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 toric*Q-divisor such that K**X*+ Δ isQ-Cartier with index
*not divisible by p and let φ*Δ*: F*_{∗}^{e}*O**X**→ O**X* be the toric map corresponding
*to Δ. We identify all ideals I of**O**X* *with φ*Δ*(F*_{∗}^{e}*I) = 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 φ*Δand 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*^{p}^{e}*· 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*

*φ*^{n}

a^{t(p}^{ne}^{−1)}*· I*

*= I,*

*where φ*^{n}*denotes 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 Society*c
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*

_{∗}

^{e}*O*

*X*

*→ O*

*X*

*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(* *) = φ**c**(x*^{−w}*· ) for some w ∈ M, where φ**c* is the canonical
splitting^{1} *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*
*I**τ* :=*x*^{v}*|v ∈ relint(*_{1}_{−p}^{w}*e* *+ τ** ^{}*)

*∩ S for some τ*

^{}*⊇ τ . Our main result is as follows:*

**Main Theorem (Theorem 3.5).**

*n>0**φ*^{n}

a^{t(p}^{ne}^{−1)}*· I*

*= I if and only if I =*

*τ**∈G**I**τ* *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>0**φ*^{n}

a^{t(p}^{ne}^{−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 J*^{0}*= 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}^{w}*e*.

*x*^{0}*y*^{0}

1−pe*w*

*x*^{3}*y*^{2}

*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*

1*The canonical splitting is the map which sends a monomial x*^{m}*to x*^{m/p}^{e}*if m/p** ^{e}*is an integer,

*and otherwise sends x*

*to zero.*

^{m}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}^{w}*e*):

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 K**X**+ 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*_{∗}^{e}*O**X* *→ O**X* *be the toric map*
*corresponding to Δ. Then an ideal I* *⊂ O**X* *satisﬁes*

*n>0**φ*^{n}

a^{t(p}^{ne}^{−1)}*· I*

*= I*
*if and only if there exists a toric log resolution π : X*^{}*→ X of (X, Δ, a) such that*
a*·O**X** ^{}* =

*O*

*X*

*(−G) and an eﬀective divisor E on X*

^{}

^{}*with π(E)⊂ Supp Δ∪Sing X ∪*

*V (a) such that*

*I = π*_{∗}*O**X** ^{}*(K

*X*

^{}*− π*

^{∗}*(K*

*X*+ 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*_{∗}^{e}*M → M*
where*M = O**X*(K*X**− π*^{∗}*(K**X**+ 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 = O**X*(K*X**− π*^{∗}*(K**X**+ 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 of*

^{}*O*

*X*.

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*^{p}^{e}*x) = 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*

_{∗}

^{e}*M to denote the R-module isomorphic to R as an Abelian group but with*

*the R-action, r.m = r*

^{p}

^{e}*m. From this perspective, a p*

*-linear map is simply an*

^{−e}*R-linear map F*

_{∗}

^{e}*R→ R. For the rest of the paper, all p*

^{−e}*-linear maps φ : R→ R*

*will be written as R-linear maps φ : F*

_{∗}

^{e}*R→ R.*

*Once a map φ : F*_{∗}^{e}*R→ 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*_{∗}^{e}*I) = 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**≥0**C**e* *such that each φ* *∈ C**e* is a map
*φ : F*_{∗}^{e}*R→ R. Furthermore, the multiplication of elements of C corresponds to the*
*composition of maps. In other words, for φ∈ C**e* *and ψ∈ C**d*, we deﬁne:

*φ· ψ := φ ◦ (F*_{∗}^{e}*ψ).*

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

If one uses End*e**(R) to denote Hom**R**(F*_{∗}^{e}*R, R), thenC**R*:=

*e*End*e**(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 by*C*^{φ,a}* ^{t}*, i.e.

*C*^{φ,a}* ^{t}* :=

*n*

*φ*^{n}*· F*_{∗}* ^{ne}*a

^{t(p}

^{ne}*=*

^{−1)}*n**≥0*

*C**n*^{φ,a}^{t}*,*

*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 use*C** ^{φ}* to denote

*C*

^{φ,a}*.*

^{t}Furthermore, we deﬁne

*C*+^{φ,a}* ^{t}* :=

*n>0*

*C**n*^{φ,a}^{t}*.*

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

* Deﬁnition 2.2. If an ideal I satisﬁesC*+

^{φ,a}

^{t}*(I) = I, i.e.*

*n>0**φ*^{n}*(F*_{∗}* ^{ne}*a

^{t(p}

^{ne}

^{−1)}*·I) =*

*I, then I is calledC*

^{φ,a}*-ﬁxed. According to the terminology of [Bli09], these ideals*

^{t}*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,a}* ^{t}* is the sub-
Cartier-algebra of

*C*

*R*

*generated by φ; we will denote this sub-Cartier-algebra by*

*C*

*. In this special case, one of the main theorems in [BB09] guarantees that there*

^{φ}*are only ﬁnitely many ideals I of R satisfying φ(F*

_{∗}

^{l}*I) = I (i.e.*

*C*+

^{φ}*(I) = I). However,*the ﬁniteness of

*C*

^{φ,a}*-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.*

^{t}**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*_{∗}^{e}*ω**R* *→ ω**R* *for each e > 0 – the*
*Grothendieck trace map. If R is Gorenstein and local, then ω**R* *∼= R and thus we*
*obtain a canonical p*^{−e}*-linear map φ*^{e}*: F*_{∗}^{e}*R→ R. In this case, the smallest non-zero*
*ideal J such that φ*^{e}*(F*_{∗}^{e}*J ) = 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 of*C*^{φ,a}* ^{t}*-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,a}^{t}*-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 Δ**φ* *as in sub-*
*section 2.3 below, then the test ideal*^{3} *τ (R, Δ**φ**, a*^{t}*) is the unique smallest*
*non-zero* *C*^{φ,a}^{t}*-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*_{∗}^{e}*d*^{p}^{e}^{−1}*· ). Then*
*an ideal J* *⊆ R is C*^{φ,a}^{t}*-ﬁxed if and only if dJ isC*^{ψ,a}^{t}*-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*_{∗}^{e}*J )* *⊆ 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,a}* ^{t}*-ﬁxed so that

*n>0**φ** ^{n}*(a

^{t(p}

^{ne}

^{−1)}*J ) = J .*

Then

*n>0*

*ψ** ^{n}*(a

^{t(p}

^{ne}

^{−1)}*dJ ) =*

*n>0*

*φ*^{n}*(F*_{∗}^{ne}*d*^{p}^{ne}* ^{−1}*a

^{t(p}

^{ne}

^{−1)}*dJ )*

=

*n>0*

*φ*^{n}*(F*_{∗}^{ne}*d*^{p}* ^{ne}*a

^{t(p}

^{ne}

^{−1)}*J )*

*= d*

*n>0*

*φ** ^{n}*(a

^{t(p}

^{ne}

^{−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 many*C*^{φ,a}* ^{t}*-ﬁxed ideals.

*Remark 2.5. Suppose that R is a normal domain and φ : F*_{∗}^{e}*R→ R is an R-linear*
*map as in Proposition 2.3. One can always extend φ to a map ¯φ : F*_{∗}^{e}*K(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*
*φ**c* *: F*_{∗}*R* *→ R. Then the fractional ideal generated by* _{x+1}^{1} *is φ**c*-ﬁxed. Indeed, a
*generating set of this fractional ideal over R** ^{p}* is

*x*^{i}*x+1*

0*≤i≤p−1*. Notice that
*φ**c*

*x*^{i}*x+1*

*= φ**c*

*(x+1)*^{p}^{−1}*x*^{i}*(x+1)*^{p}

=_{x+1}^{1} *φ**c*

*x*^{p}* ^{−1+i}*+

_{p}

_{−1}1

*x*^{p}* ^{−1+i−1}*+

*· · · + x*

^{i}*= b*^{x}_{x+1}^{i/p}

*for some constant b, which proves the claim. Clearly the fractional ideal generated*
by _{x+1}^{1} is not toric. Of course, the same statement holds for the fractional ideal
generated by _{x+λ}^{1} *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*_{∗}^{e}*R* *→ R, we can*
*compose φ with itself to obtain a map φ*^{2} *= φ◦ F*_{∗}^{e}*φ : F*_{∗}^{2e}*R* *→ R and similarly*
*construct φ*^{n}*for any positive integer n. While a priori, we may have an inﬁnite*
*descending chain of ideals φ*^{n}*(F*_{∗}^{ne}*R)* *⊇ φ*^{n+1}*(F*_{∗}^{(n+1)e}*R)* *⊇ . . . . 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 φ**^{n}*(F*_{∗}^{ne}*R) = φ*^{n+1}*(F*_{∗}^{(n+1)e}*R) = . . . 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 ψ**n**: F*_{∗}^{ne}*R→ R by the formula ψ**n*( ) =
*φ*^{n}*(F*_{∗}^{ne}*d· ), we have that S(ψ**n**) is φ-ﬁxed.*

*Proof. We ﬁrst claim that S(ψ**n*)*⊆ S(ψ**n+1**) for any n (compare also with [FST11,*
*Proposition 14.10(1)]). To prove this claim, ﬁrst notice that for any α : F*_{∗}^{e}*R→ R,*
*S(α) = S(α*^{m}*). Thus, in order to show the claim for ψ, it suﬃces to show that*
*ψ*_{n}^{n+1}*(F*_{∗}^{(n+1)ne}*J )⊆ ψ**n+1*^{n}*(F*_{∗}^{(n+1)ne}*J ) for every ideal J . However, because*

*1 + p** ^{ne}*+

*· · ·+p*

^{n}^{2}

*=*

^{e}*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}*we obtain that*

^{−1)(n+1)e}*ψ*^{n+1}*n* *(F*_{∗}^{(n+1)ne}*J ) = φ*^{n(n+1)}*(F*_{∗}^{(n+1)ne}*d*^{1+p}^{ne}^{+}^{···+p}^{n2 e}*J )*

*⊆ φ*^{n(n+1)}*(F*_{∗}^{(n+1)ne}*d*^{1+p}^{(n+1)e}^{+}^{···+p}*(n−1)(n+1)e*

*J ) = ψ*_{n+1}^{n}*(F*_{∗}^{(n+1)ne}*J ),*
which proves the claim.

*We choose n, which stabilizes this chain S(ψ**n**) = S(ψ**n+1**). Suppose now m > 0*
*is such that S(ψ**n**) = ψ*^{m}_{n}*(F*_{∗}^{m(ne)}*R). Then*

*φ(S(ψ**n*))

*= φ(F*_{∗}^{e}*ψ*^{m+1}_{n}*(F*_{∗}^{(m+1)ne}*R))*

*= ψ**n+1*

*F*_{∗}^{(n+1)e}*(ψ*_{n}^{m}*(F*_{∗}^{mne}*R))*

*= ψ**n+1*

*F*_{∗}^{(n+1)e}*S(ψ**n*)

*= ψ**n+1*

*F*_{∗}^{(n+1)e}*S(ψ**n+1*)

*= S(ψ**n+1*)

*= S(ψ**n**),*

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 K**X* + Δ is *Q-Cartier with index not divisible by p > 0.*

Further suppose that Δ is an eﬀective*Q-divisor such that (p*^{e}*− 1)Δ is integral and*
*(p*^{e}*− 1)(K**X**+ Δ) is Cartier for some e. Then there is a bijection of sets:*

Eﬀective*Q-divisors Δ on X such*
*that (p*^{e}*− 1)(K**X*+ Δ) is Cartier

*←→*

Line bundles*L and non-zero*
elements of Hom_{O}_{X}*(F*_{∗}^{e}*L , O**X*)

*∼.*

*The equivalence relation on the right side identiﬁes two maps φ*1 *: F*_{∗}^{e}*L*1 *→ O**X*

*and φ*2*: F*_{∗}^{e}*L*2*→ O**X* *if there is an isomorphism γ :L*1*→ L*2 and a commutative
diagram

*F*_{∗}^{e}*L*1
*φ*_{1}

*−−−−→ O**X*
*F*_{∗}^{e}*γ*

⏐⏐

⏐⏐^{id}

*F*_{∗}^{e}*L*2
*φ*_{2}

*−−−−→ O**X*

*.*

Given Δ, set*L = O**X*((1*− p*^{e}*)(K**X*+ Δ)). Then observe that

*F*_{∗}^{e}*O**X**((p*^{e}*− 1)Δ) ∼= F*_{∗}^{e}*H om**O**X*(L , O*X*((1*− p*^{e}*)K**X**)) ∼*=*H om**O**X**(F*_{∗}^{e}*L , O**X**).*

*The choice of a section η* *∈ O**X**((p*^{e}*− 1)Δ) corresponding to (p*^{e}*− 1)Δ thus gives a*
*map φ*Δ*: F*_{∗}^{e}*L → O**X*. The choice depends on various isomorphisms selected, but
this is harmless for our purposes.

*For the converse direction, an element φ∈ H om*_{O}*X**(F*_{∗}^{e}*L , O**X**) ∼= F*_{∗}^{e}*L** ^{−1}*((1

*−*

*p*

^{e}*)K*

*X*

*) determines an F*

_{∗}

^{e}*O*

*X*-linear map

*F*_{∗}^{e}*O**X*
1*→φ*

*−−−→ H om**O**X**(F*_{∗}^{e}*L , O**X*)*−→ F*^{∼}_{∗}^{e}*L** ^{−1}*((1

*− p*

^{e}*)K*

*X*)

*which corresponds to an eﬀective Weil divisor D such that* *O**X**(D) ∼*= *L** ^{−1}*((1

*−*

*p*

^{e}*)K*

*X*). Set Δ

*φ*=

_{p}*e*

^{1}

*−1*

*D.*

*Remark 2.8. Explicitly, suppose φ : F*_{∗}^{e}*R* *→ R is an R-linear map and d ∈ R.*

*Deﬁne a new map ψ(* *) = φ(F*_{∗}^{e}*d· ). Then D**ψ**= D**φ*+_{p}*e*^{1}*−1**div(d).*

**Deﬁnition 2.9. Suppose that φ : F**_{∗}^{e}*R→ 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
to*O**X**). 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}*)(K**X*+ Δ) = div*X**(x** ^{w}*)

*for some w∈ M. Then a map φ*Δcorresponding to Δ can be expressed as
*φ*Δ( *) = φ**c**(x*^{−w}*· ),*

*where φ**c* *is the canonical splitting on R deﬁned by*
*φ**c**(x** ^{v}*) =

*x*^{pe}* ^{v}* if

_{p}

^{v}*e*

*∈ M,*0 otherwise.

Let us explain these claims carefully since this identiﬁcation is critical for what
*follows. Our ﬁrst claim is that the map φ**c* corresponds to the torus invariant
divisor Δ*c* =*−K**X**. It is suﬃcient to show that φ**c* ﬁxes every height-one prime
torus invariant ideal (which implies that Δ*φ** _{c}* contains each torus invariant divisor
as a component) and does not ﬁx any other height-one ideal (which implies that
Δ

*φ*

*c*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}*)(K*

*X*+ Δ) is Cartier and thus is equal to div

*X*

*(x*

*). Therefore*

^{w}(1*− p*^{e}*)(K**X*+ Δ) = div*X**(x** ^{w}*) + 0 = div

*X*

*(x*

*) + (1*

^{w}*− p*

^{e}*)(K*

*X*+ (−K

*X*

*)).*

Dividing through by (1*− p** ^{e}*) gives us Δ = (−K

*X*) +

_{p}*e*

^{1}

*−1*div

*X*

*(x*

*). However, it*

^{−w}*is easy to see that given any map β : F*

_{∗}

^{e}*O*

*X*

*→ O*

*X*

*and any element d∈ F*

_{∗}

^{e}*Frac R,*

*the map α(*

*) = β(d· ), if it is indeed a map α : F*

_{∗}

^{e}*O*

*X*

*→ O*

*X*, has associated divisor Δ

*α*= Δ

*β*+

_{p}*e*

^{1}

*−1*div

*X*

*(d). In other words, the map φ*Δ as described above does indeed correspond to Δ.

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

*j=1**d**j**D**j* *such that the D**j*’s are distinct
prime Weil divisors, we deﬁne

*D
=*

*r*
*j=1*

*d**j**
D**j* and*D =*

*r*
*j=1*

*d**j**D**j**,*

*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. x*1 *<x ≤ x). We also*
deﬁne

*D** ^{≥k}*=

*d*_{j}*≥k*

*d**j**D**j**.*

3. *C*^{φ,a}* ^{t}*-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}*)(K**X*+ Δ) = div*X**(x*^{w}*) for some positive integer e and some element w∈ M.*

*Denote φ**c**: F*_{∗}^{e}*O**X**→ O**X* *as the canonical splitting on X. Consider a p** ^{−e}*-linear

*map φ*Δ(

*) = φ*

*c*

*(x*

^{−w}*) (w is determined by Δ; when Δ is clear we will simply*

*write φ), i.e.*

*φ(x** ^{u}*) =

*x*^{u−w}* ^{pe}* if

^{u}

_{p}

^{−w}*e*

*∈ 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,a}^{t}*generated by φ and*
a^{t}*, i.e. the ideals I satisfying*

*n>0*

*φ*^{n}

a^{t(p}^{ne}^{−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, I** ^{n}* =

*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*

*{(s*1*u*1+*· · · + s**l**u**l**) + (t*1*v*1+*· · · + t**m**v**m*)*| l, m > 0,*

*Σs**i**= 1, s**i**≥ 0, x*^{u}^{i}*∈ I, Σt**j**= 1, t**j**≥ 0, x*^{v}^{j}*∈ J}.*

*By allowing repeats in the u**i* *and v**j* *and possibly making some s**i**, t**j* = 0, we may
*assume that l = m and that s**i**= t**i* *for all i. Thus Newt(I) + Newt(J ) is equal to*

*{s*1*(u*1*+ v*1) +*· · · + s**l**(u**l**+ v**l*)*| l > 0, Σs**i**= 1, s**i**≥ 0, x*^{u}^{i}*∈ I, x*^{v}^{i}*∈ J},*
*which is clearly equal to Newt(IJ ) as desired.*

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

*n**i**a**i**for some a**i**∈ M*
*with x*^{a}^{i}*∈ I, n**i* *> 0, and*

*n**i* *= 1. For each i, since x*^{a}^{i}*∈ I, there exist g**i* *∈ Γ*
*and s**i* *∈ S such that a**i* *= g**i**+ s**i*. Therefore,

*n**i**a**i* = (

*n**i**g**i*) + (
*n**i**s**i*) *∈*
*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**^{t}*be as above. If I is* *C*^{φ,a}^{t}*-ﬁxed, then I is monomial*
*and{v|x*^{v}*∈ I} ⊆* _{1}_{−p}^{w}*e* *+ t Newt(a).*

*Proof. For any element h =*

*c**i**x*^{m}^{i}*∈ I, ﬁx any term x*^{m}^{i}*and set m = m**i*. Since

*n>0*

*φ*^{n}

a^{t(p}^{en}^{−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(p}^{en}^{−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*

*. Then*

^{m}*m =* *a*^{}*+ m*^{}*−*^{p}^{en}_{p}*e**−1*^{−1}*w*
*p*^{en}^{}*.*

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

*m =*

*k**−1*

*i=1* *p** ^{e}*(

^{}

^{k}*j=i+1*

*n*

^{(j)}*)a*

^{(i)}*+ a*^{(k)}*+ m*^{(k)}*−*^{p}* ^{e}*(

^{k}*j=1*

*n(j)*)

_{−1}*p*^{e}*−1* *w*

*p*^{e(}^{}^{k}^{j=1}^{n}^{(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*^{e}^{}^{k}^{j=1}^{n}* ^{(j)}*+

*p*

^{e}^{}

^{k}

^{j=1}

^{n}

^{(j)}*− 1*

*p*^{e}*− 1* *w*

=

_{k}_{−1}

*i=1*

*p** ^{e}*(

^{}

^{k}*j=i+1*

*n*

^{(j)}*)a*

^{(i)}

*+ a*^{(k)}*+ m*^{(k)}

*∈*

_{k}_{−1}

*i=1*

*p** ^{e}*(

^{}

^{k}*j=i+1*

*n*

*) Newt*

^{(j)}
a^{t}

*p*^{(en(i) )}*−1*

+ Newt

a^{t}

*p*^{en(k)−1}

*=t*

^{k}^{−1}

*i=1*

*p** ^{e}*(

^{}

^{k}*j=i+1*

*n*

*)*

^{(j)}*p*^{(en}^{(i)}^{)}*− 1*

*+ (p*^{(en}^{(k)}^{)}*− 1)*

*· Newt(a)*

*=t*

^{k}^{−1}

*i=1*

*p** ^{e}*(

^{}

^{k}*j=i*

*n*

^{(j)}*) − p*

*(*

^{e}^{}

^{k}*j=i+1*

*n*

*)*

^{(j)}*+ (p*^{(en}^{(k)}^{)}*− 1)*

*· Newt(a)*

*=t*

*p** ^{e}*(

^{}

^{k}*j=1*

*n*

^{(j)}*) − p*

^{(en}

^{(k)}^{)}

*+ (p*^{(en}^{(k)}^{)}*− 1)*

*· Newt(a)*

*=t*

*p*^{e(}^{}^{k}^{j=1}^{n}^{(j)}^{)}*− 1*

*· Newt(a).*

*Dividing by p*^{e}^{}^{k}^{j=1}^{n}^{(j)}*and letting k go to inﬁnity, we see that m∈* _{1}_{−p}^{w}*e**+t Newt(a).*

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

*x*^{(p}^{en}^{−1)m+}^{pen−1}^{pe−1}^{w}*∈ a*^{t(p}^{en}^{−1)}*.*

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

*n 0. So x*^{m}*∈ I, as desired.*

*Given any face F of t Newt(a), we will use relint(*_{1}_{−p}^{w}*e**+ F ) to denote the relative*
interior of _{1}_{−p}^{w}*e**+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 y*0*, y*1*∈*
*A∩ C such that x = δy*0+ (1*− δ)y*1 *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:

*σ*0

*σ*1

*σ*2

*σ*3

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

*relint σ*1 *relint σ*2 *relint σ*3

**Proposition 3.4. Suppose I is**C^{φ,a}^{t}*-ﬁxed and ﬁx a face F of t Newt(a). If{v|x*^{v}*∈*
*I} ∩ (*_{1}_{−p}^{w}*e* *+ F )= ∅, then {v|x*^{v}*∈ I} ⊇ relint(*_{1}_{−p}^{w}*e* *+ F )∩ S.*

*Proof. Choose v*0 *∈ {v|x*^{v}*∈ I} ∩ (*_{1}_{−p}^{w}*e* *+ F ) and v*_{0}^{}*∈ relint(*_{1}_{−p}^{w}*e* *+ F )∩ S such*
*that v*0*= v** ^{}*0

*. Choose α*1

*, . . . , α*

*k*

*∈*

_{1}

_{−p}

^{w}*e*

*+ F so that v*

_{0}

^{}*= n*0

*v*0+

*k*

*i=1**n**i**α**i*, where
*0 < n**i**< 1 and**k*

*i=0**n**i* *= 1. For n 0, p*^{en}*n*0*> 1 and*
*p*^{en}*v*_{0}^{}*− v*0*= (p*^{en}*n*0*− 1)v*0+

*k*
*i=1*

*p*^{en}*n**i**α**i* *∈ (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*^{p}^{en}^{v}^{0}^{}^{−v}^{0}^{+}^{pen−1}^{pe}^{−1}^{w}*∈ a*^{t(p}^{en}^{−1)}*, and hence x*^{v}^{0}^{}*= φ*^{n}*(x*^{p}^{en}^{v}^{}^{0}^{−v}^{0}^{+}^{pen−1}^{pe}^{−1}^{w}*·*

*x*^{v}^{0})*∈ I, as desired.*

Now, we are ready to describe the*C*^{φ,a}* ^{t}*-ﬁxed ideals. Set

*F={faces of t Newt(a)},*

*(Here, we assume t Newt(a)∈ F .) For F ∈ F , denote I*

*F*:=

*x*

^{v}*|v ∈ relint(*

_{1}

_{−p}

^{w}*e*+

*F*

*)*

^{}*∩ S for some F*

^{}*⊇ F .*

**Theorem 3.5. I is***C*^{φ,a}^{t}*-ﬁxed if and only if I =*

*F**∈G**I**F* *for some non-empty*
*G ⊆ F .*

*Proof. First, suppose*

*n>0**φ*^{n}

a^{t(p}^{en}^{−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}^{w}*e* *+ F )= ∅. Then I ⊇*

*F**∈G**I**F*

*by Proposition 3.4. By Proposition 3.2, x*^{v}*∈ I implies v ∈ relint(*_{1}_{−p}^{w}*e* *+ F )∩ S for*
*some F* *∈ G . So I ⊆*

*F**∈G**I**F*.
*Conversely, suppose I =*

*F**∈G**I**F* for some*G ⊆ F . We ﬁrst show that*

*n>0*

*φ*^{n}

a^{t(p}^{en}^{−1)}*· I*

*⊇ I.*

*Let x*^{v}*∈ I; then v ∈ relint(*_{1}_{−p}^{w}*e* *+ F )* *⊆* _{1}_{−p}^{w}*e* *+ 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*^{(p}^{en}^{−1)v+}^{pen−1}^{pe−1}^{w}*∈ a*^{t(p}^{en}^{−1)}*.*
*So x*^{v}*= φ*^{n}

*x*^{(p}^{en}^{−1)v+}^{pen−1}^{pe−1}^{w}*· x*^{v}

*∈ φ*^{n}

a^{t(p}^{en}^{−1)}*· I*

*, as desired.*

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

*n>0*

*φ*^{n}

a^{t(p}^{en}^{−1)}*· I**F*

*⊆ I**F*

*for any F . Suppose x*^{u}*∈*

*n>0**φ*^{n}

a^{t(p}^{en}^{−1)}*· I**F*

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

*p*^{en}*u = α + β−p*^{en}*− 1*
*p*^{e}*− 1* *w*

*for some α and β such that x*^{α}*∈ a*^{t(p}^{en}^{−1)}*and x*^{β}*∈ I**F**. In particular, α* *∈*
Newt(a^{t(p}^{en}^{−1)}*) = t(p*^{en}*− 1) Newt(a) (by Lemma 3.1) and β ∈* _{1}_{−p}^{w}*e* *+ relint γ for*
*some γ⊇ F (because x*^{β}*∈ I**F*). Therefore,

*u−* *w*

1*− p** ^{e}* =

*α*

*p** ^{en}* +

*(β−*

_{1}

_{−p}

^{w}*e*)

*p*

^{en}*∈* *(p*^{en}*− 1)*

*p*^{en}*t Newt(a) +* 1

*p*^{en}*relint γ*

*⊆ t Newt(a).*

*In particular, u−*_{1}_{−p}^{w}*e* *∈ relint τ for some unique face τ in F . We claim that τ ⊇ γ.*

*This implies that x*^{u}*∈ I**τ**⊆ I**γ**⊆ I**F*, which ﬁnishes the proof.

*To see why τ* *⊇ γ, write u −*_{1}_{−p}^{w}*e* = ^{(p}^{en}_{p}*en*^{−1)}*a +*_{p}*en*^{1} *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*
let*F**τ*be the set of all maximal faces in*F \ {t Newt(a)} which contain τ. For each*