S}}_1 \otimes {\mathcal{S}}_2)= \mu({\mathcal{S}}_1) +
\mu({\mathcal{S}}_2)$ [@maruyamagrauertmuelich Lemma 1.5(4)]. The minimal slope of ${\mathcal{S}}$, $\mu_{\min} ({\mathcal{S}})$, is given by $$\mu_{\min} ({\mathcal{S}}) = \inf \{ \mu({\mathcal{Q}}):\, {\mathcal{S}}{\rightarrow}{\mathcal{Q}}{\rightarrow}0
\mbox{ is a torsion-free quotient sheaf} \} \, .$$ If ${\mathcal{S}}_1
{ \subset \ldots \subset }{\mathcal{S}}_k= {\mathcal{S}}$ is the Harder-Narasimhan filtration of ${\mathcal{S}}$ [@maruyamagrauertmuelich Proposition 1.13], then $\mu_{\min} ({\mathcal{S}}) = \mu({\mathcal{S}}/{\mathcal{S}}_{k-1})$. If ${\mathcal{L}}$ is an invertible sheaf and $\mu_{\min}({\mathcal{S}}) > \deg({\mathcal{L}})$, then there does not exist any non-trivial sheaf homomorphism ${\mathcal{S}}{\rightarrow}{\mathcal{L}}$. The sheaf ${\mathcal{S}}$ is called semistable if $ \mu({\mathcal{S}})
=\mu_{\min}({\mathcal{S}})$.

Suppose now that the characteristic of $K$ is positive and let $F^{e}\!:Y {\rightarrow}Y$ denote the $e$-th absolute Frobenius morphism. We denote the pull-back of ${\mathcal{S}}$ under this morphism by ${\mathcal{S}}^q
=F^{e*}({\mathcal{S}})$, $q=p^{e}$. The slope behaves like $\mu({\mathcal{S}}^q) = q
\mu ({\mathcal{S}})$ (this follows from [@maruyamagrauertmuelich Lemma 1.6], for which it is enough to assume that the finite mapping is flat in codimension one; note that we compute the slope always with respect to ${\mathcal{O}}_Y(1)$, not with respect to $F^{*e}({\mathcal{O}}_Y(1))={\mathcal{O}}_Y(q)$). It may however happen that $\mu_{\min}
({\mathcal{S}}^q) < q \mu_{\min} ({\mathcal{S}})$. Therefore it is useful to consider the number (compare [@langersemistable]) $$\bar{\mu}_{\min} ({\mathcal{S}})=\liminf_{q=p^{e}}  \mu_{\min} ({\mathcal{S}}^q)/q \, .$$ This limit exists, since there exists for some number $k$ a surjection $\oplus_j {\mathcal{O}}( \beta_j) {\rightarrow}{\mathcal{S}}(k) $ such that all $
\beta_j$ are positive. Then ${\mathcal{S}}(k)$ is a quotient of an ample bundle and so all its quotients have positive degree. This holds also for all its Frobenius pull-backs, hence $\mu_{\min}
(({\mathcal{S}}(k))^q) \geq 0$ and the limit is $\geq 0$. Thus $\mu_{\min}
({\mathcal{S}}^q) \geq -q k \deg ({\mathcal{O}}_Y(1))$ for all $q$. Moreover, a theorem of Langer implies that this limit is even a rational number, see [@langersemistable]. The sheaf ${\mathcal{S}}$ is called strongly semistable if $\mu({\mathcal{S}})=\bar{\mu}_{\min} ({\mathcal{S}})$; equivalently, if all Frobenius pull-backs ${\mathcal{S}}^q$ are semistable.

The degree of the variety $Y$ (with respect to $H$) is by definition the top self intersection number $\deg(Y) = \deg({\mathcal{O}}_Y(1))=H^t$. In the following we will impose on a polarized variety $(Y, {\mathcal{O}}_Y(1))$ of dimension $t$ the condition that $H^i(Y,{\mathcal{O}}_Y(m))=0 $ for $i=1
{ , \ldots , }t-1$ and all $m$. If $Y= {\operatorname{Proj}}R$, where $R$ is a standard-graded Cohen Macaulay ring, this property holds true due to [@brunsherzog Theorem 3.5.7].

\[sheafproposition\] Let $Y$ denote a normal projective variety of dimension $t \geq 1$ over an algebraically closed field $K$ of positive characteristic $p$. Let ${\mathcal{O}}_Y(1)$ denote a very ample invertible sheaf on $Y$ such that $H^i(Y,{\mathcal{O}}(m))=0 $ for $i=1
{ , \ldots , }t-1$. Suppose that the dualizing sheaf $\omega_Y$ on $Y$ is invertible. Let ${\mathcal{S}}$ denote a torsion-free coherent sheaf on $Y$. Suppose that the stalk ${\mathcal{S}}_y$ is free for every non-smooth point $y \in Y$. Let $$\cdots {\longrightarrow}{\mathcal{G}}_3 {\longrightarrow}{\mathcal{G}}_2 {\longrightarrow}{\mathcal{S}}{\longrightarrow}0$$ denote an exact complex of sheaves, where ${\mathcal{G}}_j$ has type ${\mathcal{G}}_j=
\bigoplus_{(k,j) } {\mathcal{O}}_Y(- \alpha_{k,j})$. Set ${\mathcal{S}}_j={\operatorname{im}}({\mathcal{G}}_{j+1}
{\rightarrow}{\mathcal{G}}_{j})= \ker( {\mathcal{G}}_{j} {\rightarrow}{\mathcal{G}}_{j-1})$, $j \geq 2$, and ${\mathcal{S}}_1={\mathcal{S}}$. Fix $i=1 { , \ldots , }t$. Then for $$m > - q \frac{  \bar{\mu}_{\min}( {\mathcal{S}}_{t-i+1}) }{\deg(Y)} + \frac{\deg(\omega_Y)}{\deg(Y)}$$ we have $H^i (Y, {\mathcal{S}}^q (m)) =0$.

Note first that the Frobenius acts flat on the exact complex and on the corresponding short exact sequences $0 {\rightarrow}{\mathcal{S}}_{j+1} {\rightarrow}{\mathcal{G}}_{j+1} {\rightarrow}{\mathcal{S}}_j {\rightarrow}0$. This can be checked locally and is true for the smooth points of $Y$. Over a singular point $y \in Y$ the sheaf ${\mathcal{S}}$ is free, so these short exact sequences split locally in a neighborhood of such a point and hence all the ${\mathcal{S}}_j$ are also free in $y$. So also in these points the Frobenius preserves the exactness of the complex.

Due to our assumption on ${\mathcal{O}}_Y(1)$ we have $H^i(Y,{\mathcal{G}}_j(m))=0 $ for $i=1 { , \ldots , }t-1$ and all $m$ and all $j \geq 2$. Hence from the short exact sequences $0 {\rightarrow}{\mathcal{S}}_{j+1}(m) {\rightarrow}{\mathcal{G}}_{j+1}(m) {\rightarrow}{\mathcal{S}}_j(m) {\rightarrow}0$ we can infer that $$\begin{aligned}
& & H^i (Y, {\mathcal{S}}_j(m)) \cong H^{i+1}(Y, {\mathcal{S}}_{j+1}(m)) \, \, \,
\mbox{ isomorphisms  for  } i=1 { , \ldots , }{t}-2, \cr & & H^{{t}-1} (Y, {\mathcal{S}}_j(m)) \subseteq H^{ {t}}(Y, {\mathcal{S}}_{j+1}(m)) \, \, \,
\mbox{ injection for } t \geq 2 , \cr & & H^{{t}} (Y, {\mathcal{G}}_{j+1}
(m)) {\rightarrow}H^{{t}} (Y, {\mathcal{S}}_j(m)) \, \, \, \mbox{ surjection}.\end{aligned}$$ The same is true if we replace $S_j$ and $G_j$ by their Frobenius pull-backs $S_j^q$ and $G_ j^q$. For $i=1 { , \ldots , }{t}$ we find $$H^{i}(Y, {\mathcal{S}}^q_1 (m)) \!\cong \! H^{i+1}(Y, {\mathcal{S}}^q_2 (m)) \! { \cong \ldots \cong }\!
H^{{t}-1} (Y, {\mathcal{S}}^q_{ {t}-i} (m)) \! \subseteq \! H^{{t}} (Y, {\mathcal{S}}^q _{{t}-i+1}(m)) .$$ So we only have to look at $H^{t}(Y, {\mathcal{S}}^q_{t-i+1}(m))$, which is by Serre duality dual to ${\operatorname{Hom}}({\mathcal{S}}^q_{t-i+1}(m), \omega_Y)$, see [@hartshornealgebraic Theorem III.7.6]. Suppose now that $m$ fulfills the numerical condition. Then $$\begin{aligned}
\mu_{\min}({\mathcal{S}}^q_{t-i+1}(m)) &=& \mu_{\min}({\mathcal{S}}^q_{t-i+1})+ m \deg(Y) \cr
&\geq & q \bar{\mu}_{\min} ({\mathcal{S}}_{t-i+1}) + m \deg (Y) \cr
&>& q  \bar{\mu}_{\min}( {\mathcal{S}}_{t-i+1})
+ \big(- q \frac{ \bar{\mu}_{\min}( {\mathcal{S}}_{t-i+1}) }{\deg(Y)}
+ \frac{\deg(\omega_Y)}{\deg(Y)} \big) \deg(Y) \cr
&=& \deg(\omega_Y)  \, .\end{aligned}$$ So for these $m$ there are no non-trivial mappings from ${\mathcal{S}}_{t-i+1}^q(m)$ to $\omega_Y$ and therefore $H^{t}(Y, {\mathcal{S}}_{t-i+1}^q(m))=0$.

The dualizing sheaf $\omega_Y$ on the projective variety $Y
\subseteq {\mathbb{P}}^N$ is invertible under the condition that $Y$ is locally a complete intersection in ${\mathbb{P}}^N$ and in particular if $Y$ is smooth (see [@hartshornealgebraic Theorem III.7.11 and Corollary III.7.12]. If $\omega_Y$ is not invertible, but torsion-free, then we may replace $\deg(\omega_Y)$ by $\mu_{\max}(\omega_Y)$ to get the same statement as in Proposition \[sheafproposition\].

An inclusion bound for tight closure {#inclusion}
====================================

We first fix the following situation, with which we will deal in this section.

\[situation\] Let $K$ denote an algebraically closed field of characteristic $p >0$. Let $R$ denote a standard-graded normal Cohen-Macaulay domain of dimension $t+1 \geq 2$ over $K$ with corresponding projective normal variety $Y={\operatorname{Proj}}R$. Suppose that the dualizing sheaf $\omega_Y$ of $Y$ is invertible. Let $I
\subseteq R$ denote a homogeneous $R_+$-primary ideal. Let $$\cdots  {\longrightarrow}F_2= \bigoplus_{(k,2)}  R(- \alpha_{k,2})
{\longrightarrow}F_1= \bigoplus_{(k,1)} R(- \alpha_{k,1}) {\longrightarrow}I {\longrightarrow}0 \, ,$$ denote a homogeneous complex of graded $R$-modules which is exact on $D(R_+)$. Let $$\cdots {\longrightarrow}{\mathcal{G}}_2=\bigoplus_{(k,2)} {\mathcal{O}}(- \alpha_{(k,2)}) {\longrightarrow}{\mathcal{G}}_1= \bigoplus_{(k,1)} {\mathcal{O}}(- \alpha_{k,1})
 {\longrightarrow}{\mathcal{O}}_Y {\longrightarrow}0 \,$$ denote the corresponding exact complex of sheaves on $Y$. Denote by ${\operatorname{Syz}}_j = {\operatorname{kern}}({\mathcal{G}}_j {\rightarrow}{\mathcal{G}}_{j-1})$ the locally free kernel sheaves on $Y$, and set ${\operatorname{Syz}}_j(m)= {\operatorname{Syz}}_j \otimes {\mathcal{O}}_Y(m)$. Let $\nu =- \bar{\mu}_{\min} ({\operatorname{Syz}}_t) /\deg (Y)$, where $t$ is the dimension of $Y$.

\[theoreminclusion\] Suppose the situation and notation described in \[situation\]. Then for all prime powers $q=p^{e}$ we have the inclusion $R_{> q \nu + \frac{\deg(\omega_Y) }{\deg(Y)}}
\subseteq I^{[q]}$.

Since $I$ is primary all the syzygy sheaves occurring in the resolution on $Y$ are locally free and hence we may apply Proposition \[sheafproposition\]. Fix a prime power $q=p^{e}$. Let $h \in R$ denote a homogeneous element of degree $m > q \nu +
\frac{\deg(\omega_Y) }{\deg(Y)}$. This gives via the short exact sequence on $Y$, $$0 {\longrightarrow}{\operatorname{Syz}}(f_1^q { , \ldots , }f_n^q)(m) {\longrightarrow}\bigoplus_{i=1}^n {\mathcal{O}}_Y(m -qd_i)
\stackrel{f_1^q { , \ldots , }f_n^q}{{\longrightarrow}} {\mathcal{O}}_Y(m) {\longrightarrow}0$$ rise to a cohomology class $\delta(h) \in H^1(Y,{\operatorname{Syz}}(f_1^q { , \ldots , }f_n^q)(m))$, where $${\operatorname{Syz}}(f_1^q { , \ldots , }f_n^q)(m) =(F^{e*}({\operatorname{Syz}}(f_1
{ , \ldots , }f_n))) (m)= {\mathcal{S}}^q (m) \, ,$$ ${\mathcal{S}}= {\mathcal{S}}_1= {\operatorname{Syz}}(f_1
{ , \ldots , }f_n)$. It is enough to show that $\delta (h)=0$, for then $h \in I^{[q]} \Gamma(D(R_+), {\mathcal{O}}) =I^{[q]}$, since $R$ is normal. But this follows from Proposition \[sheafproposition\] applied to ${\mathcal{S}}= {\operatorname{Syz}}(f_1 { , \ldots , }f_n)$ and $i=1$.

\[resolutionremark\] We do not insist that the “resolution” of the ideal is exact on the whole ${\operatorname{Spec}}R$ nor that it is minimal, but it is likely that a minimal resolution will give us in general a better bound $\nu$. For example we can always use the Koszul complex given by ideal generators of the $R_+$-primary ideal $I$.

The next theorem gives an inclusion bound for tight closure. Recall that the tight closure of an ideal $I \subseteq R$ in a noetherian domain containing a field of positive characteristic $p$ is by definition the ideal $$I^* = \{f \in R: \exists 0 \neq  c \in R \mbox{ such that } cf^q \in I^{[q]}
\mbox{ for all } q=p^{e} \} \, .$$ See [@hunekeapplication] for basic properties of this closure operation.

\[tightinclusion\] Suppose the situation described in \[situation\]. Then we have the inclusion $R_{\geq  \nu} \subseteq
I^*$.

Let $f \in R$ be a homogeneous element of degree $\deg (f)=m
\geq \nu =- \bar{\mu}_{\min}({\operatorname{Syz}}_t) /\deg(Y)$. Due to the definition of tight closure we have to show that $cf^q \in I^{[q]}$ holds for some $c \neq 0$ and all prime powers $q$. Let $c \neq 0$ be any homogeneous element of degree $ >  \deg (\omega_Y)/\deg (Y)$. Then $\deg (cf^q) = qm +\deg(c) > q \nu + \deg(\omega_Y)/\deg (Y)$ and therefore $cf^q \in I^{[q]}$ by Theorem \[theoreminclusion\].

Suppose that $R$ fulfills the condition of the situation described in \[situation\] and let $I=(f_1 { , \ldots , }f_n)$ denote an ideal generated by a full regular system of homogeneous parameters of degree $\deg(f_i)=d_i$ (so $n=t+1$). Then the Koszul resolution of these elements gives a resolution on $Y={\operatorname{Proj}}R$ such that the top-dimensional syzygy bundle is invertible, namely $${\operatorname{Syz}}_t (m)={\mathcal{G}}_{t+1}(m)= {\mathcal{O}}_Y(m-d_1 - \cdots -d_{t+1} ) \,$$ Then Theorem \[tightinclusion\] gives the known (even without the condition Cohen-Macaulay) inclusion bound $R_{\geq d_1 { + \ldots + }d_n} \subseteq (f_1 { , \ldots , }f_n)^*$, see [@hunekeparameter Theorem 2.9].

The next easiest case is then the $R_+$-primary homogeneous ideal $I$ has finite projective dimension (it is again enough to impose the exactness only on $D(R_+)$). In this case the resolution on $Y$ looks like $$0 {\longrightarrow}{\mathcal{G}}_{t+1} {\longrightarrow}{\mathcal{G}}_t { {\longrightarrow}\ldots {\longrightarrow}}{\mathcal{G}}_1 {\longrightarrow}{\mathcal{O}}_Y{\longrightarrow}0$$ and the top-dimensional syzygy bundle is ${\operatorname{Syz}}_t = {\mathcal{G}}_{t+1}
=\bigoplus_k {\mathcal{O}}_Y( - \alpha_{k,t+1})$, and therefore $$\mu_{\min}({\operatorname{Syz}}_t)= \deg(Y) \min_k \{ - \alpha_{k,t+1}\} = -
\deg(Y) \max_k \{ \alpha_{k,t+1} \} \, .$$ The corresponding inclusion bound was proved in [@hunekesmithkodaira Theorem 5.11]. Such a situation arises for example if $I$ is generated by a set of monomials in a system of homogeneous parameters.

The following easy corollary unifies two known inclusion bounds for tight closure given by K. Smith (see [@smithgraded Propositions 3.1 and 3.3]), namely that $R_{\geq \sum_{i=1}^n \deg (f_i)}
\subseteq I^*$ and that $R_{\geq \dim(R) \max _i \{ \deg(f_i)\}}
\subseteq I^*$.

\[tightcorollary\] Suppose the situation described in \[situation\] and suppose that the homogeneous $R_+$-primary ideal $I=(f_1 { , \ldots , }f_n)$ is generated by homogeneous elements of degree $d_i= \deg(f_i)$. Set $d= \max _{1 \leq i_1 < \ldots <
i_{\dim (R)} \leq n}( d_{i_1} { + \ldots + }d_{i_{\dim(R)}} )$. Then $R
_{\geq d} \subseteq I^*$.

We consider the Koszul resolution of $I=(f_1 { , \ldots , }f_n)$, which is exact outside the origin. This gives the surjection $$\bigoplus _{1 \leq i_1 < \ldots < i_{\dim (R)} \leq n}
{\mathcal{O}}(-d_{i_1} { - \ldots - }d_{i_{\dim(R)}} ) {\longrightarrow}{\operatorname{Syz}}_{\dim(R)-1 } {\longrightarrow}0$$ which shows that $$\begin{aligned}
\bar{\mu}_{\min} ({\operatorname{Syz}}_{\dim(R)-1}) &\geq&  \bar{\mu}_{\min}
\big(\bigoplus _{1 \leq i_1 < \ldots < i_{\dim (R)} \leq n}
{\mathcal{O}}(-d_{i_1} { - \ldots - }d_{i_{\dim(R)}}) \big) \cr &=& - \max _{1
\leq i_1 < \ldots < i_{\dim (R)} \leq n} \{ d_{i_1} { + \ldots + }d_{i_{\dim(R)}}   \} \deg (Y) \, .\end{aligned}$$ Hence $\nu =- \bar{\mu}_{\min} ( {\operatorname{Syz}}_{\dim(R)-1})/ \deg (Y) \leq
\max \{d_{i_1} { + \ldots + }d_{i_{\dim(R)}} \}$ and Theorem \[tightinclusion\] applies.

If the dimension of $R$ is two, then Theorem \[tightinclusion\] was proved in [@brennerslope Theorem 6.4] using somewhat more geometric methods. In this case $Y=  {\operatorname{Proj}}R$ is a smooth projective curve and the top syzygy bundle is just the first syzygy bundle, and the result also holds in characteristic zero for solid closure. See [@brennerslope] and [@brennercomputationtight] for concrete computations of the number $\nu$ in this case. It is in general difficult to compute the number $\nu$ of the theorem, as it is difficult to compute the minimal slope of a locally free sheaf.

The following corollary gives an inclusion bound for tight closure under the condition that the top-dimensional syzygy bundle is strongly semistable. In the two-dimensional situation this bound is exact, in the sense that below this bound an element belongs to the tight closure only if it belongs to the ideal itself, see [@brennerslope Theorem 8.4].

\[topstable\] Suppose the situation described in \[situation\] and let $I=(f_1 { , \ldots , }f_n)$ be generated by homogeneous elements of degree $d_i=\deg(f_i)$. Let $F_\bullet {\rightarrow}I$ denote the Koszul complex and suppose that the top-dimensional syzygy bundle ${\operatorname{Syz}}_t$ is strongly semistable. Set $d= (\dim (R)-1) (d_1 { + \ldots + }d_n)/
(n-1)$. Then $R_{\geq d} \subseteq I^*$.

The condition strongly semistable means that $\mu({\operatorname{Syz}}_t)= \bar{\mu}_{\min}({\operatorname{Syz}}_t)$. So we only have to compute the degree and the rank of ${\operatorname{Syz}}_t$. It is easy to compute that $\det({\operatorname{Syz}}_t)= {\mathcal{O}}_Y(\binom{n-2}{t-1}(- \sum_{i=1}^n d_i))$, hence $$\deg( {\operatorname{Syz}}_t) = \binom{n-2}{t-1}(- \sum_{i=1}^n d_i) \deg (Y)$$ and ${\operatorname{rk}}({\operatorname{Syz}}_t)= \binom{n-1}{t}$. Therefore $$\mu( {\operatorname{Syz}}_t)=  \binom{n-2}{t-1} (- \sum_{i=1}^n d_i) \deg (Y) / \binom{n-1}{t}
= \frac{t}{n-1} (- \sum_{i=1}^n d_i) \deg (Y)$$ and $\nu = \frac{t}{n-1} ( \sum_{i=1}^n d_i)$.

As the proofs of Theorem \[tightinclusion\] and Proposition \[sheafproposition\] show, Corollary \[topstable\] is also true under the weaker condition that there does not exist any non-trivial mapping ${\operatorname{Syz}}_t^q {\rightarrow}{\mathcal{L}}$ to any invertible sheaf ${\mathcal{L}}$ contradicting the semistability of ${\operatorname{Syz}}_t^q$ for all $q=p^{e}$.

Theorem \[tightinclusion\] applies in particular when $R$ is a normal complete intersection domain. Let $R=K[X_1 { , \ldots , }X_N]/(H_1
{ , \ldots , }H_r)$, where $H_j$ are homogeneous forms of degree $\delta_j$. Then $\omega_Y = {\mathcal{O}}(\sum_j \delta_j -N)$. Therefore the number $ \deg(\omega_Y)/ \deg(Y)= \sum_j \delta_j -N$ is just the $a$-invariant of $R$.

We want to apply Corollary \[topstable\] to the computation of the tight closure $(x^a,y^a,z^a,w^a)^*$ in $R=K[x,y,z,w]/(H)$, where $H$ is supposed to be a polynomial of degree $4$ defining a smooth projective (hyper-)surface $$Y= V_+(H)={\operatorname{Proj}}R \subset {\mathbb{P}}^3 ={\operatorname{Proj}}K[x,y,z,w]$$ of degree $4$; hence $Y$ is a $K 3$ surface. Our result will only hold true for generic choice of $H$. We look at the Koszul complex on ${\mathbb{P}}^3$ defined by $x^a,y^a,z^a,w^a$ and break it up to get $$0 {\longrightarrow}{\operatorname{Syz}}_2 \cong \bigwedge^2 {\operatorname{Syz}}{\longrightarrow}\bigoplus_6 {\mathcal{O}}_{{\mathbb{P}}^3}(-2a)
{\longrightarrow}\bigoplus_4 {\mathcal{O}}_{{\mathbb{P}}^3}(-a) {\longrightarrow}{\mathcal{O}}_{{\mathbb{P}}^3} {\longrightarrow}0 \, .$$ Suppose first that $K$ is an algebraically closed field of characteristic $0$. It is easy to see that the syzygy bundle ${\operatorname{Syz}}={\operatorname{Syz}}(x^a,y^a,z^a,w^a)$ is semistable on ${\mathbb{P}}^3$ [@brennerlookingstable Corollary 3.6 or Corollary 6.4]. Therefore also the exterior power ${\operatorname{Syz}}_2 \cong \bigwedge^2 {\operatorname{Syz}}$ is semistable on ${\mathbb{P}}^3$. By the restriction theorem of Flenner [@flennerrestriction Theorem 1.2] it follows that the restriction ${\operatorname{Syz}}_2 \!|_Y$ is also semistable on the generic hypersurface $Y=V_+(H)$.

On the other hand, due to the Theorem of Noether (see [@haramp §IV.4]), every curve on the generic surface of degree $4$ in ${\mathbb{P}}^3$ is a complete intersection and $R=K[x,y,z,w]/(H)$ is a factorial domain for generic $H$ of degree $4$. It follows that the cotangent bundle $\Omega_Y$ on $Y=V_+(H)$ is semistable. For the semistability of a rank two bundle we only have to look at mappings ${\mathcal{L}}{\rightarrow}\Omega_Y$, where ${\mathcal{L}}$ is invertible. But since ${\mathcal{L}}=
{\mathcal{O}}_Y(k)$, the semistability follows, since $Y$ is a $K3$ surface and so $\Omega_Y$ has degree $0$ but does not have any global non-trivial section (see [@griffithsharris IV. 5]).

So for $H$ generic the relevant second syzygy bundle ${\operatorname{Syz}}_2 \!|_Y$ and the cotangent bundle $\Omega_Y$ are both semistable in characteristic $0$. Since the ${\mathbb{Q}}$-rational points are dense in ${\mathbb{A}}^N_K$, there exist also such polynomials $H$ with rational coefficients and then also with integer coefficients. We consider such a polynomial $H$ with integer coefficients as defining a family of quartics over ${\operatorname{Spec}}{\mathbb{Z}}$. Since semistability is an open property, we infer that the second syzygy bundle and the cotangent bundle are also semistable on $Y_p=V_+(H_p)$ for $p \gg 0$.

By the semistability of $\Omega_{Y_p}$ ($p \gg 0$), the maximal slope of $\Omega_{Y_p}$ is $\leq 0$. A theorem of Langer [@langersemistable Corollary 2.4 and Corollary 6.3] shows then that every semistable bundle on $Y_p$ is already strongly semistable. Hence the second syzygy bundle is also strongly semistable. Therefore we are in the situation of Corollary \[topstable\] and we compute $d=8a/3$. Thus $$R_{8a/3}\subseteq (x^a,y^a,z^a,w^a)^*$$ holds in $R=K[x,y,z,w]/(H)$ for $H$ generic of degree $4$ and for $p \gg 0$. The first non-trivial instance is for $a=3$. In fact for the (non-generic) Fermat quartic $x^4+y^4+z^4+w^4=0$ it was proved by Singh in [@singhcomputation Theorem 4.1] directly that $x^2y^2z^2w^2 \in
(x^3,y^3,z^3,w^3)^*$.

For the next corollary we recall the definition of the Frobenius closure. Suppose that $R$ is a noetherian ring containing a field of positive characteristic $p >0$, and let $I$ denote an ideal. Then the Frobenius closure of $I$ is defined by $$I^F =\{ f \in R:\, \exists q=p^{e} \mbox{ such that } f^q \in I^{[q]} \} \, .$$ It is easy to see that the Frobenius closure of an ideal is contained in its tight closure.

\[frobenius\] Suppose the situation described in \[situation\]. Then $R_{> \nu} \subseteq I^F$, the Frobenius closure of $I$.

Let $f$ denote a homogeneous element of degree $m=\deg(f) >
\nu=- \bar{\mu}_{\min} ({\operatorname{Syz}}_t)/\deg(Y)$. Then we just have to take a prime power $q=p^{e}$ such that $\deg(f^q)=qm >q \nu+
\deg(\omega_Y)/ \deg(Y)$ holds. Then $f^q \in I^{[q]}$ holds due to Theorem \[theoreminclusion\].

Corollary \[frobenius\] is not true for $R_{\geq \nu}$ instead of $R_{> \nu }$. This is already clear for parameter ideals in dimension two, say for $(x,y)$ in $R=K[x,y,z]/(H)$, where $H$ defines a smooth projective curve $Y= {\operatorname{Proj}}R =V_+(H) \subset
{\mathbb{P}}^2$. Here we have the resolution $$0 {\longrightarrow}{\mathcal{O}}_Y(-2) \cong {\operatorname{Syz}}(x,y)(0) {\longrightarrow}{\mathcal{O}}_Y(-1) \oplus {\mathcal{O}}_Y(-1) \stackrel{x,y}{{\longrightarrow}}
{\mathcal{O}}_Y {\longrightarrow}0 \, .$$ Hence we get $\nu =2$, but an element of degree two (say $z^2$) does not in general belong to the Frobenius closure of $(x,y)$.

A problem of Katzman and Sharp (see [@katzmansharpfrobenius]) asks in its strongest form: does there exist a number $b$ such that whenever $f \in I^F$ holds, then already $f^{p^b} \in I^{[p^b]}$ holds. A positive answer (together with the knowledge of a bound for the number $b$) to this question would give a finite test to check whether a given element $f$ belongs to the Frobenius closure $I^F$ or not. For those elements which belong to $I^{F}$ because of Corollary \[frobenius\] (due to degree reasons, so to say), the answer is yes, at least in the sense that for $f$ fulfilling $\deg(f) \geq \nu + \epsilon$ ($\epsilon
>0$) we have $ \deg (f^q) = q \deg(f) \geq q \nu + q \epsilon$, so the condition $q \epsilon > \deg(\omega_Y)/\deg(Y)$ is sufficient to ensure that $f^q \in I^{[q]}$. It is however possible that elements of degree $\deg(f) \leq \nu$ belong to the Frobenius closure.

The Castelnuovo-Mumford regularity of Frobenius powers {#cmregularity}
======================================================

We recall briefly the notion of the Castelnuovo-Mumford regularity following [@brodmannsharp Definition 15.2.9]. Let $R$ denote a standard-graded ring and let $M$ denote a finitely generated graded $R$-module. Then the Castelnuovo-Mumford regularity of $M$ (or regularity of $M$ for short) is $${\operatorname{reg}}(M) = \sup \{ {\operatorname{end}}(H^i_{R_+} (M)) +i :\, 0 \leq i \leq \dim M \} \, ,$$ where ${\operatorname{end}}(N)$ of a graded $R$-module $N$ denotes the maximal degree $e$ such that $N_e \neq 0$. For a number $l$ we define the regularity ${\operatorname{reg}}^l (M)$ at and above level $l$ by $${\operatorname{reg}}^l(M)= \sup \{ {\operatorname{end}}(H^i_{R_+} (M))+i :\, l \leq i \leq \dim M \} \, ,$$

A question of M. Katzman raised in [@katzmanfrobenius Introduction] asks how the regularity of the Frobenius powers $I^{[q]}$ behaves, in particular whether there exists a linear bound ${\operatorname{reg}}(I^{[q]}) \leq C_1 q + C_0$. Such a linear bound for the regularity of the Frobenius powers of an ideal was recently given by M. Chardin in [@chardinregularitypowers Theorem 2.3]. The following theorem gives a better linear bound for the regularity of Frobenius powers of $I$ in terms of the slope of the syzygy bundles.

\[regularitybound\] Let $K$ denote an algebraically closed field of positive characteristic $p$. Let $R$ denote a standard-graded normal Cohen-Macaulay $K$-domain of dimension $t+1 \geq 2$. Let $I=(f_1 { , \ldots , }f_n) \subseteq R$ denote a homogeneous ideal generated by homogeneous elements of degree $d_i =\deg(f_i)$. Suppose that the dualizing sheaf $\omega_Y$ on $Y={\operatorname{Proj}}R$ is invertible. Suppose that the points $y \in \sup ({\mathcal{O}}_Y/{\mathcal{I}})$ are smooth points of $Y$. Let $F_\bullet {\rightarrow}I$ denote a graded free resolution with corresponding exact complex of sheaves on $Y$, $
{\mathcal{G}}_\bullet {\rightarrow}{\mathcal{I}}\subseteq {\mathcal{O}}_Y $. Set ${\operatorname{Syz}}_j = \ker({\mathcal{G}}_j
{\rightarrow}{\mathcal{G}}_{j-1})$. Then we have for the Castelnuovo-Mumford regularity of the Frobenius powers $I^{[q]}$ the linear bound ${\operatorname{reg}}(I^{[q]}) \leq C_1q + C_0 $, where $$C_1= \max \{ d_i, i=1 { , \ldots , }n, \,
- \frac{ \bar{\mu}_{\min}({\operatorname{Syz}}_j)}{\deg(Y)},\, j=1 { , \ldots , }t =\dim(Y) \} \mbox{ and }$$ $$C_0 = \max \{ {\operatorname{reg}}(R), \frac{\deg(\omega_Y)}{\deg(Y) } \}
\, .$$

The ideal generators define for $q=p^{e}$ the homogeneous short exact sequences $$0 {\longrightarrow}{\operatorname{Syz}}(f_1^q { , \ldots , }f_n^q) {\longrightarrow}\bigoplus_{i=1}^n R(-qd_i)
\stackrel{f_1^q { , \ldots , }f_n^q}{ {\longrightarrow}} I^{[q]} {\longrightarrow}0$$ of graded $R$-modules. It is an easy exercise [@brodmannsharp Exc. 15.2.15] to show that for a short exact sequence $0
{\rightarrow}L {\rightarrow}M {\rightarrow}N {\rightarrow}0$ we have ${\operatorname{reg}}(N) \leq \max \{ {\operatorname{reg}}^1(L) -1,
{\operatorname{reg}}(M) \}$. We have ${\operatorname{reg}}(R(-qd)) = {\operatorname{reg}}(R) +qd$ and $${\operatorname{reg}}( \bigoplus_{i=1}^n R(-qd_i)) = \max_i \{{\operatorname{reg}}(R(-qd_i)) \}
={\operatorname{reg}}(R) + q \max_i \{ d_i \} \, ,$$ which gives the first terms in the definition of $C_1$ and $C_0$ respectively. Hence it is enough to give a linear bound for ${\operatorname{reg}}^1({\operatorname{Syz}}(f_1^q { , \ldots , }f_n^q))$. Moreover, the long exact local cohomology sequence associated to the above short exact sequence gives $${\longrightarrow}H^0_{R_+}(I^{[q]}) {\longrightarrow}H^1_{R_+} ({\operatorname{Syz}}(f_1^q { , \ldots , }f_n^q)) {\longrightarrow}\bigoplus_{i=1}^n
H^1_{R_+} (R(-qd_i)) {\longrightarrow}\, .$$ The term on the right is $0$, since $R$ is Cohen-Macaulay, and the term on the left is $0$, since $R$ is a domain. Therefore $H^1_{R_+} ({\operatorname{Syz}}(f_1^q { , \ldots , }f_n^q))
=0$ and we have to find a linear bound for $ {\operatorname{reg}}^2({\operatorname{Syz}}(f_1^q
{ , \ldots , }f_n^q)) ={\operatorname{reg}}^1({\operatorname{Syz}}(f_1^q { , \ldots , }f_n^q))$. We have $H^i_{R_+}({\operatorname{Syz}}(f_1^q { , \ldots , }f_n^q) )= H^{i-1} (D(R_+), {\operatorname{Syz}}(f_1^q
{ , \ldots , }f_n^q) \widetilde{\, } \, )$ for $i \geq 2$ due to the long exact sequence relating local cohomology with sheaf cohomology. Denote now by ${\operatorname{Syz}}(f_1^q { , \ldots , }f_n^q) $ the corresponding torsion-free sheaf on $Y={\operatorname{Proj}}R$. On $Y$ we have the short exact sequences of sheaves $$0 {\longrightarrow}{\operatorname{Syz}}(f_1^q { , \ldots , }f_n^q)
{\longrightarrow}\bigoplus_{i=1}^n {\mathcal{O}}_Y(-qd_i) {\longrightarrow}{\mathcal{I}}^{[q]} {\longrightarrow}0  \, .$$ We may compute the cohomology as $$H^{i}(D_+(R), {\operatorname{Syz}}(f_1^q { , \ldots , }f_n^q) \widetilde{\,} \,  )_m
= H^i(Y, {\operatorname{Syz}}(f_1^q { , \ldots , }f_n^q) (m)) \, .$$ Note that the syzygy bundle ${\operatorname{Syz}}(f_1 { , \ldots , }f_n)$ is free by assumption in the singular points of $Y$. Hence we are in the situation of Proposition \[sheafproposition\] with ${\mathcal{S}}={\operatorname{Syz}}(f_1 { , \ldots , }f_n)$; therefore $H^{i} (Y, {\operatorname{Syz}}(f_1^q { , \ldots , }f_n^q)(m))=0$ ($i=1 { , \ldots , }t$) holds for $m> \max_{j=1 { , \ldots , }t} \{- q \frac{ \bar{\mu}_{\min}
({\operatorname{Syz}}_j)}{ \deg(Y)} \} + \frac{\deg(\omega_Y)}{\deg(Y)}$, which proves the theorem.

The Castelnuovo-Mumford regularity of a standard-graded Cohen-Macaulay domain $R$ is just ${\operatorname{reg}}(R)= {\operatorname{end}}(H^{\dim ( R)}_{R_+} (R)) + \dim (R)$. The end of the top-dimensional local cohomology module of a graded ring is also called its $a$-invariant, see [@brodmannsharp 13.4.7], hence ${\operatorname{reg}}(R)=a + \dim (R)$. If $R$ is Gorenstein, then $R(a)$ is the canonical module of $R$ and $\omega_Y={\mathcal{O}}_Y(a)$ is the dualizing sheaf on $Y={\operatorname{Proj}}R$. So in this case the quotient $\deg (\omega_Y) / \deg (Y)= a \deg (Y)/ \deg (Y) =a$ equals also the $a$-invariant.

The surjection $\bigoplus_{(k,j+1)} {\mathcal{O}}_Y(- \alpha_{k,j+1}) {\rightarrow}{\operatorname{Syz}}_j {\rightarrow}0$ gives at once the bound $\bar{\mu}_{\min} ({\operatorname{Syz}}_j)
\geq \bar{\mu}_{\min}(\bigoplus_{(k,j+1)} {\mathcal{O}}_Y(- \alpha_{k,j+1}))
\!= \! - \max \{\alpha_{k,j+1} \} \deg(Y)$. Therefore we get for the constant $C_1$ coming from Theorem \[regularitybound\] the estimate $C_1 \leq \max \{ \alpha_{k,j}:\, j=1 { , \ldots , }t+1 = \dim
(R) \,  \} =C_1'$. This number $C_1'$ is the coefficient for the linear bound which M. Chardin has obtained in [@chardinregularitypowers Theorem 2.3]. This bound corresponds to the inclusion bounds for tight closure of K. Smith which we obtained in Corollary \[tightcorollary\]. The following standard example of tight closure theory shows already the difference between the Chardin-Smith bound and the slope bound.

Consider the ideal $I=(x^2,y^2,z^2)$ in $R=K[z,y,z]/(x^3+y^3+z^3)$, ${\rm char} (K) \neq 3$. We compute the bound coming from Theorem \[regularitybound\] for the regularity of the Frobenius powers $I^{[q]}=(x^{2q}, y^{2q},z^{2q})$. We first observe that we may consider the curve equation $0=x^3+y^3+z^3=xx^2+yy^2+zz^2$ as a global section of the syzygy bundle of degree $3$. Since this section has no zero on $Y= {\operatorname{Proj}}R$, we get the short exact sequence $$0 {\longrightarrow}{\mathcal{O}}_Y {\longrightarrow}{\operatorname{Syz}}(x^2,y^2,z^2)(3) {\longrightarrow}{\mathcal{O}}_Y {\longrightarrow}0 \, .$$ This shows that the syzygy bundle is strongly semistable and therefore $\bar{\mu}_{\min} ( {\operatorname{Syz}}(x^2,y^2,z^2)(0))= -6 \deg(Y)/2=
-9 $. So $C_1=3$ and we get altogether the bound ${\operatorname{reg}}( I^{[q]})
\leq 3q+2$.

Since $\, {\operatorname{Syz}}(x^2,y^2,z^2)(3)\, $ is not generated by its global sections, because the section just mentioned is the only section. Hence a surjection $ \bigoplus_k {\mathcal{O}}(-\alpha_k) {\rightarrow}{\operatorname{Syz}}(x^2,y^2,z^2)(0)$ is only possible for $ \max _k \{ \alpha_k \}
\geq 4$. So the linear bound for the regularity which you get by considering only the degrees in a resolution is worse than the slope bound.
