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AMERICAN MATHEMATICAL SOCIETY Volume 366, Number 4, April 2014, Pages 1773–1795 S 0002-9947(2013)05856-4

Article electronically published on November 25, 2013

CARTIER MODULES ON TORIC VARIETIES

JEN-CHIEH HSIAO, KARL SCHWEDE, AND WENLIANG ZHANG

Abstract. Assume that X is an aﬃne toric variety of characteristic p > 0.

Let Δ be an eﬀective toricQ-divisor such that KX+ Δ isQ-Cartier with index not divisible by p and let φΔ: FeOX→ OX be the toric map corresponding to Δ. We identify all ideals I ofOX with φΔ(FeI) = I combinatorially and also in terms of a log resolution (giving us a version of these ideals which can be deﬁned in characteristic zero). Moreover, given a toric ideal a, we identify all ideals I ﬁxed by the Cartier algebra generated by φΔand a; this answers a question by Manuel Blickle in the toric setting.

1. Introduction

Suppose that R is a ring of characteristic p > 0 and F : R→ R is the Frobenius map which we always assume is a ﬁnite map. If φ : R → R is a splitting of the Frobenius, then there are ﬁnitely many ideals I such that φ(I) ⊆ I; see [KM09], [Sch09]. These ideals are called φ-compatible and are an interesting and useful collection of objects to study in their own right (they are closely related to the characteristic zero notion of “log canonical centers”).

Much more generally, suppose that R is a reduced ring and φ : R → R is an additive map that satisﬁes the condition φ(rpe · x) = rφ(x) for all r, x ∈ R (for example, a splitting of the Frobenius). In [BB09], M. Blickle and G. B¨ockle generalize the above-mentioned ﬁniteness results and show that there are ﬁnitely many I ⊆ R such that φ(I) = I (we call such ideals φ-ﬁxed). In [Bli09], M. Blickle generalized the notion of φ-ﬁxed ideals to include the additional data of an ideal a to a formal real power t ≥ 0, in fact, he generalized these ﬁxed ideals in even greater settings. However, the full ﬁniteness results still remain illusive. Explicitly, it is natural to study ideals I such that



n>0

φn



at(pne−1)· I

= I,

where φndenotes the eth iterate of φ and J denotes the integral closure of an ideal.

We call such ideals φ, at-ﬁxed.

These φ-ﬁxed ideals are exactly the ideals for which one has certain Fujita-type global generation statements [Sch11a, Section 6]. In particular, identifying these ideals might be very useful in the problem of lifting sections for projective varieties

Received by the editors August 18, 2011 and, in revised form, April 10, 2012.

2010 Mathematics Subject Classiﬁcation. Primary 14M25, 13A35, 14F18, 14B05.

The ﬁrst author was partially supported by the NSF grant DMS #0901123.

This research was initiated at the Commutative Algebra MRC held in June 2010. Support for this meeting was provided by the NSF and AMS.

The second author was supported by an NSF postdoctoral fellowship and also by NSF grant DMS #1064485.

The third author was partially supported by the NSF grant DMS #1068946.

2013 American Mathematical Societyc Reverts to public domain 28 years from publication 1773

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in characteristic p > 0. However, very few examples of the sets of φ-ﬁxed ideals, let alone φ, at-ﬁxed ideals, are known. In this paper we compute these ideals in the toric setting. In other words, X = Spec k[S] is an aﬃne toric variety, φ : FeOX → OX

is a toric map and a is a monomial ideal. Here S = M ∩ σ for some lattice M and some rational convex polyhedral cone σ in MR. Because φ is a toric map, we can write φ( ) = φc(x−w· ) for some w ∈ M, where φc is the canonical splitting1 of Fe: k[S]→ k[S]. Finally let t = a/b be a rational number such that p does not divide b. Set P = t Newt(a) and F = {faces of P }. For τ ∈ F , denote Iτ :=xv|v ∈ relint(1−pwe + τ)∩ S for some τ⊇ τ . Our main result is as follows:

Main Theorem (Theorem 3.5). 

n>0φn



at(pne−1)· I

= I if and only if I =



τ∈GIτ for someG ⊆ F , where the sum is taken over all positive integers n > 0 such that t(pne− 1) is an integer.

As an immediate corollary, we obtain:

Corollary. Suppose that (k[S], φ, at) is as above. Then there are only ﬁnitely many ideals I such that 

n>0φn



at(pne−1)· I

= I, where we again sum over n such that t(pne− 1) is an integer.

This answers a question by Manuel Blickle ([Bli09, Question 5.4]) in the toric setting. Even in the case when t = 0 (we deﬁne J0= R for any ideal J ) or a = R, i.e. we have a pair (R, φ) instead of a triple (R, φ, at), our result identiﬁes all φ- ﬁxed ideals. We illustrate our Main Theorem in this special case below; in this case σ = Newt(a).

In the following diagram, circles represent the monomials of the semi-group ring k[S] and solid lines represent the boundaries of σ. Given φ as above, we consider the vector 1−pwe.

x0y0

1−pew

x3y2

The φ-ﬁxed ideals will each be generated by monomials contained in the interior or boundary of the above “dotted” region and we can explicitly identify them pictorially. Explicitly, as our Main Theorem says, each of the φ-ﬁxed ideals will

1The canonical splitting is the map which sends a monomial xmto xm/peif m/peis an integer, and otherwise sends xmto zero.

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be generated by all monomials contained in one of the following shaded regions (in each region, the open circle corresponds to the point 1−pwe):

I

w 1−pe

II III IV V

While each of the ideals associated with these diﬀerent bodies are potentially dif- ferent, in many cases (depending on the particular w) they are the same.

As we have already noted, in the case that φ is a Frobenius splitting, the φ- ﬁxed ideals are closely related to log canonical centers (a notion deﬁned by using a resolution of singularities). It is thus natural to ask if these ideals I such that φ(I) = I are also related to a notion deﬁned using a resolution of singularities. At least in the toric setting, we identify a class of ideals, deﬁned using a resolution of singularities which coincides with the φ-ﬁxed ideals I. Our main result on relating φ-ﬁxed ideals and resolutions of singularities is the following.

Theorem (Theorem 5.5). Let X be an aﬃne toric variety of characteristic p > 0 and let Δ be an eﬀective torus-invariant Q-divisor on X such that KX+ Δ isQ- Cartier. Let a be a toric ideal and t a non-negative rational number which can be written without p in its denominator. Let φ : FeOX → OX be the toric map corresponding to Δ. Then an ideal I ⊂ OX satisﬁes 

n>0φn



at(pne−1)· I

= I if and only if there exists a toric log resolution π : X → X of (X, Δ, a) such that a·OX =OX(−G) and an eﬀective divisor E on Xwith π(E)⊂ Supp Δ∪Sing X ∪ V (a) such that

I = πOX( KX− π(KX+ Δ)− tG + εE ) for 1  ε > 0.

Perhaps this result should not be unexpected; we explain the motivation for this result in the special case that a = R. Note that φ extends to a map φ : FeM → M whereM = OX( KX− π(KX+ Δ) + E ) for every eﬀective Q-divisor E on X, [HW02, Proof #2 of Theorem 3.3], [Sch10, Theorem 6.7]. Now choose 1 ε > 0, and it follows that the fractional ideal

I = OX( KX− π(KX+ Δ) + εE )

is φ-ﬁxed. The ideals discussed in Theorem 5.5 are thus exactly the push forwards of φ-ﬁxed fractional ideals whose push forwards are honest (non-fractional) ideals ofOX.

2. Preliminaries for fixed ideals

Throughout this paper, all rings are Noetherian and excellent and possess dual- izing complexes and all schemes are separated. Mostly we will work in the setting of toric varieties where all these conditions are automatic.

In this section, R is a normal F -ﬁnite domain of characteristic p > 0 which admits an additive map φ : R → R satisfying the relation that φ(rpex) = rφ(x)

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for all r, x ∈ R. Such a φ is called a p−e-linear map. Typical examples of such maps are the maps that split the Frobenius map F : R → R. It can be diﬃcult to distinguish the source and target of this map. Thus for any R-module M , we use FeM to denote the R-module isomorphic to R as an Abelian group but with the R-action, r.m = rpem. From this perspective, a p−e-linear map is simply an R-linear map FeR→ R. For the rest of the paper, all p−e-linear maps φ : R→ R will be written as R-linear maps φ : FeR→ R.

Once a map φ : FeR→ R has been ﬁxed, R ﬁts into the theory of Cartier mod- ules developed by Blickle and B¨ockle [BB09]. One of the main theorems in [BB09]

guarantees that there are only ﬁnitely many ideals I of R satisfying φ(FeI) = I.

We will call such an I a φ-ﬁxed ideal (according to the terminology of [Bli09], these ideals are also called F -pure Cartier-submodules of (R, φ)). Moreover, the theory of Cartier modules in [BB09] was generalized in [Bli09] which we will review brieﬂy.

2.1. Cartier algebras.

Deﬁnition 2.1 ([Sch11b]; cf. [Bli09]). Let R be a commutative Noetherian ring of characteristic p > 0. An algebra of p−e-linear maps on R, or simply a Cartier algebra on R, is an N-graded ring C := 

e≥0Ce such that each φ ∈ Ce is a map φ : FeR→ R. Furthermore, the multiplication of elements of C corresponds to the composition of maps. In other words, for φ∈ Ce and ψ∈ Cd, we deﬁne:

φ· ψ := φ ◦ (Feψ).

We call the pair (R,C) F -pure if there exists a surjective φ ∈ Cefor some e > 0.

If one uses Ende(R) to denote HomR(FeR, R), thenCR:=

eEnde(R) will be an example of an R-Cartier-algebra. In this paper, we will focus on the subring determined by an ideal a, a non-negative real number t and a p−e-linear map φ, denoted byCφ,at, i.e.

Cφ,at :=

n

φn· Fneat(pne−1)=

n≥0

Cnφ,at,

where the sum is taken over all n such that t(pne− 1) is an integer. As mentioned before, J is used to denote the integral closure of J ; see [HS06]. If a = R or t = 0, then we useCφ to denoteCφ,at.

Furthermore, we deﬁne

C+φ,at :=

n>0

Cnφ,at.

Finally, we state the deﬁnition of our main object of study in this paper.

Deﬁnition 2.2. If an ideal I satisﬁesC+φ,at(I) = I, i.e. 

n>0φn(Fneat(pne−1)·I) = I, then I is calledCφ,at-ﬁxed. According to the terminology of [Bli09], these ideals are also called F -pure Cartier-submodules of the triple (R, φ, at).

Note that when either t = 0 or a = R, this Cartier-algebra Cφ,at is the sub- Cartier-algebra of CR generated by φ; we will denote this sub-Cartier-algebra by Cφ. In this special case, one of the main theorems in [BB09] guarantees that there are only ﬁnitely many ideals I of R satisfying φ(FlI) = I (i.e. C+φ(I) = I). However, the ﬁniteness of Cφ,at-ﬁxed ideals was left as an open question in [Bli09, Question 5.4], to which our Main Theorem (Theorem 3.5) gives a partial positive answer.

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2.2. Fixed ideals. Fixed ideals have appeared throughout commutative algebra and representation theory. Indeed, if φ : FR→ R is a splitting of the Frobenius, then a compatibly-φ-split ideal J ⊆ R is an ideal such that φ(FJ ) = J . Alternately, note that there is always an R-linear map φe : FeωR → ωR for each e > 0 – the Grothendieck trace map. If R is Gorenstein and local, then ωR ∼= R and thus we obtain a canonical p−e-linear map φe: FeR→ R. In this case, the smallest non-zero ideal J such that φe(FeJ ) = J is the (big) test ideal of R, [Sch09]. Furthermore, if φ is surjective, the largest such proper ideal is the splitting prime of Aberbach and Enescu, [AE05]. In this section, we develop the theory ofCφ,at-ﬁxed ideals.

We mention the following results about the set of φ-ﬁxed ideals which we will need.

Proposition 2.3. Suppose that (R, φ, at) is a triple as above with R an F -ﬁnite domain.2

(i) The set of Cφ,at-ﬁxed ideals of R is closed under sum.

(ii) If φ is surjective, then the set of φ-ﬁxed ideals is closed under intersection.

(iii) If φ is surjective, then any φ-ﬁxed ideal is a radical ideal.

(iv) If R is a normal domain and φ corresponds to a Q-divisor Δφ as in sub- section 2.3 below, then the test ideal3 τ (R, Δφ, at) is the unique smallest non-zero Cφ,at-ﬁxed ideal of R.

(v) There are ﬁnitely many φ-ﬁxed ideals.4

(vi) For an element d ∈ R, deﬁne a new map ψ( ) = φ(Fedpe−1· ). Then an ideal J ⊆ R is Cφ,at-ﬁxed if and only if dJ isCψ,at-ﬁxed.

Proof. Part (i) is trivial from the deﬁnition. Part (ii) follows from the observation that if φ is surjective, then any ideal J satisfying the condition φ(FeJ ) ⊆ J is automatically φ-ﬁxed, and the set of these ideals is closed under intersection. Part (iii) is again easy. Part (iv) is an easy exercise based upon [Sch11b]. Part (v) is, as mentioned, one of the main results of [BB09].

To prove (vi), suppose ﬁrst that J isCφ,at-ﬁxed so that

n>0φn(at(pne−1)J ) = J .

Then 

n>0

ψn(at(pne−1)dJ ) =

n>0

φn(Fnedpne−1at(pne−1)dJ )

=

n>0

φn(Fnedpneat(pne−1)J )

= d

n>0

φn(at(pne−1)J )

= dJ.

The converse statement merely reverses this. 

Remark 2.4. We will see in the toric setting that the set of φ-ﬁxed ideals is closed under intersection for any φ. It would be interesting to discover if this holds more generally.

2We make this hypothesis only for simplicity; most of what follows below can be generalized outside of this setting with minimal work.

3The reader may take this to be the deﬁnition of the test ideal if they are not already familiar with it.

4As our main result shows, in the toric setting there are also ﬁnitely manyCφ,at-ﬁxed ideals.

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Remark 2.5. Suppose that R is a normal domain and φ : FeR→ R is an R-linear map as in Proposition 2.3. One can always extend φ to a map ¯φ : FeK(R)→ K(R) where K(R) is the fraction ﬁeld of R. While it is true that there are only ﬁnitely many φ-ﬁxed ideals of R, in general there are inﬁnitely many φ-ﬁxed fractional ideals of K(R). For example, consider the ring R = k[x] with the canonical splitting φc : FR → R. Then the fractional ideal generated by x+11 is φc-ﬁxed. Indeed, a generating set of this fractional ideal over Rp is

 xi x+1

0≤i≤p−1. Notice that φc

 xi x+1



= φc

(x+1)p−1xi (x+1)p



=x+11 φc

xp−1+i+p−1

1

xp−1+i−1+· · · + xi

= bxx+1i/p

for some constant b, which proves the claim. Clearly the fractional ideal generated by x+11 is not toric. Of course, the same statement holds for the fractional ideal generated by x+λ1 for any λ∈ k as well as for many other ideals.

We now give a method for constructing φ-ﬁxed ideals. While we will not use it directly, an analog of this result for ideals deﬁned using the resolution of singularities is the key observation which allows us to characterize φ-ﬁxed ideals via a resolution;

compare with Proposition 4.5. We ﬁrst recall that given φ : FeR → R, we can compose φ with itself to obtain a map φ2 = φ◦ Feφ : F2eR → R and similarly construct φn for any positive integer n. While a priori, we may have an inﬁnite descending chain of ideals φn(FneR) ⊇ φn+1(F(n+1)eR) ⊇ . . . . It is a theorem of Gabber that this chain eventually stabilizes; see [Gab04] (also see [Bli09] for a generalization and [HS77] for the local dual statement in the geometric setting).

Deﬁnition 2.6. The stable image φn(FneR) = φn+1(F(n+1)eR) = . . . will be denoted by S(φ) ⊆ R. It is automatically φ-ﬁxed (and it is, by construction, the largest φ-ﬁxed ideal).

In [FST11], S(φ) was denoted by σ(φ). We avoid this notation because we are already utilizing σ.

Proposition 2.7. Suppose that R is a domain and d∈ R is a non-zero element.

Then for any n 0, if we deﬁne a map ψn: FneR→ R by the formula ψn( ) = φn(Fned· ), we have that S(ψn) is φ-ﬁxed.

Proof. We ﬁrst claim that S(ψn)⊆ S(ψn+1) for any n (compare also with [FST11, Proposition 14.10(1)]). To prove this claim, ﬁrst notice that for any α : FeR→ R, S(α) = S(αm). Thus, in order to show the claim for ψ, it suﬃces to show that ψnn+1(F(n+1)neJ )⊆ ψn+1n (F(n+1)neJ ) for every ideal J . However, because

1 + pne+· · ·+pn2e= p(n+1)ne− 1

pne− 1 pn(n+1)e− 1

p(n+1)e− 1 = 1 + p(n+1)e+· · ·+p(n−1)(n+1)e we obtain that

ψn+1n (F(n+1)neJ ) = φn(n+1)(F(n+1)ned1+pne+···+pn2 eJ )

⊆ φn(n+1)(F(n+1)ned1+p(n+1)e+···+p(n−1)(n+1)e

J ) = ψn+1n (F(n+1)neJ ), which proves the claim.

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We choose n, which stabilizes this chain S(ψn) = S(ψn+1). Suppose now m > 0 is such that S(ψn) = ψmn(Fm(ne)R). Then

φ(S(ψn))

= φ(Feψm+1n (F(m+1)neR))

= ψn+1



F(n+1)enm(FmneR))



= ψn+1



F(n+1)eS(ψn)



= ψn+1



F(n+1)eS(ψn+1)



= S(ψn+1)

= S(ψn),

which proves the proposition. 

2.3. The relation between φ and Q-divisors. Later, we will relate the φ-ﬁxed ideals with the ideals coming from a resolution of singularities in characteristic 0 (e.g. multiplier ideals, Fujino’s non-LC ideal, and the ideals deﬁning arbitrary unions of log canonical centers). This relation comes from a correspondence between pairs (X = Spec R, Δ) and certain p−e-linear maps φ : R → R in the theory of F -singularities. The reader is referred to [Sch09] for a detailed account of this correspondence; also see [ST12].

Suppose that (X, Δ) is a pair where X is a variety of ﬁnite type over an F - ﬁnite ﬁeld k such that KX + Δ is Q-Cartier with index not divisible by p > 0.

Further suppose that Δ is an eﬀectiveQ-divisor such that (pe− 1)Δ is integral and (pe− 1)(KX+ Δ) is Cartier for some e. Then there is a bijection of sets:

EﬀectiveQ-divisors Δ on X such that (pe− 1)(KX+ Δ) is Cartier

←→

Line bundlesL and non-zero elements of HomOX(FeL , OX)



∼.

The equivalence relation on the right side identiﬁes two maps φ1 : FeL1 → OX

and φ2: FeL2→ OX if there is an isomorphism γ :L1→ L2 and a commutative diagram

FeL1 φ1

−−−−→ OX Feγ

⏐⏐

 ⏐⏐id

FeL2 φ2

−−−−→ OX

.

Given Δ, setL = OX((1− pe)(KX+ Δ)). Then observe that

FeOX((pe− 1)Δ) ∼= FeH omOX(L , OX((1− pe)KX)) ∼=H omOX(FeL , OX).

The choice of a section η ∈ OX((pe− 1)Δ) corresponding to (pe− 1)Δ thus gives a map φΔ: FeL → OX. The choice depends on various isomorphisms selected, but this is harmless for our purposes.

For the converse direction, an element φ∈ H omOX(FeL , OX) ∼= FeL−1((1 pe)KX) determines an FeOX-linear map

FeOX 1→φ

−−−→ H omOX(FeL , OX)−→ F eL−1((1− pe)KX)

which corresponds to an eﬀective Weil divisor D such that OX(D) ∼= L−1((1 pe)KX). Set Δφ= pe1−1D.

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Remark 2.8. Explicitly, suppose φ : FeR → R is an R-linear map and d ∈ R.

Deﬁne a new map ψ( ) = φ(Fed· ). Then Dψ= Dφ+pe1−1div(d).

Deﬁnition 2.9. Suppose that φ : FeR→ R corresponds to a divisor Δ. Then we deﬁne S(R, Δ) to be S(φ), where S(φ) is deﬁned in the paragraph before Proposition 2.7.

We illustrate the above construction by the case where X is an aﬃne toric variety (which we recall has a trivial Picard group, so that every line bundle is isomorphic toOX). Let M be a lattice and σ be a rational convex polyhedral cone in MR. Set R = k[σ∩ M]. Suppose Δ is an eﬀective Q-divisor on X = Spec R as above. Then we can write

(1− pe)(KX+ Δ) = divX(xw)

for some w∈ M. Then a map φΔcorresponding to Δ can be expressed as φΔ( ) = φc(x−w· ),

where φc is the canonical splitting on R deﬁned by φc(xv) =

xpev if pve ∈ M, 0 otherwise.

Let us explain these claims carefully since this identiﬁcation is critical for what follows. Our ﬁrst claim is that the map φc corresponds to the torus invariant divisor Δc =−KX. It is suﬃcient to show that φc ﬁxes every height-one prime torus invariant ideal (which implies that Δφc contains each torus invariant divisor as a component) and does not ﬁx any other height-one ideal (which implies that Δφc is toric). While both these statements are well known to experts, we point out that the ﬁrst statement is simply [Pay09, Proposition 3.2], while the second is a very special case of the proof of Proposition 3.2 below. So now suppose that Δ is a torus invariant divisor such that (1− pe)(KX+ Δ) is Cartier and thus is equal to divX(xw). Therefore

(1− pe)(KX+ Δ) = divX(xw) + 0 = divX(xw) + (1− pe)(KX+ (−KX)).

Dividing through by (1− pe) gives us Δ = (−KX) +pe1−1divX(x−w). However, it is easy to see that given any map β : FeOX→ OX and any element d∈ FeFrac R, the map α( ) = β(d· ), if it is indeed a map α : FeOX → OX, has associated divisor Δα= Δβ+pe1−1divX(d). In other words, the map φΔ as described above does indeed correspond to Δ.

Notation. For an R-Weil divisor D = r

j=1djDj such that the Dj’s are distinct prime Weil divisors, we deﬁne

D =

r j=1

dj Dj andD =

r j=1

djDj,

where for each real number x, the round-up (resp. round-down) x (resp. x) denotes the integer deﬁned by x ≤ x < x + 1 (resp. x1 <x ≤ x). We also deﬁne

D≥k= 

dj≥k

djDj.

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3. Cφ,at-fixed ideals on toric varieties

In this section, we compute the ﬁxed ideals of certain Cartier algebras (in the sense of Blickle) on toric varieties.

We ﬁx M to be a lattice and σ to be a rational convex polyhedral cone in MR, set S = σ∩ M, R = k[S], and X = Spec(k[S]). Let d denote the dimension of X, and let a be a monomial ideal on X. Let Δ be a toric divisor on X such that (1−pe)(KX+ Δ) = divX(xw) for some positive integer e and some element w∈ M.

Denote φc: FeOX→ OX as the canonical splitting on X. Consider a p−e-linear map φΔ( ) = φc(x−w ) (w is determined by Δ; when Δ is clear we will simply write φ), i.e.

φ(xu) =

xu−wpe if up−we ∈ M, 0 otherwise.

Given a rational number t > 0 that can be written without p in its denominator, we describe all the ideals I ﬁxed by the Cartier algebra Cφ,at generated by φ and at, i.e. the ideals I satisfying



n>0

φn



at(pne−1)· I

= I,

where the sum runs through all n > 0 such that t(pne− 1) is an integer.

Lemma 3.1. Let R be a normal toric algebra, and let I, J be monomial ideals of R. Denote Newt(I) as the Newton polytope of an ideal I. Then

(1) I =xv | v ∈ Newt(I) ∩ M [Ful93].

(2) Newt(I) + Newt(J ) = Newt(IJ ); in particular, In =xv|v ∈ (n Newt(I)) ∩ M for any positive integer n.

(3) If I = xg|g ∈ Γ , then Newt(I) = Conv(Γ) + σ, where Conv(Γ) is the convex hull of Γ.

Proof. We simply prove (2) and (3).

For (2), by deﬁnition Newt(I) = Conv{a ∈ M|xa ∈ I}. Then note that Newt(I) + Newt(J ) is simply

{(s1u1+· · · + slul) + (t1v1+· · · + tmvm)| l, m > 0,

Σsi= 1, si≥ 0, xui∈ I, Σtj= 1, tj≥ 0, xvj ∈ J}.

By allowing repeats in the ui and vj and possibly making some si, tj = 0, we may assume that l = m and that si= ti for all i. Thus Newt(I) + Newt(J ) is equal to

{s1(u1+ v1) +· · · + sl(ul+ vl)| l > 0, Σsi= 1, si≥ 0, xui ∈ I, xvi∈ J}, which is clearly equal to Newt(IJ ) as desired.

We now prove (3). Every point in Newt(I) is of the form

niaifor some ai∈ M with xai ∈ I, ni > 0, and 

ni = 1. For each i, since xai ∈ I, there exist gi ∈ Γ and si ∈ S such that ai = gi+ si. Therefore, 

niai = (

nigi) + ( nisi) Conv(Γ) + σ. So Newt(I)⊆ Conv(Γ) + σ. The proof of the other containment is

similar to (2) above. 

Proposition 3.2. Let R, φ, atbe as above. If I is Cφ,at-ﬁxed, then I is monomial and{v|xv ∈ I} ⊆ 1−pwe + t Newt(a).

(10)

Proof. For any element h =

cixmi∈ I, ﬁx any term xmi and set m = mi. Since



n>0

φn



at(pen−1)· I

= I,

where the sum is taken over all integers n > 0 such that t(pen− 1) is an integer, there exist n > 0, a ∈ Newt(at(pen−1)), and m ∈ S such that xm is a term of some element in I and that φn(xa· xm) = xm. Then

m = a+ mpenpe−1−1w pen .

Repeating the same process k times with m in place of m, we can write

m =

k−1

i=1 pe(kj=i+1n(j))a(i)

+ a(k)+ m(k)pe(kj=1n(j))−1

pe−1 w

pe(kj=1n(j)) ,

where a(i)∈ Newt

 at



p(en(i) )−1



and m(k)∈ S.

By Lemma 3.1 and the fact that Newt(I) + S = Newt(I), we have mpekj=1n(j)+pekj=1n(j)− 1

pe− 1 w

=

k−1



i=1

pe(kj=i+1n(j))a(i)



+ a(k)+ m(k)

k−1



i=1

pe(kj=i+1n(j)) Newt

 at



p(en(i) )−1



+ Newt

 at

 pen(k)−1



=t

k−1

i=1

pe(kj=i+1n(j))

p(en(i))− 1

+ (p(en(k))− 1)



· Newt(a)

=t

k−1

i=1

pe(kj=in(j)) − pe(kj=i+1n(j)) 

+ (p(en(k))− 1)



· Newt(a)

=t

 

pe(kj=1n(j)) − p(en(k))

+ (p(en(k))− 1)



· Newt(a)

=t



pe(kj=1n(j))− 1

· Newt(a).

Dividing by pekj=1n(j)and letting k go to inﬁnity, we see that m∈ 1−pwe+t Newt(a).

Now for any n > 0 such that t(pen− 1) is an integer, (pen− 1)m + ppene−1−1w∈ t(pen− 1) Newt(a), and hence by Lemma 3.1

x(pen−1)m+pen−1pe−1w∈ at(pen−1).

Notice that for any term xmj in h∈ I, φn(x(pen−1)m+pen−1pe−1 w· xmj) = xm+mj −mpen . Therefore, φn(x(pen−1)m−pen−1pe−1 w· h) ∈ I is a non-zero constant multiple of xmfor

n 0. So xm∈ I, as desired. 

(11)

Given any face F of t Newt(a), we will use relint(1−pwe+ F ) to denote the relative interior of 1−pwe+F . By the relative interior relint(C) of a convex set C with positive dimension, we mean the interior of C in the aﬃne hull of C. The relative interior of a convex set C can be characterized algebraically as follows.

Theorem 3.3 (Theorem 3.5 in [Brø83]). For any convex set C, the following are equivalent:

(1) x∈ relint(C);

(2) for any line A in the aﬃne hull of C with x∈ A, there are points y0, y1 A∩ C such that x = δy0+ (1− δ)y1 for some δ∈ (0, 1);

(3) for any point y∈ C with y = x, there exists z ∈ C such that x = δy+(1−δ)z for some δ∈ (0, 1).

For our purpose, the relative interior of a point is the point itself. For example, there are four faces of the following polyhedron:

σ0

σ1

σ2

σ3

The relative interiors of the three positive dimensional faces are pictured as below.

relint σ1 relint σ2 relint σ3

Proposition 3.4. Suppose I isCφ,at-ﬁxed and ﬁx a face F of t Newt(a). If{v|xv I} ∩ (1−pwe + F )= ∅, then {v|xv∈ I} ⊇ relint(1−pwe + F )∩ S.

Proof. Choose v0 ∈ {v|xv ∈ I} ∩ (1−pwe + F ) and v0 ∈ relint(1−pwe + F )∩ S such that v0= v0. Choose α1, . . . , αk 1−pwe+ F so that v0 = n0v0+k

i=1niαi, where 0 < ni< 1 andk

i=0ni = 1. For n 0, penn0> 1 and penv0 − v0= (penn0− 1)v0+

k i=1

penniαi ∈ (pen− 1)( w

1− pe + F )

pen− 1

1− pe w + t(pen− 1) Newt(a).

By Lemma 3.1 xpenv0−v0+pen−1pe−1w∈ at(pen−1), and hence xv0= φn(xpenv0−v0+pen−1pe−1w·

xv0)∈ I, as desired. 

(12)

Now, we are ready to describe theCφ,at-ﬁxed ideals. SetF={faces of t Newt(a)}, (Here, we assume t Newt(a)∈ F .) For F ∈ F , denote IF :=xv|v ∈ relint(1−pwe + F)∩ S for some F⊇ F .

Theorem 3.5. I is Cφ,at-ﬁxed if and only if I =

F∈GIF for some non-empty G ⊆ F .

Proof. First, suppose

n>0φn



at(pen−1)· I

= I, where the sum is taken over all integers n > 0 such that t(pen−1) is an integer. Let G be the subset of F consisting of all the faces F satisfyingv|xv ∈ I ∩ relint(1−pwe + F )= ∅. Then I ⊇

F∈GIF

by Proposition 3.4. By Proposition 3.2, xv ∈ I implies v ∈ relint(1−pwe + F )∩ S for some F ∈ G . So I ⊆

F∈GIF. Conversely, suppose I =

F∈GIF for someG ⊆ F . We ﬁrst show that



n>0

φn



at(pen−1)· I

⊇ I.

Let xv ∈ I; then v ∈ relint(1−pwe + F ) 1−pwe + t Newt(a) for some face F of t Newt(a). For any n > 0 such that t(pen− 1) is an integer,

(pen− 1)v ∈ pen− 1

1− pe w + t(pen− 1) Newt(a).

Therefore by Lemma 3.1,

x(pen−1)v+pen−1pe−1w∈ at(pen−1). So xv= φn



x(pen−1)v+pen−1pe−1w· xv

∈ φn

at(pen−1)· I

, as desired.

For the other containment, notice that it suﬃces to show that



n>0

φn



at(pen−1)· IF

⊆ IF

for any F . Suppose xu

n>0φn



at(pen−1)· IF



for some F . Then there exists n > 0 such that t(pen− 1) is an integer and that

penu = α + β−pen− 1 pe− 1 w

for some α and β such that xα ∈ at(pen−1) and xβ ∈ IF. In particular, α Newt(at(pen−1)) = t(pen− 1) Newt(a) (by Lemma 3.1) and β ∈ 1−pwe + relint γ for some γ⊇ F (because xβ∈ IF). Therefore,

u− w

1− pe = α

pen +(β−1−pwe) pen

(pen− 1)

pen t Newt(a) + 1

penrelint γ

⊆ t Newt(a).

In particular, u−1−pwe ∈ relint τ for some unique face τ in F . We claim that τ ⊇ γ.

This implies that xu∈ Iτ⊆ Iγ⊆ IF, which ﬁnishes the proof.

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