If φ : R → R is a splitting of the Frobenius, then there are finitely many ideals I such that φ(I

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: F_∗^eOX→ OX be the toric map corresponding to Δ. We identify all ideals I ofOX with φΔ(F_∗^eI) = 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(r^pe · 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



a^t(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, a^t-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

in characteristic p > 0. However, very few examples of the sets of φ-fixed ideals, let alone φ, a^t-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 : F_∗^eOX → 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 M_R. Because φ is a toric map, we can write φ( ) = φc(x^−w· ) for some w ∈ M, where φc is the canonical splitting¹ of F^e: 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 :=x^v|v ∈ relint(_1−p^we + τ^)∩ S for some τ^⊇ τ . Our main result is as follows:

Main Theorem (Theorem 3.5). 

n>0φⁿ



a^t(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(p^ne− 1) is an integer.

As an immediate corollary, we obtain:

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

n>0φⁿ



a^t(pne−1)· I

= I, where we again sum over n such that t(p^ne− 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, a^t), 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−p^we.

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 x^mto x^m/peif m/p^eis an integer, and otherwise sends x^mto zero.

be generated by all monomials contained in one of the following shaded regions (in each region, the open circle corresponds to the point _1−p^we):

I

w 1−p^e

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 : F_∗^eOX → OX be the toric map corresponding to Δ. Then an ideal I ⊂ OX satisfies 

n>0φⁿ



a^t(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 = π_∗OX^(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^ : F_∗^eM → 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(r^pex) = rφ(x)

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 F_∗^eM to denote the R-module isomorphic to R as an Abelian group but with the R-action, r.m = r^pem. From this perspective, a p^−e-linear map is simply an R-linear map F_∗^eR→ R. For the rest of the paper, all p^−e-linear maps φ : R→ R will be written as R-linear maps φ : F_∗^eR→ R.

Once a map φ : F_∗^eR→ 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(F_∗^eI) = 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 : F_∗^eR→ 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 ◦ (F_∗^eψ).

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(F_∗^eR, 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 byC^φ,at, i.e.

C^φ,at :=

n

φⁿ· F_∗^nea^t(pne−1)=

n≥0

Cn^φ,at,

where the sum is taken over all n such that t(p^ne− 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 useC^φ to denoteC^φ,at.

Furthermore, we define

C+^φ,at :=

n>0

Cn^φ,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(F_∗^nea^t(pne−1)·I) = I, then I is calledC^φ,at-fixed. According to the terminology of [Bli09], these ideals are also called F -pure Cartier-submodules of the triple (R, φ, a^t).

Note that when either t = 0 or a = R, this Cartier-algebra C^φ,at is the sub- Cartier-algebra of CR generated by φ; we will denote this sub-Cartier-algebra by C^φ. In this special case, one of the main theorems in [BB09] guarantees that there are only finitely many ideals I of R satisfying φ(F_∗^lI) = I (i.e. C+^φ(I) = I). However, the finiteness of C^φ,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.

2.2. Fixed ideals. Fixed ideals have appeared throughout commutative algebra and representation theory. Indeed, if φ : F_∗R→ R is a splitting of the Frobenius, then a compatibly-φ-split ideal J ⊆ R is an ideal such that φ(F∗J ) = J . Alternately, note that there is always an R-linear map φ^e : F_∗^eωR → 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 φ^e: F_∗^eR→ R. In this case, the smallest non-zero ideal J such that φ^e(F_∗^eJ ) = 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 ofC^φ,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, a^t) is a triple as above with R an F -finite domain.²

(i) The set of C^φ,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, a^t) is the unique smallest non-zero C^φ,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(F_∗^ed^pe−1· ). Then an ideal J ⊆ R is C^φ,at-fixed if and only if dJ isC^ψ,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(F_∗^eJ ) ⊆ 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 isC^φ,at-fixed so that

n>0φⁿ(a^t(pne−1)J ) = J .

Then 

n>0

ψⁿ(a^t(pne−1)dJ ) =

n>0

φⁿ(F_∗^ned^pne−1a^t(pne−1)dJ )

=

n>0

φⁿ(F_∗^ned^pnea^t(pne−1)J )

= d

n>0

φⁿ(a^t(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 manyC^φ,at-fixed ideals.

Remark 2.5. Suppose that R is a normal domain and φ : F_∗^eR→ R is an R-linear map as in Proposition 2.3. One can always extend φ to a map ¯φ : F_∗^eK(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 : F_∗R → R. Then the fractional ideal generated by _x+1¹ is φc-fixed. Indeed, a generating set of this fractional ideal over R^p 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

x^p−1+i+_p−1

1

x^p−1+i−1+· · · + xⁱ

= b^x_x+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 : F_∗^eR → R, we can compose φ with itself to obtain a map φ² = φ◦ F_∗^eφ : F_∗^2eR → R and similarly construct φⁿ for any positive integer n. While a priori, we may have an infinite descending chain of ideals φⁿ(F_∗^neR) ⊇ 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(F_∗^neR) = 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: F_∗^neR→ R by the formula ψn( ) = φⁿ(F_∗^ned· ), 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 : F_∗^eR→ R, S(α) = S(α^m). Thus, in order to show the claim for ψ, it suffices to show that ψ_nⁿ⁺¹(F_∗^(n+1)neJ )⊆ ψn+1ⁿ (F_∗^(n+1)neJ ) for every ideal J . However, because

1 + p^ne+· · ·+p^n2e= p^(n+1)ne− 1

p^ne− 1 ≥ p^n(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)ned^{1+pne+···+pn2 e}J )

⊆ φⁿ⁽ⁿ⁺¹⁾(F_∗^(n+1)ned^{1+p(n+1)e+···+p}(n−1)(n+1)e

J ) = ψ_n+1ⁿ (F_∗^(n+1)neJ ), which proves the claim.

We choose n, which stabilizes this chain S(ψn) = S(ψn+1). Suppose now m > 0 is such that S(ψn) = ψ^m_n(F_∗^m(ne)R). Then

φ(S(ψn))

= φ(F_∗^eψ^m+1_n (F_∗^(m+1)neR))

= ψn+1



F_∗^(n+1)e(ψ_n^m(F_∗^mneR))



= 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 (p^e− 1)Δ is integral and (p^e− 1)(KX+ Δ) is Cartier for some e. Then there is a bijection of sets:

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

←→

Line bundlesL and non-zero elements of Hom_OX(F_∗^eL , OX)



∼.

The equivalence relation on the right side identifies two maps φ1 : F_∗^eL1 → OX

and φ2: F_∗^eL2→ OX if there is an isomorphism γ :L1→ L2 and a commutative diagram

F_∗^eL1 φ₁

−−−−→ OX F_∗^eγ

⏐⏐

 ⏐⏐^id

F_∗^eL2 φ₂

−−−−→ OX

.

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

F_∗^eOX((p^e− 1)Δ) ∼= F_∗^eH omOX(L , OX((1− p^e)KX)) ∼=H omOX(F_∗^eL , OX).

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

For the converse direction, an element φ∈ H om_OX(F_∗^eL , OX) ∼= F_∗^eL⁻¹((1− p^e)KX) determines an F_∗^eOX-linear map

F_∗^eOX 1→φ

−−−→ H omOX(F_∗^eL , OX)−→ F^∼ _∗^eL⁻¹((1− p^e)KX)

which corresponds to an effective Weil divisor D such that OX(D) ∼= L⁻¹((1− p^e)KX). Set Δφ= _pe¹−1D.

Remark 2.8. Explicitly, suppose φ : F_∗^eR → R is an R-linear map and d ∈ R.

Define a new map ψ( ) = φ(F_∗^ed· ). Then Dψ= Dφ+_pe¹−1div(d).

Definition 2.9. Suppose that φ : F_∗^eR→ 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 M_R. Set R = k[σ∩ M]. Suppose Δ is an effective Q-divisor on X = Spec R as above. Then we can write

(1− p^e)(KX+ Δ) = divX(x^w)

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(x^v) =

x^pev if _p^ve ∈ 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 Δφ_c 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− p^e)(KX+ Δ) is Cartier and thus is equal to divX(x^w). Therefore

(1− p^e)(KX+ Δ) = divX(x^w) + 0 = divX(x^w) + (1− p^e)(KX+ (−KX)).

Dividing through by (1− p^e) gives us Δ = (−KX) +_pe¹−1divX(x^−w). However, it is easy to see that given any map β : F_∗^eOX→ OX and any element d∈ F_∗^eFrac R, the map α( ) = β(d· ), if it is indeed a map α : F_∗^eOX → 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 andD =

r j=1

djDj,

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= 

d_j≥k

djDj.

3. C^φ,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 M_R, 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−p^e)(KX+ Δ) = divX(x^w) for some positive integer e and some element w∈ M.

Denote φc: F_∗^eOX→ 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(x^u) = 

x^u−wpe if ^u_p^−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 C^φ,at generated by φ and a^t, i.e. the ideals I satisfying



n>0

φⁿ



a^t(pne−1)· I

= I,

where the sum runs through all n > 0 such that t(p^ne− 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 =x^v | v ∈ Newt(I) ∩ M [Ful93].

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

(3) If I = x^g|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|x^a ∈ I}. Then note that Newt(I) + Newt(J ) is simply

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

Σsi= 1, si≥ 0, x^ui∈ I, Σtj= 1, tj≥ 0, x^vj ∈ 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, x^ui ∈ I, x^vi∈ 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 x^ai ∈ I, ni > 0, and 

ni = 1. For each i, since x^ai ∈ 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, a^tbe as above. If I is C^φ,at-fixed, then I is monomial and{v|x^v ∈ I} ⊆ _1−p^we + t Newt(a).

Proof. For any element h =

cix^mi∈ I, fix any term x^mi and set m = mi. Since



n>0

φⁿ



a^t(pen−1)· I

= I,

where the sum is taken over all integers n > 0 such that t(p^en− 1) is an integer, there exist n^ > 0, a^ ∈ Newt(a^t(pen−1)), and m^ ∈ S such that x^m is a term of some element in I and that φ^n(x^a· x^m) = x^m. Then

m = a^+ m^−^pen_pe−1⁻¹w p^en .

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

m =

k−1

i=1 p^e(^kj=i+1n^(j))a⁽ⁱ⁾

+ a^(k)+ m^(k)−^pe(^kj=1n(j))₋₁

p^e−1 w

p^e(kj=1n(j)) ,

where a⁽ⁱ⁾∈ Newt

 a^t



p^{(en(i) )}−1



and m^(k)∈ S.

By Lemma 3.1 and the fact that Newt(I) + S = Newt(I), we have mp^ekj=1n(j)+p^ekj=1n(j)− 1

p^e− 1 w

=

_k−1



i=1

p^e(^kj=i+1n^(j))a⁽ⁱ⁾



+ a^(k)+ m^(k)

∈

_k−1



i=1

p^e(^kj=i+1n^(j)) Newt

 a^t



p^{(en(i) )}−1



+ Newt

 a^t

 p^en(k)−1



=t

^k⁻¹

i=1

p^e(^kj=i+1n^(j))

p^(en(i))− 1

+ (p^(en(k))− 1)



· Newt(a)

=t

^k⁻¹

i=1

p^e(^kj=in^(j)) − p^e(^kj=i+1n^(j))

+ (p^(en(k))− 1)



· Newt(a)

=t

 

p^e(^kj=1n^(j)) − p^(en(k))

+ (p^(en(k))− 1)



· Newt(a)

=t



p^e(kj=1n(j))− 1

· Newt(a).

Dividing by p^ekj=1n(j)and letting k go to infinity, we see that m∈ _1−p^we+t Newt(a).

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

x^{(pen−1)m+pen−1pe−1w}∈ a^t(pen−1).

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

n 0. So x^m∈ I, as desired. 

Given any face F of t Newt(a), we will use relint(_1−p^we+ F ) to denote the relative interior of _1−p^we+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 isC^φ,at-fixed and fix a face F of t Newt(a). If{v|x^v∈ I} ∩ (_1−p^we + F )= ∅, then {v|x^v∈ I} ⊇ relint(_1−p^we + F )∩ S.

Proof. Choose v0 ∈ {v|x^v ∈ I} ∩ (_1−p^we + F ) and v₀^ ∈ relint(_1−p^we + F )∩ S such that v0= v^0. Choose α1, . . . , αk∈ _1−p^we+ F so that v₀^ = n0v0+k

i=1niαi, where 0 < ni< 1 andk

i=0ni = 1. For n 0, p^enn0> 1 and p^env₀^ − v0= (p^enn0− 1)v0+

k i=1

p^enniαi ∈ (p^en− 1)( w

1− p^e + F )

⊆ p^en− 1

1− p^e w + t(p^en− 1) Newt(a).

By Lemma 3.1 x^{penv0−v0+pen−1pe−1w}∈ a^t(pen−1), and hence x^v0= φⁿ(x^{penv0−v0+pen−1pe−1w}·

x^v0)∈ I, as desired. 

Now, we are ready to describe theC^φ,at-fixed ideals. SetF={faces of t Newt(a)}, (Here, we assume t Newt(a)∈ F .) For F ∈ F , denote IF :=x^v|v ∈ relint(_1−p^we + F^)∩ S for some F^⊇ F .

Theorem 3.5. I is C^φ,at-fixed if and only if I =

F∈GIF for some non-empty G ⊆ F .

Proof. First, suppose

n>0φⁿ



a^t(pen−1)· I

= I, where the sum is taken over all integers n > 0 such that t(p^en−1) is an integer. Let G be the subset of F consisting of all the faces F satisfyingv|x^v ∈ I ∩ relint(_1−p^we + F )= ∅. Then I ⊇

F∈GIF

by Proposition 3.4. By Proposition 3.2, x^v ∈ I implies v ∈ relint(_1−p^we + 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



a^t(pen−1)· I

⊇ I.

Let x^v ∈ I; then v ∈ relint(_1−p^we + F ) ⊆ _1−p^we + t Newt(a) for some face F of t Newt(a). For any n > 0 such that t(p^en− 1) is an integer,

(p^en− 1)v ∈ p^en− 1

1− p^e w + t(p^en− 1) Newt(a).

Therefore by Lemma 3.1,

x^{(pen−1)v+pen−1pe−1w}∈ a^t(pen−1). So x^v= φⁿ



x^{(pen−1)v+pen−1pe−1w}· x^v

∈ φⁿ

a^t(pen−1)· I

, as desired.

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



n>0

φⁿ



a^t(pen−1)· IF

⊆ IF

for any F . Suppose x^u∈

n>0φⁿ



a^t(pen−1)· IF



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

p^enu = α + β−p^en− 1 p^e− 1 w

for some α and β such that x^α ∈ a^t(pen−1) and x^β ∈ IF. In particular, α ∈ Newt(a^t(pen−1)) = t(p^en− 1) Newt(a) (by Lemma 3.1) and β ∈ _1−p^we + relint γ for some γ⊇ F (because x^β∈ IF). Therefore,

u− w

1− p^e = α

p^en +(β−_1−p^we) p^en

∈ (p^en− 1)

p^en t Newt(a) + 1

p^enrelint γ

⊆ t Newt(a).

In particular, u−_1−p^we ∈ relint τ for some unique face τ in F . We claim that τ ⊇ γ.

This implies that x^u∈ Iτ⊆ Iγ⊆ IF, which finishes the proof.

To see why τ ⊇ γ, write u −_1−p^we = ^(pen_pen⁻¹⁾a +_pen¹ b for some a∈ t Newt(a) and b ∈ relint γ. If τ = t Newt(a), there is nothing to do. Suppose τ = t Newt(a) and letFτbe the set of all maximal faces inF \ {t Newt(a)} which contain τ. For each