Toric algebras

ISSN: 0092-7872 print/1532-4125 online DOI:10.1080/00927872.2015.1065840

FIXED IDEALS OF CERTAIN CARTIER ALGEBRAS OVER F-SPLIT TORIC ALGEBRAS

Jen-Chieh Hsiao

Department of Mathematics, National Cheng Kung University, Taiwan

In this article, the ideals of an F -split toric algebra that are fixed by certain Cartier algebra are described, extending the result for normal toric algebras in the author’s joint work with K. Schwede and W. Zhang.

Key Words: Cartier modules; Frobenius split; Toric algebras.

2010 Mathematics Subject Classification: 14M25; 13A35; 14F18.

1. INTRODUCTION

Let S be a toric algebra over a perfect field ! of characteristic p > 0. Given a toric p^−e-linear map ! " S → S (also called a toric near splitting), a monomial ideal

" of S, and a positive rational number t that has no p in the denominator, we are interested in the ideals I of S satisfying the condition

!

n>0

!ⁿ"

"^t#pne−1$· I#

= I% (1.1)

where the sum runs through all n∈ # such that t#p^ne− 1$ ∈ # and "^t#pne−1$ is the integral closure of "^t#pne−1$ in S. These ideals are the so-called F -pure submodules of certain Cartier algebra $^!%"t in the sense of [1]. We call the ideals satisfying (1.1)

$^!%"t-fixed. When S is normal, $^!%"t-fixed ideals are described in [4]. In particular, there are only finitely many of them. This provides a positive evidence to [1, Question 5.3]. Moreover, the $^!%"t-fixed ideals of S are identified in [4] with certain multiplier-like ideals on the normal toric variety SpecS arising from birational geometry.

The goal of this article is to generalize the description of $^!%"t-fixed ideals in [4]

to F -split toric algebras. To state the result, we need the following notation, which will be fixed throughout the article.

Notation. Let ! be a perfect field of characteristic p > 0, and let & be a semigroup satisfying %&= %^d= M. We assume that 0 is the only invertible element

Received February 13, 2014; Revised January 14, 2015. Communicated by U. Walther.

Address correspondence to Jen-Chieh Hsiao, Department of Mathematics, National Cheng Kung University, No. 1 University Rd., East District, Tainan City 70101, Taiwan; E-mail:

jhsiao@mail.ncku.edu.tw

2670

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in &. Denote ' "= &_≥0& the real cone of & in M_& "= M ⊗%&, and denote $& "= '∩ M the saturation of &. For any face ( of ', denote int ( the relative interior of (. Let S "= !)&* be the semigroup algebra (toric algebra) generated by & over !.

Similarly, denote $S = !)$&*.

Fix a number e∈ #, a p^−e-linear map (or a near splitting) on S is a map ! " S→ S satisfying !#r^pes$= r!#s$ for all r% s ∈ S. A toric near splitting is a near splitting

!_a " S→ S for some suitable a ∈ p^1eM such that for +∈ &

!_a#x⁺$=

%x^a+pe+ if a+_p^+e ∈ M%

0 otherwise.

See Proposition 3.1 for a characterization of toric near splittings on toric algebras.

Fix a toric near splitting != !a, and set ,_!%"t "=

& p^ea

p^e− 1 + t Newt "

'

∩ &-

Here, Newt " is defined to be the usual Newton polyhedra Newt#"$S$ in the normal case.

For +∈ ,!%"^t, consider

J₊^!%"t "=!

n≥0

!ⁿ"

"^t#pne−1$· 'x⁺(# -

Since ! and "^t are fixed, we simply write J₊ for J₊^!%"t. For each face (⊆ t Newt ", denote

,_!%("=

& p^ea

p^e− 1 + int ( '

∩ &-

Let G₍ be the minimal subset of ,_!%( such that .x⁺ * + ∈ G(/ generates the ideal (x⁺ * + ∈ ,!%(

), and denote G "=*

(⊆t Newt "G₍.

Theorem 1.1. Assume S is F-split. An ideal I of S is $^!%"t-fixed if and only if I = +

+∈HJ₊ for some subset H ⊆ G.

When S is normal, we have J₊= J+^′ for any +% +^′ ∈ G( by [4, Proposition 3.4]. So Theorem 1.1 recovers [4, Theorem 3.5].

A simple example (Example 4.6) is computed to illustrate Theorem 1.1. We also determine the fixed ideals for some non-F -split toric algebra (Examples 4.7), showing that the situation is more subtle in the general case.

2. F-SPLIT TORIC ALGEBRAS

A ring R of prime characteristic p is called F -split if the Frobenius map F " R → R splits. The F-split toric algebras have been characterized before (see

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for example [2, Proposition 6.2]). For our purpose, we give an alternative characterization in terms of the toric language.

First, the following lemma is well known for a toric algebra S = !)&*, . Lemma 2.1 ([6, Proposition 3.4]). There exists b_i ∈ $& and faces (_i of ', i= 1% 2% - - - % m, such that

$&\& = ,m i=1

)b_i+ #(i ∩ &$* -

In particular, there exists +∈ & ∩ int ' such that + + $&⊂ &.

The proof of Lemma 2.1 basically follows from the fact that the finite S-module

$S/S admits a finite filtration such that the quotient of each step in the filtration is isomorphic to S/P for some graded associated prime P of S.

Proposition 2.2. A toric algebra S = !)&* is F-split if and only if p"

$&\&#

∩ & = ∅-

Proof. First, suppose there exists an element 0 ∈ $&\& such that p0 ∈ &. Choose +∈ )& ∩ int '* as in Lemma 2.1. Then #++ p0$ + $& ⊂ &. In particular, + + p0 and ++ #p + 1$0 are both in &. If S was F-split, there would be a splitting ! " S → S of the Frobenius map. We have

x^++p0!#x^p0$= !#x^p#++p0$+p0$= x^++#p+1$0!#1$= x^++#p+1$0-

This implies !#x^p0$= x⁰ ∈ S since S is a domain. But this contradicts the assumption that 0. &.

Conversely, if p"

$&\&#

∩ & = ∅, then for any 0 ∈ & either ⁰_p ∈ & or ⁰_p . M.

So we have the well-defined splitting ! of the Frobenius map on S:

!#x⁰$=

%x^0p if ⁰_p ∈ M%

0 otherwise.

! Example 2.3. Consider & = #²\.#k% 0$ * k is odd/. Then !)&* is split when p is odd.

Corollary 2.4. If S is F-split, then "

&$\&#

∩ int ' = ∅. In particular, the only one- dimensional F -split toric algebra is the polynomial ring of one variable.

Proof. If there is an element 0 ∈"

&$\&#

∩ int ', then by Proposition 2.2

pⁿ0∈"

$&\&#

∩ int '

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for all n∈ #. This contradicts Lemma 2.1, which guarantees that k0+ $&⊂ & for

large k∈ #. !

Corollary 2.5. Assume S is F-split. Suppose v ∈ $&\& such that v is the primitive lattice point on the ray 1_v "= &_≥0v. Then 1_v∩ & = .k#nvv$*k ∈ #/ for some nv ∈ # which is not divisible by p.

Proof. First notice that some multiple of v is in &, so 1_v∩ & ̸= ∅. Let nv be the smallest positive integer n such that nv∈ &. This implies nv is not divisible by p, for otherwise n_v = pk for some k ∈ #. By Proposition2.2,

p#kv$= nvv ∈ & 0⇒ kv ∈ &- This contradicts the minimality of n_v.

To complete the proof, assume that 0 < n₀= nv< n₁ < n₂< - - - be all the integers satisfying n_iv ∈ &. We will show by induction that each ni is divisible by n_v. Clearly, n₀is divisible by n_v. Suppose n₀% - - - % n_k−1 is divisible by n_v. Since p does not divide n_v, there exists 0≤ l ≤ p − 1 such that p*#lnv+ nk$. By Proposition 2.2 again, ^lnv_p^+nkv ∈ &. But ^lnv+n_p ^k = ^#p−1$np^k+nk < n_k implies that ^lnv+n_p ^k = ni for some i < k.

Therefore, the induction hypothesis implies that n_k= pni − lnv is divisible by n_v. ! We will use the following lemma, whose proof is evident.

Lemma 2.6. Let n ∈ #, and let p be a prime number such that p " n. Then for any e∈ #, there exists l ∈ # such that

n divides

l−1

!

i=0

p^ei-

Corollary 2.7. Assume S is F-split. If 2 ∈ $&, then for any e∈ # there exists l ∈ # such that

&

p^le− 1 p^e− 1

' 2=

-_l−1

!

i=0

p^ei .

2∈ &-

Proof. If 2 ∈ &, simply take l = 1. Suppose 2 . &. As in Corollary2.5, let v∈ $&\&

be the primitive lattice point on the ray &_≥02, let n_v ∈ # be the smallest positive integer such that n_vv ∈ &, and write 2 = n2v. Since p" nv, by Lemma2.6there exists k% l∈ # such that

kn_v = -_l−1

!

i=0

p^ei .

- Therefore,

-_l−1

!

i=0

p^ei .

2= kn2#n_vv$ ∈ &-

!

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3. TORIC NEAR SPLITTINGS

In this section, we determine all the toric near splittings on an F -split toric algebra S following [5].

Let !₀ be the canonical splitting on !)M*, namely for + ∈ M,

!₀#x⁺$=

%x^p+ if ⁺_p ∈ M%

0 otherwise.

For e∈ # and a ∈ _p^1eM, consider the near splitting !_a "!)M* → !)M* defined by

!_a#!$ = !^e0#x^pea· !$-

We say !_a is regular on S if !_a#S$⊆ S (i.e., !a induces a well-defined near splitting on S). In this case, we call !_a a near splitting of S. It is observed in [5] that the !_a’s form a basis of the space of near splittings

Hom_!)M*#F_∗^e!)M*% !)M*$

and that an element + c_i!_ai ∈ Hom!)M*#F_∗^e!)M*% !)M*$, ci ∈ !^×, lies in Hom_S#F_∗^eS% S$ if and only if each !_ai is regular on S. Here, for a module M over a ring R of positive characteristic p, we denote F_∗^eM the additive group M equipped with an R-module structure via the R-action r· m "= r^pem. Notice that in [5], the monomials in F_∗^eS are identified with the points in _p¹e&. We do not use that identification here.

We generalize Proposition 4.2 in [5] to general toric algebras.

Proposition 3.1. For a ∈ _p¹eM, the map !_a is regular on S if and only if

a∈/

v₁

0 v∈ 1

p^eM 11

11#v% v¹$ > −1 2

\ 3

0∈$&\&

&

0− 1 p^e&

' -

Here the v₁’s are the primitive lattice points on the rays of the dual cone '^∨ of ' =

&_≥0&.

Proof. We first prove the (if) part. Let

a∈/

v₁

0 v∈ 1

p^eM 11

11#v% v¹$ >−1 2

\ 3

0∈$&\&

&

0− 1 p^e&

'

%

and suppose 2∈ & with p^2e + a ∈ M. We need to show that p^2e + a ∈ & (i.e., !a#x²$= x^pe2+a ∈ S). Observe that

&

2

p^e + a% v1

'

>

&

2 p^e% v₁

'

− 1 ≥ −1

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for all v₁ and hence _p²e + a ∈ $&. This implies _p²e + a ∈ &, since otherwise

a=

& 2 p^e + a

'

− 2 p^e ∈

4& 2 p^e + a

'

− 1 p^e&

5

contradicting the assumption.

We now prove the (only if) part.

Suppose #a% v₁₀$≤ −1 for some 10. Using the notation in Lemma 2.1, we choose b = bi so that (= (i is the facet corresponding to 1₀ and such that if j ̸= i is such that (_j = ( then #bj% v₁₀$≤ #b% v1₀$. If there is no such b, just choose any b∈ %^d so that #b% v₁₀$= −1. Now, choose 0 ∈ )b + #& ∩ int ($* ⊆ #%^d\&$ so that

#0− a% v1$ > #b_j% v₁$ for all b_j ̸= b and all 1 ̸= 10 (if no such b_j exists, we require

#0− a% v1$ >0 for all 1̸= 10). Notice that #0− a% v1₀$≥ #0% v1₀$+ 1 = #b% v1₀$+ 1.

By Lemma 2.1 and by the choice of b, we must have 0− a ∈ _p¹e$& and

)#0− a$ + '* ∩ #$&\&$ = ∅-

Therefore, !_a is not regular since p^e#0− a$ ∈ & but !a#x^pe#0−a$$= x⁰ . S.

To complete the proof, we need to show that !_a is regular implies

a. 3

0∈$&\&

&

0− 1 p^e&

' -

Suppose a∈ 0 − _p^1e& for some 0 ∈ $&\&. Then p^e#0− a$ ∈ &. Since !a#x^pe#0−a$$=

x⁰ . S, we see that !a is not regular. !

We illustrate Proposition 3.1 by the following figure. Consider the toric algebra generated by the four black dots #0% 1$% #1% 1$% #2% 0$% #3% 0$ in characteristic p= 3. We have $&\& = .#1% 0$/. The gridded area (excluding x = −1 and y = −1) indicates the candidates a ∈6

v₁

0

v ∈ ¹pM 11

11#v% v¹$ > −1 2

for !_a to be a regular map.

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The small gray dots indicate the points in *_0∈$&\&"

0− ¹p&#

that should be removed from the candidate set.

4. $^!"'T-FIXED IDEALS

In this section, fix an F -split toric algebra S, a monomial ideal " of S, and a number t ∈ (>0. Given an element a∈ p^1eM such that the toric near splitting != !a

is regular on S, we have for each +∈ & either p^+e + a ∈ & or p^+e + a . M. We are going to characterize the $^!%"t-fixed ideals.

To simplify the presentation, we introduce for any n∈ % and for any + ∈ M the element

+_n "= p^ne

&

+− p^ea p^e− 1

'

+ p^ea p^e− 1- We have immediately that

+_n− + = #p^ne− 1$

&

+− p^ea p^e− 1

'

- (4.1)

Also, if n≥ 0, then +n = #+n− +$ + + ∈ M, and one can also check that

!ⁿ#x⁺ⁿ$= x⁺- (4.2)

Our goal is to describe the ideals I ⊆ S satisfying (1.1)

!

n>0

!ⁿ"

"^t#pne−1$· I#

= I%

where the sum runs through all n∈ # such that t#p^ne− 1$ ∈ # and "^t#pne−1$ is the integral closure of "^t#pne−1$ in S.

When S is normal, it is well known that the integral closure of a monomial ideal J ⊆ S is

J = 'x⁺ * + ∈ Newt#J$ ∩ &(%

where Newt#J$ is the convex hull of the set .+∈ M * x⁺ ∈ J/. For general toric algebras S, define Newt#J$ "= Newt#J$S$. The following lemma is valid without the F-split assumption on S.

Lemma 4.1. Let I% J be monomial ideals of a toric algebra S. Then the following statements are true:

(1) J = 'x⁺* + ∈ Newt#J$ ∩ &(-

(2) Newt#I$+ Newt#J$ = Newt#IJ$. In particular, Iⁿ = 'x⁺ * + ∈ #nNewt#I$ ∩ &$( for any n∈ #.

(3) If I = 'x^g * g ∈ ,(, then Newt#I$ = Conv#,$ + ' where Conv#,$ is the convex hull of ,.

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Proof. When S is normal, these are proved in [4, Lemma 3.1]. All these statements extend readily to general S according to [3, Proposition 1.6.1] which states that if R⊂ S is an integral ring extension and I is an ideal of R then IS ∩ R = I. !

One can also extend [4, Proposition 3.2] to F -split toric algebras.

Proposition 4.2. If an ideal I ⊆ S is $^!%"t-fixed, then I ⊆ I!%"^t "= 'x⁺ * + ∈ ,!%"^t( where

,_!%"t "=

& p^ea

p^e− 1 + t Newt "

'

∩ &-

Moreover, I is a monomial ideal.

Proof. The first statement is valid without the assumption that S is F-split. It follows from Lemma 4.1 and the first half of the proof of [4, Proposition 3.2].

Indeed, for any element

h=!

c_jx^+j ∈ I #cj ̸= 0$%

fix a term x^+j and set += +j. Since +_n>0!ⁿ"

"^t#pne−1$· I#

= I% there exists n1 >0, +^′ ∈ &, and 01 ∈ t#p^n1e− 1$ Newt#"$ ∩ &, such that x^+′ ∈ I and +n₁ = 01+ +^′. Here +_n1 is defined as in (4.1) so that !#x⁺ⁿ¹$= x⁺. Repeating the same process k times for +^′ in the place of +, for 2≤ i ≤ k we can find ni ∈ #, 0i ∈ t#p^nie− 1$ Newt#"$ ∩ &, and +^#i$ ∈ & such that

+^+k

i=1n_i =

k−1

!

i=1

#0_i$_ni+1+ 0k+ +^#k$-

By definition (4.1), #0_i$_ni+1 = p^ni+1e"

0_i− p^pee−1^a

#+p^pee−1^a , so

#0_i$_ni+1 ∈ 4

#1− p^ni+1e$ p^ea

p^e− 1 + tp^ni+1e#p^nie− 1$ Newt#"$

5 - Hence, by Lemma 4.1(3)

+^+k

i=1n_i = p^+ki=1ni+−"

p^+ki=1ni − 1# & p^ea p^e− 1

'

=

k−1

!

i=1

#0_i$_ni+1 + 0k+ +^#k$

∈ 7-

k−

!k i=1

p^ni+1e

. p^ea

p^e− 1 + t"

p^+ki=1nie− 1#

Newt#"$

8 -

Dividing by p^+ki=1ni and letting k go to infinity, we see that +∈ ,!%"^t as desired.

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For the second statement, since #p^e− 1$"

+− p^pee−1^a

#∈ )t#p^e − 1$ Newt#"$* ∩ $&

and since S is F -split, by Corollary 2.7 there exists l∈ # such that t#p^le− 1$ ∈ # and

+_l− + =

(4.1)#p^le− 1$

&

+− p^ea p^e− 1

'

∈9

t#p^le− 1$ Newt#"$:

∩ &- Therefore,

!^l#x^+l−+· h$ ∈ !^l"

"^t#ple−1$· I#

⊆ I- By (4.2),

!^l#x^+l−+· h$ =!

j

c

1 ple

j x⁺⁺

+j −+

ple = c

1 ple

i x⁺-

Since l can be chosen arbitrarily large, we must have x⁺ ∈ I. Moreover, since + = +i

is arbitrary chosen, we conclude that I is a monomial ideal. ! We will often need the following lemma which is observed in the proof of Proposition4.2.

Lemma 4.3. For any + ∈ ,!%"^t (i.e.x⁺∈ I!%"^t), there exists l∈ # such that t#p^le− 1$ ∈ # and

x^+l−+= x^#ple−1$

"

+−_pe−1^{pe a}#

∈ "^t#ple−1$⊆ S- Proposition 4.4. The ideal I_!%"t = 'x⁺* + ∈ ,!%"^t( is $^!%"t-fixed.

Proof. We need to show

!

n>0

!ⁿ"

"^t#pne−1$· I!%"^t

#= I!%"^t-

For the containment #⊆$, fix any n ∈ # so that t#p^ne− 1$ ∈ #, and let x⁰ ∈ "^t#pne−1$· I!%"^t-

Then 0∈;

t#p^ne− 1$ Newt#"$ + p^pee−1^a + t Newt "<

=;

p^ea

p^e−1 + tp^neNewt "<

. So either

!ⁿ#x⁰$= 0 or

!ⁿ#x⁰$_(4.2)= x⁰⁻ⁿ = x^p−ne

"

0−_pe−1^{pe a}# +_pe−1^{pe a} %

where 0_−n ∈ ,!%"^t. Therefore, !ⁿ#x⁰$∈ I!%"^t as desired.

Conversely, let x⁰ ∈ I!%"^t, and by Lemma4.3 there exists l∈ # such that x^0l−0 ∈ "^t#ple−1$-

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Therefore,

x⁰ = !^l#x^0l$= !^l#x^0l−0· x⁰$∈ !^l"

"^t#ple−1$· I!%"^t

#

- !

We now describe the basic $^!%"t-fixed ideals.

As in the introduction, consider for each +∈ ,!%"^t the ideal J₊ = J+^!%"t =!

n≥0

!ⁿ"

"^t#pne−1$· 'x⁺(#

It follows from Proposition 4.4that 'x⁺( ⊆ J+ ⊆ I!%"^t- Proposition 4.5.

(1) J₊ is $^!%"t-fixed.

(2) If I is $^!%"t-fixed, then for any x⁺∈ I we have J+⊆ I. In particular, if I = 'x⁺¹% - - - % x^+k( is $^!%"t-fixed, then I =+k

i=1J_+i- Proof. (1) By construction of J₊, we have

!

n>0

!ⁿ"

"^t#pne−1$· J+

#⊆ J+-

Conversely, let x⁰ ∈ 'x⁺(. By Lemma4.3, there exists l∈ # such that t#p^le− 1$ ∈ # and x^0l−0 ∈ "^t#ple−1$- So

x⁰_(4.2)= !^l#x^0l−0· x⁰$∈ !^l"

"^t#ple−1$· 'x⁺(#

⊆ !^l"

"^t#ple−1$· J+

# - Therefore,

J₊= 7

'x⁺( +!

n>0

!ⁿ"

"^t#pne−1$· 'x⁺(#8

⊆!

n>0

!ⁿ"

"^t#pne−1$· J+

#-

(2) For the first statement, observe again that J₊ =!

n≥0

!ⁿ"

"^t#pne−1$· 'x⁺(#

⊆!

n≥0

!ⁿ"

"^t#pne−1$· I#

= I-

The second statement follows immediately. !

Now, we are ready to describe all $^!%"t-fixed ideals.

Recall that for each face (⊆ t Newt ", ,_!%("=

&

p^ea

p^e− 1 + int ( '

∩ &%

G₍ is the minimal subset of ,_!%( such that .x⁺* + ∈ G(/ generates the ideal (x⁺ * + ∈ ,!%(

), and G "=*

(⊆t Newt "G₍.

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Proof of Theorem1.1. The (if) part follows from Proposition 4.5#1$ and the fact that sums of $^!%"t-fixed ideals are $^!%"t-fixed.

Conversely, if I is $^!%"t-fixed, then by Proposition 4.2 I is a monomial ideal satisfying I⊆ I!%"^t. Let

H = .+ ∈ G * x⁺ ∈ I/- Then I ⊇+

+∈HJ₊ by Proposition 4.5#2$. We claim that H ̸= ∅ and I = !

+∈H

J₊-

Let x⁰ ∈ I. By Proposition 4.2, there is a unique face (⊆ t Newt#"$ such that 0 ∈ ,_!%(. So there exists an +∈ G(such that 0 ∈ + + &, namely x⁰ ∈ 'x⁺( ⊆ J+. It suffices to prove that +∈ H-

If + = 0, there is nothing to do. Suppose + ̸= 0, and let 3 be the face of ' such that 0− + ∈ #int 3$ ∩ &. On each ray (one-dimensional face) of 3, choose an element in &. Denote these elements by 4₁% - - - % 4_k. Then there exists l∈ # such that

l#0− +$ =

!k i=1

n_i4_i% for some n_i ∈ #-

On the other hand, since

+₁− + = #p^e− 1$

&

+− p^ea p^e− 1

'

∈ $&%

we must have +₁− + ∈ #int 3^′$∩ $& for some unique face 3^′ of '. Note that if (= t Newt ", then +1− + ∈ int ' and hence 3^′ = ' ⊇ 3. Also, if ( # t Newt ", then

0̸= 0 − + ∈ )& ∩ #int ( − int ($ ∩ int 3*

implies that ( contains a translation of 3. In this case, since +∈ ,!%(, we still have 3^′ ⊇ 3. So we may choose an element in & on each ray of 3^′ that is not in 3 (if exists). Denote these elements 4_k+1% - - - % 4_k′ (k≤ k^′). Write

+₁− + =

k^′

!

i=1

q_i4_i for some q_i ∈ (>0-

Choose u% v∈ # such that p " v and that p^uvq_i ∈ # for i = 1% - - - % k^′. By Lemma2.6, there exists n∈ # such that the following statements hold:

(1) t#p^ne− 1$ ∈ #;

(2) v devides ^p_p^nee−1⁻¹; (3) "

p^ne−1 p^e−1

#

p^uq_i ≥ p^un_i, for i= 1% - - - % k.

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In particular, by (4.1) p^u)#+_n− +$ − l#0 − +$* =

4&

p^ne− 1 p^e− 1

'

p^u#+₁− +$ − p^ul#0− +$

5

=

!k i=1

4&p^ne− 1 p^e− 1

'

p^uq_i− p^un_i 5

4_i+

k^′

!

i=k+1

4&p^ne − 1 p^e− 1

' p^uq_i

5 4_i

∈ &-

Since S is F -split, we have #+_n− +$ − l#0 − +$ ∈ & by Proposition 2.2. Hence +_n− 0 ∈ )#l − 1$#0 − +$ + &* ⊆ &-

Notice also that, since ( contains a translation of 3, for l sufficiently large

+_n− 0 = #+n− +$ − #0 − +$ = #p^ne − 1$

&

+− p^ea p^e− 1

'

−

!k i=1

n_i l 4_i

∈ #p^ne − 1$int ( −

!k i=1

n_i l 4_i

⊆ #p^ne − 1$int (

⊆ t#p^ne− 1$ Newt "- Therefore, we have x⁺ⁿ⁻⁰ ∈ "^t#pne−1$ and

x⁺= !#x⁺ⁿ$= !#x⁺ⁿ⁻⁰· x⁰$∈ !"

"^t#pne−1$· I#

⊆ I

as desired. !

Example 4.6. As in Example 2.3, consider S = !)&* where

&= #²\.#k% 0$ * k is odd/-

Assume char!= p = 3 and e = 1. Again, the gray dots in the following figure indicate the points a with !_a not regular.

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Consider a= #²₃%²₃$. Then _p^pa₋₁ = #1% 1$ and G= 3

(⊆'

G₍= .#1% 1$% #1% 2$% #2% 1$% #3% 1$% #2% 2$% #3% 2$/-

The following figure illustrates the points in G as well as the sets _p^pa₋₁ + int ( for (⊆ '.

By Theorem1.1, the nontrivial fixed ideals are as follows:

J_#1%1$ = 'xy(% J#2%1$ = 'x²y(% J#3%1$ = 'x³y% x²y²(%

J_#1%2$ = 'xy²% x²y²(% J#2%2$ = J#3%2$= 'x²y²% x³y²( = smallest%

J_#1%1$+ J#2%1$= 'xy% x²y( = largest%

J_#2%1$+ J#3%1$ = 'x²y% x³y(%

J_#2%1$+ J#1%2$ = 'x²y% xy²(%

J_#1%2$+ J#3%1$ = 'xy²% x²y²% x³y(%

J_#2%1$+ J#3%1$+ J#1%2$= 'x²y% x³y% xy²(-

Observe that since (₁∩"

$&\&#

̸= ∅, the generators in G(₁ produce multiple fixed ideals, J_#2%1$ and J_#3%1$, whereas the generators in G_' produce only one fixed ideal, J_#2%2$ = J#3%2$.

Similarly, one can easily verify the following table.

a _p−1^pa Nontrivial !_a fixed ideals

#¹₃%¹₃$ #¹₂%¹₂$ 'xy% x²y(

#²₃%¹₃$ #1%¹₂$ 'xy% x²y(% 'x²y% x³y(

#¹₃%²₃$ #¹₂%1$ 'xy% x²y(% 'xy²% x²y²(

#²₃%0$ #1% 0$ 'x²(% 'xy% x²y(% 'x²y% x³y(

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In the following examples, we will see that many conclusions in this section are false for non-F -split toric algebras.

Example 4.7. Again, by Example 2.3 the toric algebra in characteristic p= 2 generated by & = #²\.#k% 0$ * k is odd/ is not F-split. By Proposition 3.1, the set

0

#u% v$∈ 1 2M

11

1 u > −1% v > 0

230&

u− 1 2%0' 11

1 u ≥ 0 2

consists of all a with !_a regular. Notice that in the following figure the points

#2k% 0$∈ &% k ≥ 0, are corresponding to nonregular maps. This does not happen for F-split toric algebras.

We consider the following cases.

(1) a==

−¹₂%0>

%_p^pa₋₁ = #−1% 0$. Observe that x^2k . !a#S$for all k≥ 0. In particular, as a generator in G, x^2k does not produce any fixed ideals. Also, even though

& ⊆ p^pa−1+ ', the toric algebra S is not fixed by !a. The only nontrivial !_a-fixed ideal is 'y% xy(.

(2) a==₁

2%0>

%_p^pa₋₁ = #1% 0$. Again, x^2k . !a#S$for all k≥ 0 and the only nontrivial

!_a-fixed ideals are 'xy% x²y(% 'x²y% x³y(.

(3) a==

−¹2%1>

%_p^pa₋₁ = #−1% 2$. The set of generators is G = .#0% 2$% #1% 2$% #0% 3$%

#1% 3$/. One can easily check that J_#1%2$ = 'y% xy²( and J#0%3$= J#1%3$ = 'y³% xy³( are !_a-fixed ideals, but J_#0%2$ = 'y²( is not fixed by !a.

ACKNOWLEDGEMENT

The author was partially supported by NSC in Taiwan under grant 101-2115- M-006-011-MY2.

REFERENCES

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