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arXiv:0708.2394v2 [math.AC] 26 Nov 2007
F-THRESHOLDS, TIGHT CLOSURE, INTEGRAL CLOSURE, AND
MULTIPLICITY BOUNDS
CRAIG HUNEKE, MIRCEA MUSTAT¸ ˘
A, SHUNSUKE TAKAGI, AND KEI-ICHI WATANABE
Abstract. The F-threshold cJ(a) of an ideal awith respect to the ideal Jis a
positive characteristic invariant obtained by comparing the powers of awith the
Frobenius powers of J. We show that under mild assumptions, we can detect
the containment in the integral closure or the tight closure of a parameter ideal
using F-thresholds. We formulate a conjecture bounding cJ(a) in terms of the
multiplicities e(a) and e(J), when aand Jare zero-dimensional ideals, and J
is generated by a system of parameters. We prove the conjecture when Jis a
monomial ideal in a polynomial ring, and also when aand Jare generated by
homogeneous systems of parameters in a Cohen-Macaulay graded k-algebra.
Introduction
Let Rbe a Noetherian ring of positive characteristic p. For every ideal ain R, and
for every ideal Jwhose radical contains a, one can define asymptotic invariants that
measure the containment of the powers of ain the Frobenius powers of J. These
invariants were introduced in the case of a regular local F-finite ring in [MTW], where
it was shown that they coincide with the jumping exponents for the generalized test
ideals of Hara and Yoshida [HY]. In this paper we work in a general setting, and show
that the F-thresholds still capture interesting and subtle information. In particular,
we relate them to tight closure and integral closure, and to multiplicities.
If aand Jare as above, we define for every positive integer e
νJ
a(pe) := max{r|ar6⊆ J[pe]},
where J[q]is the ideal generated by the pe-powers of the elements of J. We put
cJ
+(a) := lim sup
e→∞
νJ
a(pe)
pe,cJ
−(a) := lim inf
e→∞
νJ
a(pe)
pe,
and if these two limits coincide, we denote their common value by cJ(a), and call it
the F-threshold of awith respect to J.
Our first application of this notion is to the description of the tight closure and
of the integral closure for parameter ideals. Suppose that (R, m) is a d-dimensional
Noetherian local ring of positive characteristic, and that Jis an ideal in Rgenerated
by a full system of parameters. We show that under mild conditions, for every ideal
I⊇J, we have I⊆J∗if and only if cI
+(J) = d(and in this case cI
−(J) = d, too).
We similarly show that under suitable mild hypotheses, if I⊇J, then I⊆Jif and
2000 Mathematics Subject Classification. Primary 13A35; Secondary 13B22, 13H15, 14B05.
1
2 C. Huneke, M. Mustat¸˘a, S. Takagi, and K.-i. Watanabe
only if cJ
+(I) = d. For the precise statements, see Corollary 3.2 and Theorem 3.3
below.
As we have mentioned, if Ris regular and F-finite, then it was shown in [MTW]
that the F-thresholds of an ideal acoincide with the jumping exponents for the
generalized test ideals of [HY]. In order to recover such a result in a more general
setting, we develop a notion of F-threshold for the ideal acorresponding to a sub-
module Nof a module M, such that anN= 0 for some n. We then show that under
suitable hypotheses on a local ring R, one can again recover the jumping exponents
for the generalized test ideals of an ideal ain Rfrom the F-thresholds of awith re-
spect to pairs (E, N ), where Nis a submodule of the injective hull Eof the residue
field (see Corollary 4.4).
We study the connection between F-thresholds and multiplicity, and formulate
the following conjecture: if (R, m) is a d-dimensional Noetherian local ring of char-
acteristic p > 0, aand Jare m-primary ideals in R, with Jgenerated by a system
of parameters, then
e(a)≥dd
cJ
−(a)de(J).
The case J=m(when Ris in fact regular) was proved in [TW]. We mention
that in this case cm(a) is related via reduction mod pto a fundamental invariant in
birational geometry, the log canonical threshold lct(a) (see loc. cit. for the precise
relation between these two invariants). The corresponding inequality between the
multiplicity and the log canonical threshold of awas proved in [dFEM], and plays
a key role in proving that for small values of n, no smooth hypersurface of degree n
in Pnis rational (see [Cor] and [?]).
We prove our conjecture when both aand Jare generated by homogeneous sys-
tems of parameters in a graded Cohen-Macaulay k-algebra (cf. Corollary 5.9). More-
over, we prove it also when Ris regular and J= (xa1
1,...,xan
n), for a regular system
of parameters x1,...,xn. The proof of this latter case follows the ideas in [TW]
and [dFEM], reducing to the case of a monomial ideal a, and then using the explicit
interpretation of the invariants involved in terms of the Newton polyhedron of a.
On the other hand, the proof of the homogeneous case is based on new ideas that
we expect to be useful also in attacking the general case of the conjecture. In fact, we
prove the following stronger statement. Suppose that aand Jare ideals generated
by homogeneous systems of parameters in a d-dimensional graded Cohen-Macaulay
k-algebra, where kis a field of arbitrary characteristic. If aN⊆Jfor some N, then
e(a)≥d
d+N−1d
e(J).
The paper is structured as follows. In the first section we recall some basic notions
of tight closure theory, and review the definition of generalized test ideals from [HY].
In §2 we introduce the F-thresholds and discuss some basic properties. The third
F-thresholds, tight closure, integral closure, and multiplicity bounds 3
section is devoted to the connections with tight closure and integral closure. We
introduce the F-thresholds with respect to pairs of modules in §4, and relate them
to the jumping exponents for the generalized test ideals. In the last section we
discuss inequalities involving F-thresholds and multiplicities. In particular, we state
here our conjecture and prove the above-mentioned special cases.
1. Preliminaries
In this section we review some definitions and notation that will be used through-
out the paper. All rings are Noetherian commutative rings with unity. For a ring
R, we denote by R◦the set of elements of Rthat are not contained in any minimal
prime ideal. Elements x1,...,xrin Rare called parameters if they generate an ideal
of height r. The integral closure of an ideal ais denoted by a. The order of a nonzero
element fin a Noetherian local ring (R, m) is the largest rsuch that f∈mr. For
a real number u, we denote by ⌊u⌋the largest integer ≤u, and by ⌈u⌉the smallest
integer ≥u.
Let Rbe a ring of characteristic p > 0, and let F:R→Rdenote the Frobenius
map which sends x∈Rto xp∈R. The ring Rviewed as an R-module via the
e-times iterated Frobenius map Fe:R→Ris denoted by e
R. We say that Ris
F-finite if 1
Ris a finitely generated R-module. We also say that Ris F-pure if the
Frobenius map is pure, that is, FM= 1M⊗F:M=M⊗RR→M⊗R1
Ris injective
for any R-module M. For every ideal Iin R, and for every q=pe, we denote by I[q]
the ideal generated by the qth powers of all elements of I.
If Mis an R-module, then we put Fe(M) := e
R⊗RM. Hence in Fe(M) we have
u⊗(ay) = uape⊗yfor every a∈R. Note that the e-times iterated Frobenius map
Fe
M:M→Fe(M) is an R-linear map. The image of z∈Mvia this map is denoted
by zq:= Fe
M(z). If Nis a submodule of M, then we denote by N[q]
M(or simply
by N[q]) the image of the canonical map Fe(N)→Fe(M) (note that if N=Iis a
submodule of M=R, then this is consistent with our previous notation for I[q]).
First, we recall the definitions of classical tight closure and related notions. Our
references for classical tight closure theory and for F-rational rings are [HH] and[FW],
respectively; see also the book [Hu].
Definition 1.1. Let Ibe an ideal in a ring Rof characteristic p > 0.
(i) The Frobenius closure IFof Iis defined as the ideal of Rconsisting of all
elements x∈Rsuch that xq∈I[q]for some q=pe. If Ris F-pure, then
J=JFfor all ideals J⊆R. The tight closure I∗of Iis defined to be the
ideal of Rconsisting of all elements x∈Rfor which there exists c∈R◦such
that cxq∈I[q]for all large q=pe.
(ii) We say that c∈R◦is a test element if for all ideals J⊆Rand all x∈J∗,
we have cxq∈I[q]for all q=pe≥1. Every excellent and reduced ring Rhas
a test element.
4 C. Huneke, M. Mustat¸˘a, S. Takagi, and K.-i. Watanabe
(iii) If N⊆Mare R-modules, then the tight closure N∗
Mof Nin Mis defined
to be the submodule of Mconsisting of all elements z∈Mfor which there
exists c∈R◦such that czq∈N[q]
Mfor all large q=pe. The test ideal τ(R)
of Ris defined to be τ(R) = \
M
AnnR(0∗
M), where Mruns over all finitely
generated R-modules. If M=R/I, then AnnR(0∗
M) = (I:I∗). That is,
τ(R)J∗⊆Jfor all ideals J⊆R. We say that Ris F-regular if τ(RP) = RP
for all prime ideals Pof R.
(iv) Ris called F-rational if J∗=Jfor every ideal J⊆Rgenerated by param-
eters. If Ris an excellent equidimensional local ring, then Ris F-rational if
and only if I=I∗for some ideal Igenerated by a full system of parameters
for R.
We now recall the definition of at-tight closure and of the generalized test ideal
τ(at). The reader is referred to [HY] for details.
Definition 1.2. Let abe a fixed ideal in a reduced ring Rof characteristic p > 0
such that a∩R◦6=∅, and let Ibe an arbitrary ideal in R.
(i) Let N⊆Mbe R-modules. Given a rational number t≥0, the at-tight
closure N∗at
Mof Nin Mis defined to be the submodule of Mconsisting of
all elements z∈Mfor which there exists c∈R◦such that czqa⌈tq⌉⊆N[q]
M
for all large q=pe.
(ii) The generalized test ideal τ(at) is defined to be τ(at) = \
M
AnnR(0∗at
M), where
Mruns through all finitely generated R-modules. If a=R, then the gener-
alized test ideal τ(at) is nothing but the test ideal τ(R).
(iii) Assume that Ris an F-regular ring and that Jis an ideal containing ain its
radical. The F-jumping exponent of awith respect to Jis defined by
ξJ(a) = sup{c∈R≥0|τ(ac)6⊆ J}.
If (R, m) is local, then we call the smallest F-jumping exponent ξm(a) the
F-pure threshold of aand denote it by fpt(a).
In characteristic zero, one defines multiplier ideals and their jumping exponents
using resolution of singularities (see Ch. 9 in [La]). It is known that for a given ideal
in characteristic zero and for a given t, the reduction mod p≫0 of the multiplier
ideal J(at) coincides with the generalized test ideal τ(at
p) of the reduction apof a.
Therefore the F-jumping exponent ξJ(a) is a characteristic panalogue of jumping
exponent of multiplier ideals. We refer to [BMS2], [HM], [HY], [MTW] and [TW]
for further discussions.
F-thresholds, tight closure, integral closure, and multiplicity bounds 5
2. Basic properties of F-thresholds
The F-thresholds are invariants of singularities of a given ideal ain positive char-
acteristic, obtained by comparing the powers of awith the Frobenius powers of other
ideals. They were introduced and studied in [MTW] in the case when we work in a
regular ring. In this section, we recall the definition of F-thresholds and study their
basic properties when the ring is not necessarily regular.
Let Rbe a Noetherian ring of dimension dand of characteristic p > 0. Let a
be a fixed proper ideal of Rsuch that a∩R◦6=∅. To each ideal Jof Rsuch that
a⊆√J, we associate an F-threshold as follows. For every q=pe, let
νJ
a(q) := max{r∈N|ar6⊆ J[q]}.
Since a⊆√J, this is a nonnegative integer (if a⊆J[q], then we put νJ
a(q) = 0). We
put
cJ
+(a) = lim sup
q→∞
νJ
a(q)
q,cJ
−(a) = lim inf
q→∞
νJ
a(q)
q.
When cJ
+(a) = cJ
−(a), we call this limit the F-threshold of the pair (R, a) (or simply
of a) with respect to J, and we denote it by cJ(a).
Remark 2.1.(1) (cf. [MTW, Remark 1.2]) One has
0≤cJ
−(a)≤cJ
+(a)<∞.
In fact, if ais generated by lelements and if aN⊆J, then
aN(l(pe−1)+1) ⊆(a[pe])N= (aN)[pe]⊆J[pe].
Therefore νJ
a(pe)≤N(l(pe−1) + 1) −1. Dividing by peand taking the limit gives
cJ
+(a)≤Nl.
(2) Question 1.4 in [MTW] asked whether the F-threshold cJ(a) is a rational
number (when it exists). A positive answer was given in [BMS2] and [BMS1] for
a regular F-finite ring, essentially of finite type over a field, and for every regular
F-finite ring, if the ideal ais principal. For a proof in the case of a principal ideal in
a complete regular ring (that is not necessarily F-finite), see [KLZ]. However, this
question remains open in general.
Recall that a ring extension R ֒→Sis cyclic pure if for every ideal Iin R, we
have IS ∩R=I.
Proposition 2.2 (cf. [MTW, Proposition 1.7]).Let a,Jbe ideals as above.
(1) If I⊇J, then cI
±(a)≤cJ
±(a).
(2) If b⊆a, then cJ
±(b)≤cJ
±(a). Moreover, if a⊆b, then cJ
±(b) = cJ
±(a).
(3) cJ
±(ar) = 1
rcJ
±(a)for every integer r≥1.
(4) cJ[q]
±(a) = qcJ
±(a)for every q=pe.
6 C. Huneke, M. Mustat¸˘a, S. Takagi, and K.-i. Watanabe
(5) If R ֒→Sis a cyclic pure extension, then
cJ
±(a) = cJS
±(aS).
(6) Let R ֒→Sbe an integral extension. If the conductor ideal c(S/R) :=
AnnR(S/R)contains the ideal ain its radical, then
cJ
±(a) = cJS
±(aS).
(7) cJ
+(a)≤c(resp. cJ
−(a)≥c)if and only if for every power q0of p, we have
a⌈cq⌉+q/q0⊆J[q](resp. a⌈cq⌉−q/q06⊆ J[q])for all q=pe≫q0.
Proof. For (1)–(4), see [MTW] (the proofs therein do not use the fact that Ris
regular). If R ֒→Sis cyclic pure, then νJ S
aS(q) = νJ
a(q) for every q, and we get (5).
For (6), we fix a positive integer msuch that am⊆c(S/R). By the definition of
the conductor ideal c(S/R), if (aS)n⊆(JS)[q]for some n∈Nand some q=pe,
then am+n⊆J[q]. This implies that
νJS
aS(q)≤νJ
a(q)≤νJS
aS(q) + m.
These inequalities imply (6).
In order to prove (7), suppose first that cJ
+(a)≤c. It follows from the definition
of cJ
+(a) that for every power q0of p, we can find q1such that νJ
a(q)/q < c +1
q0for
all q=pe≥q1. Thus, νJ
a(q)<⌈cq⌉+q
q0, that is,
(1) a⌈cq⌉+q/q0⊆J[q]
for all q=pe≥q1. Conversely, suppose that (1) holds for every q≥q1. This implies
νJ
a(q)≤ ⌈cq⌉+q
q0−1. Dividing by qand taking the limit gives cJ
+(a)≤c+1
q0. If
this holds for every q0, we conclude that cJ
+(a)≤c. The assertion regarding cJ
−(a)
follows from a similar argument.
We now give a variant of the definition of F-threshold. If aand Jare ideals in R,
such that a∩R◦6=∅and a⊆√J, then we put
eνJ
a(q) := max{r∈N|ar6⊆ (J[q])F}.
It follows from the definition of Frobenius closure that if u6∈ (J[q])F, then up6∈
(J[pq])F. This means that
eνJ
a(pq)
pq ≥eνJ
a(q)
q
for all q=pe. Thus,
lim
q→∞ eνJ
a(q)
q= sup
q=peeνJ
a(q)
q.
We denote this limit by ecJ(a). Note that we have ecJ(a)≤cJ
−(a).
The F-threshold cJ(a) exists in many cases.
Lemma 2.3. Let a,Jbe as above.
F-thresholds, tight closure, integral closure, and multiplicity bounds 7
(1) If J[q]= (J[q])Ffor all large q=pe, then the F-threshold cJ(a)exists, that
is, cJ
+(a) = cJ
−(a). In particular, if Ris F-pure, then cJ(a)exists.
(2) If the test ideal τ(R)contains ain its radical, then the F-threshold cJ(a)
exists and cJ(a) = cJ∗(a).
(3) If ais principal, then cJ(a)exists.
Proof. (1) follows from the previous discussion since in that case we have eνJ
a(q) =
νJ
a(q) for all q≫0.
In order to prove (2), we take an integer m≥1 such that am⊆τ(R). Then, by
the definition of τ(R), one has a2m((J∗)[q])F⊆am(J∗)[q]⊆J[q]for all q=pe. This
means that
eνJ∗
a(q)≤νJ∗
a(q)≤νJ
a(q)≤eνJ∗
a(q) + 2m.
Since ecJ∗(a) always exists, cJ(a) and cJ∗(a) also exist and these three limits are all
equal.
For (3), note that if ais principal and ar⊆J[q], then apr ⊆J[pq]. Therefore we
have νJ
a(pq) + 1
pq ≤νJ
a(q) + 1
q
for every q=pe. This implies that
lim
q→∞
νJ
a(q)
q= lim
q→∞
νJ
a(q) + 1
q= inf
q=pe
νJ
a(q)
q.
As shown in [MTW, Proposition 2.7], the F-threshold cJ(a) coincides with the
F-jumping exponent ξJ(a) when the ring is F-finite and regular. The statement in
loc. cit. requires the ring to be local, however the proof easily generalizes to the
non-local case (see [BMS1]). More precisely, we have the following
Proposition 2.4. Let Rbe an F-finite regular ring of characteristic p > 0. If ais
a nonzero ideal contained in the radical of J, then τ(acJ(a))⊆J. Going the other
way, if α∈R+, then ais contained in the radical of τ(aα)and cτ(aα)(a)≤α. In
particular, the F-threshold cJ(a)coincides with the F-jumping exponent ξJ(a).
Remark 2.5.The F-threshold cJ(a) sometimes coincide with the F-jumping exponent
ξJ(a) even when Ris singular. For example, let R=k[[X, Y , Z, W]]/(XY −Z W ),
and let mbe the maximal ideal of R. Then the F-threshold cm(m) of mwith respect
to mand the F-pure threshold (that is, the smallest F-jumping exponent) fpt(m) of
mare both equal to two.
However, cJ(a) does not agree with ξJ(a) in general. For example, let R=
k[[X, Y, Z]]/(XY −Z2) be a rational double point of type A1over a field kof
characteristic p > 2 and let mbe the maximal ideal of R. Then fpt(m) = 1 (see
[TW, Example 2.5]), whereas cm(m) = 3/2.
8 C. Huneke, M. Mustat¸˘a, S. Takagi, and K.-i. Watanabe
Remark 2.6.Suppose that mis a maximal ideal in any Noetherian ring R, and that
Jis an m-primary ideal. For every q=pewe have J[q]Rm∩R=J[q], hence for every
ideal a⊆mwe have νJ
a(q) = νJRm
aRm(q). In particular, cJ
±(a) = cJRm
±(aRm).
Example 2.7. (i) Let Rbe a Noetherian local ring of characteristic p > 0, and
let J= (x1, . . . , xd), where x1,...,xdform a full system of parameters in
R. It follows from the Monomial Conjecture (which is a theorem in this
setting, see [Ho, Prop. 3]) that (x1···xd)q−16∈ J[q]for every q. Hence
νJ
J(q)≥d(q−1) for every q, and therefore cJ
−(J)≥d. On the other hand,
cJ
+(J)≤dby Remark 2.1 (1), and we conclude that cJ(J) = d.
(ii) Let R=k[x1, . . . , xd] be a d-dimensional polynomial ring over a field kof
characteristic p > 0, and let a, J ⊆Rbe zero-dimensional ideals generated
by monomials. In order to compute cJ(a) we may assume that kis perfect,
hence we may use Proposition 2.4.
Let P(a)⊆Rd
≥0denote the Newton polyhedron of a, that is P(a) is the
convex hull of those u= (u1,...,un)∈Nnsuch that xu=xu1
1···xun
n∈a. It
follows from [HY, Thm. 6.10] that
τ(ac) = (xu|u+e∈Int(c·Pa)),
where e= (1,1, . . . , 1). We deduce that if λ(u) is defined by the condition
u+e∈∂(λ(u)·P(a)), then
cJ(a) = max{λ(u)|u∈Nn, xu6∈ J}
(note that since Jis zero-dimensional, this maximum is over a finite set). In
particular, we see that if J= (xa1
1,...,xan
n), then cJ(a) is characterized by
a= (a1,...,an)∈∂(cJ(a)·P(a)).
(iii) Let (R, m) be a d-dimensional regular local ring of characteristic p > 0, and
let J⊂Rbe an m-primary ideal. We claim that
(2) cJ(m) = max{r∈Z≥0|mr6⊆ J}+d.
In particular, cJ(m) is an integer ≥d.
Indeed, if u6∈ J, then (J:u)⊆m, hence J[q]:uq= (J:u)[q]⊆m[q], and
therefore uqmd(q−1) 6⊆ J[q]. If u∈mr, it follows that νJ
m(q)≥rq +d(q−1).
Dividing by qand passing to the limit gives cJ(m)≥r+d, hence we have
”≥” in (2). For the reverse inequality, note that if mr+1 ⊆J, then
m(r+d)q⊆(mr+1)[q]⊆J[q]
for every q=pe. Hence νJ
m(q)≤(r+d)q−1 for all q, and we get cJ(m)≤r+d.
3. Connections with tight closure and integral closure
Theorem 3.1. Let (R, m)be an excellent analytically irreducible Noetherian local
domain of positive characteristic p. Set d= dim(R), and let J= (x1,...,xd)be
F-thresholds, tight closure, integral closure, and multiplicity bounds 9
an ideal generated by a full system of parameters in R, and let I⊇Jbe another
ideal. Then Iis not contained in the tight closure J∗of Jif and only if there exists
q0=pe0such that xq0−1∈I[q0], where x=x1x2···xd.
Proof. After passing to completion, we may assume that Ris a complete local
domain. Suppose first that xq0−1∈I[q0], and by way of contradiction suppose
also that I⊆J∗. Let c∈R◦be a test element. Then for all q=pe, one has
cxq(q0−1) ∈cI[qq0]⊂J[qq0], so that c∈J[qq0]:xq(q0−1) ⊆(J[q])∗, by colon-capturing
[HH, Theorem 7.15a]. Therefore c2lies in Tq=peJ[q]= (0), a contradiction.
Conversely, suppose that I*J∗, and choose an element f∈IrJ∗. We
choose a coefficient field k, and let B=k[[x1,...,xd, f ]] be the complete sub-
ring of Rgenerated by x1,...,xd, f. Note that Bis a hypersurface singular-
ity, hence Gorenstein. Furthermore, by persistence of tight closure [HH, Lemma
4.11a], f /∈((x1,...,xd)B)∗. If we prove that there exists q0=pe0such that
xq0−1∈((x1,...,xd, f )B)[q0], then clearly xq0−1is also in I[q0]. Hence we can re-
duce to the case in which Ris Gorenstein. Since I6⊆ J∗, it follows from a re-
sult of Aberbach [Ab] that J[q]:I[q]⊆mn(q), where n(q) is a positive integer with
limq→∞ n(q) = ∞. In particular, we can find q0=pe0such that J[q0]:I[q0]⊆J.
Therefore xq0−1∈J[q0]:J⊆J[q0]: (J[q0]:I[q0]) = I[q0], where the last equality follows
from the fact that Ris Gorenstein.
Corollary 3.2. Let (R, m)be a d-dimensional excellent analytically irreducible Noe-
therian local domain of characteristic p > 0, and let J= (x1,...,xd)be an ideal
generated by a full system of parameters in R. Given an ideal I⊇J, we have
I⊆J∗if and only if cI
+(J) = d(and in this case cI(J)exists). In particular, Ris
F-rational if and only if cI
+(J)< d for every ideal I)J.
Proof. Note first that by Remark 2.1 (1), for every I⊇Jwe have cJ
+(I)≤d.
Suppose now that I⊆J∗. It follows from Theorem 3.1 that Jd(q−1) 6⊆ I[q]for every
q=pe. This gives νI
J(q)≥d(q−1) for all q, and therefore cI
−(J)≥d. We conclude
that in this case cI
+(J) = cI
−(J) = d.
Conversely, suppose that I6⊆ J∗. By Theorem 3.1, we can find q0=pe0such that
b:= (xq0
1,...,xq0
d,(x1···xd)q0−1)⊆I[q0].
If (x1,...,xd)r6⊆ b[q], then
r≤(qq0−1)(d−1) + q(q0−1) −1 = qq0d−q−d.
Therefore νb
J(q)≤qq0d−q−dfor every q, which implies cb(J)≤q0d−1. Since q0
is a fixed power of p, we deduce
cI
+(J) = 1
q0
cI[q0]
+(J)≤1
q0
cb(J)≤d−1
q0
< d.
10 C. Huneke, M. Mustat¸˘a, S. Takagi, and K.-i. Watanabe
Theorem 3.3. Let (R, m)be a d-dimensional formally equidimensional Noetherian
local ring of characteristic p > 0. If Iand Jare ideals in R, with Jgenerated by a
full system of parameters, then
(1) cJ
+(I)≤dif and only if I⊆J.
(2) If, in addition, J⊆I, then I⊆Jif and only if cJ
+(I) = d. Moreover, if
these equivalent conditions hold, then cJ(I) = d.
Proof. Note that if J⊆I, then cJ
−(I)≥cJ
−(J) = cJ(J) = d, by Example 2.7 (i).
Hence both assertions in (2) follow from the assertion in (1).
One implication in (1) is easy: if I⊆J, then by Proposition 2.2 (2) we have
cJ
+(I)≤cJ
+(J) = cJ(J) = d. Conversely, suppose that cJ
+(I)≤d. In order to show
that I⊆J, we may assume that Ris complete and reduced. Indeed, first note that
the inverse image of Jb
Rred in Ris contained in J, hence it is enough to show that
Ib
Rred ⊆Jb
Rred. Since Jb
Rred is again generated by a full system of parameters, and
since we trivially have
cJ
b
Rred (Ib
Rred)≤cJ(I)≤d,
we may replace Rby b
Rred.
Since Ris complete and reduced, we can find a test element cfor R. By Proposi-
tion 2.2 (7), the assumption cJ
+(I)≤dimplies that for all q0=pe0and for all large
q=pe, we have
Iq(d+(1/q0)) ⊆J[q].
Hence IqJq(d−1+(1/q0)) ⊆J[q], and thus
Iq⊆J[q]:Jq(d−1+(1/q0)) ⊆(Jq−d+1−(q/q0))∗,
where the last containment follows from the colon-capturing property of tight closure
[HH, Theorem 7.15a]. We get cIq⊆cR ∩Jq−d+1−(q/q0)⊆cJ q−d+1−(q/q0)−lfor some
fixed integer lthat is independent of q, by the Artin-Rees lemma. Since cis a
non-zero divisor in R, it follows that
(3) Iq⊆Jq−d+1−(q/q0)−l.
If νis a discrete valuation with center in m, we may apply νto (3) to deduce
qν(I)≥q−d+ 1 −q
q0−lν(J). Dividing by qand letting qgo to infinity gives
ν(I)≥1−1
q0ν(J). We now let q0go to infinity to obtain ν(I)≥ν(J). Since
this holds for every ν, we have I⊆J.
Example 3.4. Let (R, m) be a regular local ring of characteristic p > 0 with
dim(R) = d, and Jbe an ideal of Rgenerated by a full system of parameters.
We define ato be the maximal integer nsuch that mn6⊆ J. Then ms⊆Jif and
only if s≥a
d+ 1 since cJ(ms) = a+d
sby Example 2.7 (iii) and Proposition 2.2 (3).
F-thresholds, tight closure, integral closure, and multiplicity bounds 11
Question 3.5.Does this statement hold in a more general setting ? Can we replace
“regular” by “Cohen-Macaulay” ?
4. F-thresholds of modules
In the section we give a generalization of the notion of F-thresholds, in which
we replace the auxiliary ideal in the definition by a submodule of a given module.
We have seen in Proposition 2.4 that in a regular F-finite ring, the F-thresholds
of an ideal acoincide with the F-jumping exponents of a. This might fail in non-
regular rings, and in fact, it is often the case that fpt(a)<cJ(a) for every ideal J.
However, as Corollary 4.4 below shows, we can remedy this situation if we consider
the following more general notion of F-thresholds.
Suppose now that ais a fixed ideal in a Noetherian ring Rof characteristic p > 0.
Let Mbe an R-module, and N⊆Ma submodule such that anN= 0 for some
n > 0. We define
(1) For q=pe, let νN
M,a(q) = max{r∈N|arN[q]
M6= 0}(we put νN
M,a(q) = 0 if
aN[q]
M= 0).
(2) cN
M,+(a) = lim supq→∞
νN
M,a(q)
qand cN
M,−(a) = lim infq→∞
νN
M,a(q)
q.When cN
M,+(a) =
cN
M,−(a), we call this limit the F-threshold of awith respect to (N, M ), and
we denote it by cN
M(a).
Remark 4.1.If Jis an ideal of Rwith a⊆√J, then it is clear that νA/J
a,A/J(q) = νJ
a(q),
hence cA/J
A/J,±(a) = cJ
±(a). Thus the notion of F-threshold with respect to modules
extends our previous definition of F-thresholds with respect to ideals.
Lemma 4.2. Let R,a,Mand Nbe as in the above definition.
(1) If b⊆ais an ideal, then cN
M,±(b)≤cN
M,±(a).
(2) If N′⊆N, then cN′
M,±(a)≤cN
M,±(a).
(3) If φ:M→M′is a homomorphism of R-modules, and if N′=φ(N), then
cN′
M′,±(a)≤cN
M,±(a). If Ris regular and φis injective, then cN′
M′,±(a) =
cN
M,±(a).
(4) If Ris F-pure, then νN
M,a(q)
q≤νN
M,a(qq′)
qq′for every q, q′. Hence in this case
the limit cN
M(a)exists and it is equal to supq
νN
M,a(q)
q.
Proof. The assertions in (1) and (2) follow from definition. For (3), note that φ
induces a surjection N[q]→N′[q], which gives the first statement. Moreover, if R
is regular and φis injective, then the flatness of the Frobenius morphism implies
N[q]≃N′[q], and we have equality.
Suppose now that Ris F-pure, hence M⊗Re
Ris a submodule of M⊗Ree′
R. If
q=peand q′=pe′, and if arN[q]6= 0, then aq′rN[qq′]⊇(ar)[q′]N[qq′]6= 0. Therefore
νN
M,a(qq′)≥q′·νN
M,a(q).
12 C. Huneke, M. Mustat¸˘a, S. Takagi, and K.-i. Watanabe
Our next proposition gives an analogue of Proposition 2.4 in the non-regular case.
Proposition 4.3. Let abe a proper nonzero ideal in a local normal Q-Gorenstein
ring (R, m). Suppose that Ris F-finite and F-pure, and that the test ideal τ(R)is
m-primary. We denote by Ethe injective hull of R/m.
(1) If Nis a submodule of Esuch that a⊆pAnnR(N), and if α= cN
E(a), then
N⊆(0)∗aα
E.
(2) If αis a non-negative real number, and if we put N= (0)∗aα
E, then cN
E(a)≤α.
(3) There is an order-reversing bijection between the F-thresholds of awith re-
spect to the submodules of Eand the ideals of the form τ(aα).
Proof. For (1), note that since Ris F-pure, we have νN
E(q)≤αq for every q=pe.
This implies
a⌈αq⌉+1N[q]
E= 0,
hence for every nonzero d∈awe have da⌈αq⌉N[q]
E= 0 for all q. By definition,
N⊆(0)∗aα
E.
Suppose now that α≥0, and that N= (0)∗aα
E. By hypothesis, we can find m
such that am⊆τ(R). It follows from [HT, Cor. 2.4] that every element in τ(R)
is an aα-test element. Therefore am+⌈αq⌉N[q]
E= 0, hence νN
E,a(q)< m +αq for all
q≫0. Dividing by qand taking the limit as qgoes to infinity, gives cN
E(a)≤α.
We assume that Ris F-finite, normal and Q-Gorenstein, hence for every non-
negative twe have τ(at) = AnnR(0∗at
E). Note also that by [HT, Prop. 3.2], taking
the generalized test ideal commutes with completion. This shows that the set of
ideals of the form τ(aα) is in bijection with the set of submodules of Eof the form
(0)∗aα
E. Hence in order to prove (3) it is enough to show that the map
{(0)∗aα
E|α≥0} → {cN
E(a)|N⊆E, a⊆pAnnR(N)}
that takes Nto cN
E(a) is bijective, the inverse map taking αto (0)∗aα
E.
Suppose first that N= (0)∗aα
E, and let β= cN
E(a). It follows from (2) that β≤α,
hence (0)∗aβ
E⊆N. On the other hand, (1) gives N⊆(0)∗aβ
E, hence we have equality.
Let us now start with α= cN
E(a), and let N′= (0)∗aα
E. We deduce from (1) that
N⊆N′, hence cN′
E(a)≥α. Since (2) implies cN′
E(a)≤α, we get α= cN′
E(a), which
completes the proof of (3).
Corollary 4.4. Let abe a proper nonzero ideal in a local normal Q-Gorenstein ring
(R, m). If Ris F-finite and F-regular, then for every ideal Jin Rwe have
ξJ(a) = cN
E(a),
where Eis the injective hull of R/mand N= AnnE(J). In particular, the F-pure
threshold fpt(a)is equal to cZ
E(a), where Z= (0 : Em)is the socle of E.
F-thresholds, tight closure, integral closure, and multiplicity bounds 13
Proof. Let β:= cN
E(a). Given α≥0, Matlis duality implies that τ(aα)⊆Jif and
only if N⊆(0)∗aα
E. If this holds, then part (2) in the proposition gives
α≥c(0)∗aα
E
E(a)≥cN
E(a) = β.
Conversely, if α≥β, then
(0)∗aα
E⊇(0)∗aβ
E⊇N,
by part (1) in the proposition. This shows that cN
E(a) = ξJ(a), and the last assertion
in the corollary follows by taking J=m.
Remark 4.5.Let abe an ideal in the local ring (R, m). We have seen that cI(a)≥
cm(a) for every proper ideal I. Note also that applying Prop 4.2 (3) to the embedding
R/m ≃Z ֒→E=ER(R/m), we get cm(a) = cR/m
R/m(a)≥cZ
E(a) = fpt(a). Thus we
always have fpt(a)≤cI(a), and equality is possible only if fpt(a) = cm(a). While
this equality holds in some non-regular examples (see Remark 2.5), this seems to
happen rather rarely.
5. Connections between F-thresholds and multiplicity
Given an m-primary ideal ain a regular local ring (R, m), essentially of finite type
over a field of characteristic zero, de Fernex, Ein and the second author proved in
[dFEM] an inequality involving the log canonical threshold lct(a) and the multiplicity
e(a). Later, the third and fourth authors gave in [TW] a characteristic panalogue
of this result, replacing the log canonical threshold lct(a) by the F-pure threshold
fpt(a). We propose the following conjecture, generalizing this inequality.
Conjecture 5.1. Let (R, m)be a d-dimensional Noetherian local ring of character-
istic p > 0. If J⊆mis an ideal generated by a full system of parameters, and if
a⊆mis an m-primary ideal, then
e(a)≥d
cJ
−(a)d
e(J).
Remark 5.2.(1) When Ris regular and J=m, the above conjecture is precisely
the above-mentioned inequality, see [TW, Proposition 4.5].
(2) When Ris a d-dimensional regular local ring, essentially of finite type over
a field of characteristic zero, we can consider an analogous problem: let a, J be m-
primary ideals in Rsuch that Jis generated by a full system of parameters. Does
the following inequality hold
e(a)≥d
λJ(a)d
e(J),
where λJ(a) := max{c > 0| J(ac)6⊆ J}. This would generalize the inequality in
[dFEM], which is the special case J=m. However, this version is also open in
general.
14 C. Huneke, M. Mustat¸˘a, S. Takagi, and K.-i. Watanabe
(3) The condition in Conjecture 5.1 that Jis generated by a system of parameters
is crucial, as otherwise there are plenty of counterexamples. Suppose, for example,
that (R, m) is a regular local ring of dimension d≥2 and of characteristic p > 0.
Let a=mkand J=mℓwith k≥1, ℓ≥2 integers. It follows from Example 2.7 (3)
that cJ(a) = (d+ℓ−1)/k. Moreover, we have e(a) = kdand e(J) = ℓd, thus
e(a) = kd<(dkℓ/(d+ℓ−1))d=d
cJ(a)d
e(J).
Example 5.3. Let R=k[[X, Y, Z]]/(X2+Y3+Z5) be a rational double point of
type E8, with ka field of characteristic p > 0. Let a= (x, z) and J= (y, z). Then
e(a) = 3 and e(J) = 2. It is easy to check that cJ(a) = 5/3 and ca(J) = 5/2. Thus,
e(a) = 3 >72
25 =2
cJ(a)2
e(J),
e(J) = 2 >48
25 =2
ca(J)2
e(a).
See Corollary 5.9 below for a general statement in the homogeneous case.
We now show that Conjecture 5.1 implies an effective estimate of the multiplicity
of complete intersection F-rational rings.
Proposition 5.4. Let (R, m)be a d-dimensional F-rational local ring of charac-
teristic p > 0with infinite residue field (resp. a rational singularity over a field
of characteristic zero)which is a complete intersection. If Conjecture 5.1 (resp.
Remark 5.2 (1)) holds true for the regular case, then e(R)≤2d−1.
Proof. Let J⊆mbe a minimal reduction of m. Note that Jis generated by a full
system of parameters for R. The Brian¸con-Skoda theorem for F-rational rings (or
for rational singularities), see [HV] and [AH], gives md⊆J. Taking the quotient of
Rby J, we reduce the assertion in the proposition to the following claim:
Claim. Let (A, m) be a complete intersection Artinian local ring of characteristic
p > 0 (resp. essentially of finite type over a field of characteristic zero). If sis the
largest integer ssuch that ms6= 0, then e(A)≤2s.
We now show that the regular case of Conjecture 5.1 implies the claim in positive
characteristic (the argument in characteristic zero is entirely analogous). Write
A=S/I, where (S, n) is an n-dimensional regular local ring and I⊆Sis an ideal
generated by a full system of parameters f1,...,fnfor S. For every i, we denote by
αithe order of fi. We may assume that αi≥2 for all i.
Let n= (y1,...,yn), and let us write fi=Pjaij yj. A standard argument re-
lating the Koszul complexes on the fiand, respectively, the yi, shows that det(aij )
generates the socle of A. In particular, if
s:= max{r∈N|nr6⊆ I},
F-thresholds, tight closure, integral closure, and multiplicity bounds 15
then s≥Pn
i=1(ai−1) ≥n. On the other hand, it follows from Example 2.7
(iii) that cI(m) = s+n(the corresponding formula in characteristic zero is an
immediate consequence of the description of the multiplier ideals of the ideal of a
point). Applying Conjecture 5.1 to S, we get
1 = e(n)≥n
cI(m)n
e(I) = n
s+nn
e(I).
Note that (n/(s+n))n≥(s/(s+s))s= (1/2)s, because s≥n. Thus, we have
e(A) = e(I)≤2s.
Proposition 5.5. If (R, m)is a one-dimensional analytically irreducible local do-
main of characteristic p > 0, and if a, J are m-primary ideals in R, then
cJ(a) = e(J)
e(a).
In particular, Conjecture 5.1 holds in R.
Proof. By Proposition 2.2 (5), we may assume that Ris a complete local domain.
Since Ris one-dimensional, the integral closure Ris a DVR. Therefore we have
cJR(aR) = ordR(JR)/ordR(aR).
On the other hand, e(JR) = ordR(JR) and e(aR) = ordR(aR). Thus, by Proposi-
tion 2.2 (6),
cJ(a) = cJR(aR) = e(JR)
e(aR)=e(J)
e(a).
Theorem 5.6. If (R, m)is a regular local ring of characteristic p > 0and J=
(xa1
1,...,xad
d), with x1, . . . , xda full regular system of parameters for R, and with
a1,...,adpositive integers, then the inequality given by Conjecture 5.1 holds.
Proof. The proof follows the idea in [dFEM] and [TW], reducing the assertion to
the case when ais a monomial ideal, and then using the explicit description of the
invariants involved. We have by definition e(a) = limn→∞ d!·ℓR(R/an)
nd, hence it is
enough to show that for every m-primary ideal aof R,
(4) ℓR(R/a)≥1
d!d
cJ(a)d
e(J).
After passing to completion and using Proposition 2.2 (5) and Remark 2.6, we see
that it is enough to prove the inequality (4) in the case when R=k[x1,...,xd],
m= (x1,...,xd), ais m-primary, and J= (xa1
1,...,xad
d).
Note that e(J) = a1···ad. We fix a monomial order λon the monomials in the
polynomial ring, and use it to take a Gr¨obner deformation of a, see [Eis, Ch. 15].
This is a flat family {as}s∈ksuch that R/as∼
=R/afor all s6= 0, and such that
a0= inλ(a), the initial ideal of a.
16 C. Huneke, M. Mustat¸˘a, S. Takagi, and K.-i. Watanabe
If Iis an ideal generated by monomials, we denote by P(I) the Newton polyhedron
of I(see Example 2.7 (2) for definition). We also put Vol(P) for the volume of a
region Pin Rn, with the Euclidean metric. Since the deformation we consider is
flat, it follows that inλ(a) is also m-primary and
ℓR(R/a) = ℓR(R/inλ(a)) ≥Vol Rd
≥0rP(inλ(a)),
where the inequality follows from [dFEM, Lemma 1.3].
On the other hand, by [dFe, Prop. 5.3], we have τ(inλ(a)t)⊆inλ(τ(at)) for all
t > 0. This implies that cJ(a)≥cinλ(J)(inλ(a)). Note also that since Jis generated
by monomials, we have inλ(J) = J. Thus, we can reduce to the case when ais
generated by monomials in x1,...,xd. That is, it is enough to show that for every
m-primary monomial ideal a⊆R,
Vol Rd
≥0rP(a)≥1
d!d
cJ(a)d
a1···ad.
It follows from the description of cJ(a) in Example 2.7 (2) that we have (a1,...,ad)∈
∂(cJ(a)·P(a)). We can find a hyperplane Hq:= u1/b1+···+ud/bd= 1 passing
through the point (a1, . . . , ad) such that
H+:= (u1, . . . , ud)∈Rd
≥0|u1
b1
+···+ud
bd≥1⊇cJ(a)·P(a).
Therefore, we have
Vol Rd
≥0rP(a)≥Vol Rd
≥0r1
cJ(a)H+=b1. . . bd
d!·cJ(a)d.
On the other hand, since Hpasses through (a1,...,ad), it follows that a1/b1+···+
ad/bd= 1. Comparing the arithmetic and geometric means of {ai/bi}i, we see that
b1···bd≥dd·a1···ad.
Thus, combining these two inequalities, we obtain that
Vol Rd
≥0rP(a)≥b1···bd
d!·cJ(a)d≥1
d!d
cJ(a)d
a1···ad,
as required.
Remark 5.7.It might seem that in the above proof we have shown a stronger asser-
tion than the one in Conjecture 5.1, involving the length instead of the multiplicity.
However, the two assertions are equivalent: this follows from [Mu, Corollary 3.8]
which says that for every zero-dimensional ideal ain a d-dimensional regular local
ring R, we have
ℓR(R/a)≥e(a)
d!.
We can prove a graded version of Conjecture 5.1. In fact, we prove a more precise
statement, which is valid independently of the characteristic.
F-thresholds, tight closure, integral closure, and multiplicity bounds 17
Theorem 5.8. Let R=Ld≥0Rdbe an n-dimensional graded Cohen-Macaulay ring
with R0a field of arbitrary characteristic. If aand Jare ideals generated by full
homogeneous systems of parameters for R, and if aN⊆J, then
e(a)≥n
n+N−1n
e(J).
Corollary 5.9. Let Rbe as in the theorem, with char(R0) = p > 0. If aand Jare
ideals generated by full homogeneous systems of parameters for R, then
e(a)≥n
cJ
−(a)n
e(J).
Proof. Note that each J[q]is again generated by a full homogeneous systems of
parameters. It follows from the theorem and from the definition of νJ
a(q) that for
every q=pewe have
e(a)≥n
n+νJ
a(q)n
e(J[q]) = qn
n+νJ
a(q)n
e(J).
On the right-had side we can take a subsequence converging to n
cJ
−(a)ne(J), hence
we get the inequality in the corollary.
Proof of Theorem 5.8. Suppose that ais generated by a full homogeneous system of
parameters x1,...,xnof degrees a1≤ ·· · ≤ an, and that Jis generated by another
homogeneous system of parameters f1,...,fnof degrees d1≤ ·· · ≤ dn. Define
nonnegative integers t1,...,tn−1inductively as follows: t1is the smallest integer t
such that xt
1∈J. If 2 ≤i≤n−1, then tiis the smallest integer tsuch that
xt1−1
1···xti−1−1
i−1xt
i∈J. Note that we have by assumption N≥t1+···+tn−1−n+ 1.
We first show the following inequality for every i= 1,...,n−1:
(5) t1a1+···+tiai≥d1+···+di.
Let Iibe the ideal of Rgenerated by xt1
1, xt1−1
1xt2
2,...,xt1−1
1···xti−1−1
i−1xti
i. Note that
the definition of the integers tjimplies that Ii⊆J. The natural surjection of R/Ii
onto R/J induces a comparison map between their free resolutions (we resolve R/J
by the Koszul complex, and R/Iiby a Taylor-type complex). Note that the ith step
in the Taylor complex for the monomials Xt1
1, Xt1−1
1Xt2
2,...,Xt1−1
1···Xti−1−1
i−1Xti
iin
a polynomial ring with variables X1,...,Xn, is a free module of rank one, with a
generator corresponding to the monomial
lcm(Xt1
1, Xt1−1
1Xt2
2,...,Xt1−1
1···Xti−1
i−1Xti
i) = Xt1
1···Xti−1
i−1Xti
i
(see [Eis, Exercise 17.11]). It follows that the map between the ith steps in the
resolutions of R/Iiand R/J is of the form
R(−t1a1−· ·· − tiai)→M
1≤v1<···<vi≤n
R(−dv1− · ·· − dvi).
18 C. Huneke, M. Mustat¸˘a, S. Takagi, and K.-i. Watanabe
In particular, unless this map is zero, we have
t1a1+···+tiai≥min
1≤v1<···<vi≤n(dv1+···+dvi) = d1+···+di.
We now show that this map cannot be zero. If it is zero, then also the induced map
(6) TorR
i(R/Ii, R/bi)→TorR
i(R/J, R/bi)
is zero, where biis the ideal generated by x1,...,xi. On the other hand, using the
Koszul complex on x1, . . . , xito compute the above Tor modules, we see that the
map (6) can be identified with the natural map
(Ii:bi)/Ii→(J:bi)/J.
Since xt1−1
1···xti−1
i∈(Ii:bi), it follows that xt1−1
1···xti−1
ilies in J, a contradiction.
This proves (5).
We next prove the following inequality:
(7) t1a1+···+tn−1an−1+ (N−t1− ·· ·− tn−1+n−1)an≥d1+···+dn.
Since aN⊆J, we have
(8) (xN
1,...,xN
n): J⊆(xN
1,...,xN
n): aN= (xN
1,...,xN
n) + a(n−1)(N−1).
On the other hand, the ideal (xN
1,...,xN
n): Jcan be described as follows. If we write
xN
i=Pn
j=1 bij fj, then using the Koszul resolutions of R/J and R/(xN
1,...,xN
n) one
sees that multiplication by D= det(bij) gives an injection R/J ֒→R/(xN
1,...,xN
n),
hence J= (xN
1,...,xN
n): D. Moreover, we also get
(xN
1,...,xN
n): J= (xN
1,...,xN
n, D)
(see, for example, [PS, Prop. 2.6]; note that the statement therein requires Rto be
regular, but this condition is not used). It follows from the above description that
Dis homogeneous, and deg(D) = N(a1+···+an)−(d1+···+dn).
It follows from (8) that after possibly adding to Dan element in (xN
1,...,xN
n),
we may write
D=X
m1+···+mn=(n−1)(N−1)
cm1...mnxm1
1. . . xmn
n,
where all cm1,...,mnare homogeneous. Since xt1−1
1···xtn−1−1
n−1is not in J= (xN
1,...,xN
n): D,
we see that
D6∈ (xN
1,...,xN
n): xt1−1
1···xtn−1−1
n−1= (xN−t1+1
1,...,xN−tn−1+1
n−1, xN
n).
Thus there is some (m1,...,mn) with Pjmj= (n−1)(N−1) and mj≤N−tjfor
all j≤n−1, such that cm1...mm6= 0. We deduce that the degree of Dis at least as
large as the smallest degree of such a monomial xm1
1···xmn
n, hence
deg D=N(a1+···+an)−(d1+···+dn)
≥(N−t1)a1+···+ (N−tn−1)an−1+ (t1+···+tn−1−n+ 1)an,
which implies the inequality (7).
F-thresholds, tight closure, integral closure, and multiplicity bounds 19
To finish the proof, we will use the following claim.
Claim. Let αi, βi, γibe real numbers, for 1 ≤i≤n. If 1 = γ1≤γ2≤...≤γn, and if
γ1α1+···+γiαi≥γ1β1+···+γiβifor all i= 1,...,n, then α1+···+αn≥β1+···+βn.
Proof of Claim. Let λi=αi−βifor 1 ≤i≤n, so that γ1λ1+· ·· +γiλi≥0 for all
i= 1,...,n. We prove that λ1+···+λn≥0 by induction on n, the case n= 1
being trivial. Suppose that n > 1 and that there is isuch that λi<0 (otherwise the
assertion to prove is clear). We must have i≥2, and since γi≥γi−1, it follows that
γiλi≤γi−1λi. Let us put γ′
j=γjfor 1 ≤j≤i−1 and γ′
j=γj+1 for i≤j≤n−1.
Define also λ′
j=λjfor 1 ≤j≤i−2, λ′
i−1=λi−1+λiand λ′
j=λj+1 for i≤j≤n−1.
It is straightforward to check that γ′
1λ′
1+···+γ′
jλ′
j≥0 for all j= 1,...,n−1,
hence the induction hypothesis implies λ1+···+λn=λ′
1+···+λ′
n−1≥0.
We now set αi=tifor 1 ≤i≤n−1 and αn=N−t1−· ··−tn−1+n−1. We put
βi=di/aiand γi=ai/a1for 1 ≤i≤n. Since a1≤ ·· · ≤ an, we deduce 1 = γ1≤
··· ≤ γn. Moreover, using (5) and (7), we get γ1α1+···+γiαi≥γ1β1+···+γiβi
for 1 ≤i≤n. Using the above claim, we conclude that
N+n−1 = α1+···+αn≥β1+···+βn=d1
a1
+··· +dn
an.
Comparing the arithmetic and geometric means of {di/ai}i, we see that
(N+n−1)na1. . . an≥nnd1. . . dn.
Since e(a) = a1···anand e(J) = d1···dn, this concludes the proof.
When Jis not necessarily a parameter ideal, we can prove another inequality
involving the F-threshold cJ(a), generalizing the results in [dFEM] and [TW].
Proposition 5.10. If (R, m)is a d-dimensional regular local ring of characteristic
p > 0, and if a, J are m-primary ideals in R, then we have the following inequality:
e(a)≥d
cJ(a)d
(cJ(m)−d+ 1).
Proof. As in the proof of Theorem 5.6, we do a reduction to the monomial case. We
first see that it is enough to show that if Ris the polynomial ring k[x1,...,xd] and
m= (x1,...,xd), and a,Jare m-primary ideals, then
(9) ℓ(R/a)≥1
d!d
cJ(a)d
(cJ(m)−d+ 1).
Claim. We can find monomial ideals a1and J1such that
(10) ℓR(R/a) = ℓR(R/a1),cJ(a)≥cJ1(a1),and cJ(m) = cJ1(m).
This reduces the proof of (9) to the case when both aand Jare monomial ideals.
20 C. Huneke, M. Mustat¸˘a, S. Takagi, and K.-i. Watanabe
Proof of claim. We do a two-step deformation to monomial ideals. We consider first
a flat deformation of aand Jto a′and J′, respectively, where for an ideal I⊆R,
we denote by I′the ideal defining the respective tangent cone at the origin. We
then fix a monomial order λ, and consider a Gr¨obner deformation of a′and J′to
a1:= inλ(a′) and J1:= inλ(J′), respectively. It follows as in the proof of Theorem 5.6
that the first two conditions in (10) are satisfied. For the third condition, in light of
Example 2.7 (3) it is enough to show that
mr⊆Jiff mr⊆inλ(J).
It is clear that if mr⊆J, then mr⊆J′and mr⊆J1. For the converse,
suppose that mr⊆J1. Since J′and J1are both homogeneous ideals, and since
dimk(R/J1)r= dimk(R/J′)r(see [Eis, Ch. 15]), it follows that mr⊆J′(note that
if Iis a homogeneous ideal in R, then mr⊆Iif and only if (R/I)r= 0). We know
that ms⊆Jfor some s, hence in order to prove that mr⊆Jit is enough to show
the following assertion: if mt⊆J′and mt+1 ⊆J, then mt⊆J. It is easy to check
that (J∩mt)′=J′∩mt, and since mt+1 ⊆J, we see that J∩mtis homogeneous,
hence
mt⊆J′∩mt= (J∩mt)′=J∩mt.
We return to the proof of Proposition 5.10. From now on we assume that aand J
are m-primary monomial ideals. Arguing as in the proof of Theorem 5.6, and using
Example 2.7 (3), we see that it is enough to show
Vol Rd
≥0rP(a)≥1
d!d
cJ(a)d
(r+ 1),
where r:= max{s∈Z≥0|ms6⊆ J}. By definition, we can choose a monomial
xr1
1···xrd
dof degree rthat is not contained in J. Since τ(acJ(a))⊆Jby Proposition
2.4, this monomial cannot belong to τ(acJ(a)). Using the description of generalized
test ideals of monomial ideals (see [HY, Theorem 4.8]), this translates as
(r1+ 1,...,rd+ 1) 6∈ Int(cJ(a)·P(a)).
Therefore we can find a hyperplane H:u1/a1+···+ud/ad= cJ(a) passing through
the point (r1+ 1,...,rd+ 1) such that
(11) H+:= (u1, . . . , ud)∈Rd
≥0|u1
a1
+···+ud
ad≥cJ(a)⊇cJ(a)·P(a).
Note that we get cJ(a) = (1 + r1)/a1+···+ (1 + rd)/ad. Comparing the arithmetic
and geometric means of {(1 + ri)/ai}i, we see that
cJ(a)
dd
=1 + r1
da1
+···+1 + rd
dadd
≥(1 + r1)...(1 + rd)
a1. . . ad≥1 + r
a1. . . ad
.
F-thresholds, tight closure, integral closure, and multiplicity bounds 21
On the other hand, (11) implies
Vol Rd
≥0rP(a)≥Vol Rd
≥0r(1/cJ(a))H+
=a1. . . ad
d!
≥1
d!d
cJ(a)d
(r+ 1) .
Acknowledgment. We thank all the participants in the AIM workshop “Integral
Closure, Multiplier Ideals and Cores” for valuable comments. Especially, we are
indebted to Mel Hochster for the suggestion to introduce the invariant ecJ(a). We
are also grateful to Tommaso de Fernex for suggesting Theorem 5.6. The first
author was partially supported by NSF grant DMS-0244405. The second author
was partially supported by NSF grant DMS-0500127 and by a Packard Fellowship.
The third and fourth authors were partially supported by Grant-in-Aid for Scientific
Research 17740021 and 17540043, respectively. The third author was also partially
supported by Special Coordination Funds for Promoting Science and Technology
from the Ministry of Education, Culture, Sports, Science and Technology of Japan.
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Department of Mathematics, University of Kansas, Lawrence, KS 66045-7523,
USA
E-mail address :huneke@math.ku.edu
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA
E-mail address :mmustata@umich.edu
Department of Mathematics, Kyushu University, 6-10-1 Hakozaki, Higashi-ku,
Fukuoka-city 812-8581, Japan
E-mail address :stakagi@math.kyushu-u.ac.jp
Department of Mathematics, College of Humanities and Sciences, Nihon Univer-
sity, Setagaya-Ku, Tokyo 156-0045, Japan
E-mail address :watanabe@math.chs.nihon-u.ac.jp
