PURITY OF MONOIDS AND CHARACTERISTIC-FREE SPLITTINGS IN SEMIGROUP RINGS ALESSANDRO DE STEFANI1 JONATHAN MONTAÑO2 AND LUIS NÚÑEZ-BETANCOURT3

2025-05-06 0 0 575.97KB 20 页 10玖币

侵权投诉

PURITY OF MONOIDS AND CHARACTERISTIC-FREE SPLITTINGS IN

SEMIGROUP RINGS

ALESSANDRO DE STEFANI1, JONATHAN MONTAÑO2, AND LUIS NÚÑEZ-BETANCOURT3

ABSTRACT.

Inspired by methods in prime characteristic in commutative algebra, we introduce

and study combinatorial invariants of seminormal monoids. We relate such numbers with the

singularities and homological invariants of the semigroup ring associated to the monoid. Our results

are characteristic independent.

CONTENTS

1. Introduction 1

2. Background 3

3. Purity of M-modules and (semi)normal afﬁne monoids 5

4. Asymptotic growth of number of pure translations 13

5. Applications to afﬁne semigroup rings 15

Acknowledgments 19

References 19

1. INTRODUCTION

Frobenius splittings have inspired a large number of results in commutative algebra, algebraic

geometry, and representation theory. In this manuscript we seek to continue this approach in the

context of combinatorics of monoids. Given a monoid

M⊆Zq

⩾0

for some

q∈Z>0

, and

m∈Z>0

we study the pure

-submodules of

that are translations of

, which algebraically corresponds

to free summands of

k[1

mM]

k[M]

-module. It turns out that the purity of

M⊆1

detects both

normality and seminormality (see Proposition 3.5). The study of pure submodules, or equivalently

of free summands, of normal monoids was already initiated by other authors in order to compute the

-signature of normal afﬁne semigroup rings [21,26]. Moreover, the structure of

-module

was described by Bruns and Gubeladze [4,5] for normal monoids (see [20] for a related result in

prime characteristic).

2020 Mathematics Subject Classiﬁcation. Primary 20M32, 20M25, 13A35 ; Secondary 13C15 .

Key words and phrases. Monoids, pure maps, seminormality, numerical invariants.

The ﬁrst author was partially supported by the PRIN 2020 project 2020355B8Y “Squarefree Gröbner degenerations,

special varieties and related topics".

2The second author was supported by NSF Grant DMS #2001645/2303605.

3The third author was supported by CONACyT Grant #284598.

arXiv:2210.03358v2 [math.AC] 1 Sep 2023

2 A. DE STEFANI, J. MONTAÑO, AND L. NÚÑEZ-BETANCOURT

In this manuscript we study combinatorial numerical invariants of a seminormal monoid. Our

key motivation is that seminormality for a monoid can be seen as a characteristic-free version of

F-purity for afﬁne semigroup rings. For more information and examples on seminormal monoids

we direct the interested reader to Li’s thesis on this subject [17].

In Deﬁnition 3.19 we introduce the notion of pure threshold of a seminormal monoid

, denoted

mpt(M)

, which is motivated by the

-pure threshold in prime characteristic. This number can

be described as the largest degree of a pure translation of

inside the cone

R⩾0M

or, equivalently,

for some

. We show that

mpt(M)

gives an upper bound for the Castelnuovo-Mumford

regularity

reg(k[M])

deﬁned in terms of local cohomology, and the Castelnuovo-Mumford regularity

Reg(k[M]) deﬁned in terms of graded Betti numbers of k[M](see Section 2for more details).

Theorem A (Theorem 5.4).Let

be a seminormal monoid with a minimal set of generators

{γ1,...,γu}. Then, ai(k[M]) ⩽−mpt(M). As a consequence,

reg(k[M]) = max{ai(k[M]) −i}⩽dim(k[M]) −mpt(M) = rank(M)−mpt(M).

Moreover, if we present

S/I

, where

S=k[x1,...,xu]

and each

has degree

di:=deg(xi) = |γi|

the degree of γifor i=1,...,u, and I⊆Sis a homogeneous ideal, then

Reg(k[M]) = sup{βS

i(M)−i|i∈Z}⩽rank(M) +

∑

i=1

(di−1)−mpt(M).

Theorem Aallows us to give an upper bound for the degrees of generators of the deﬁning

ideal

. We also show that

mpt(M)

is a rational if

is a normal (see Proposition 4.4). Despite

mpt(M)

being inspired by

-pure thresholds, these numbers do not always coincide (see Example

3.23 and Remark 3.24). In addition,

mpt(M)

is deﬁned independently of the ﬁeld

and so it

is a characteristic-free invariant, while the

-pure threshold is only deﬁned when

has prime

characteristic.

We introduce the pure prime ideal

P(M)

, and the pure prime face

, of a seminormal monoid

(see Corollary 3.28, and Deﬁnitions 3.26 and 3.29). The former emulates the splitting prime

ideal of an

-pure ring, while the latter is related to the quotient of a ring by its splitting prime.

In fact, the submonoid

FM∩M

is normal (see Corollary 3.28). We note that the rank of

FM∩M

is a monoid version of the splitting dimension and so we call it the pure dimension and denote

it by

mpdim(M)

. It turns out that this rank is equal to the rank of

if and only if

is normal,

and it is non-negative if and only if

is seminormal (see Corollary 3.30). Therefore, in some

sense,

mpdim(M)

measures how far a seminormal monoid is from being normal. Furthermore,

mpdim(M)is related to the depth of k[M]as the following theorem shows.

Theorem B (Theorem 5.7).If Mis a seminormal monoid, then mpdim(M)⩽depth(k[M]).

We point out that Theorem Brecovers Hochster’s result that normal semigroup rings are Cohen-

Macaulay [14].

PURITY OF MONOIDS AND CHARACTERISTIC-FREE SPLITTINGS IN SEMIGROUP RINGS 3

Finally, we consider the growth of the number of disjoint pure translations of

varies. More speciﬁcally, if m∈Z>0is such that 1

mM∩ZM=M, we deﬁne

Vm(M):=α∈1

mM|(α+M)⊆1

mMis pure.

Theorem C (Theorem 4.6).Let

be a seminormal monoid,

A(M) = {m∈Z>0|1

mM∩ZM=M}

and s=mpdim(M). Then,

mpr(M):=lim

t→∞

|Vmt(M)|

exists and it is positive for every increasing sequence

mt∈A(M)

. Furthermore, if

is normal,

then mpr(M)∈Q>0.

We call the limit in Theorem Cthe pure ratio of

. If the ﬁeld has prime characteristic, this

number coincides with the splitting ratio of

k[M]

[1]. A consequence of Theorem Cis that the

value of the

-splitting ratio depends only on the structure of

, and so it is independent of the

characteristic of the ﬁeld as long as

k[M]

-pure. Finally, using this result we give a monoid

version of a celebrated Theorem of Kunz [16, Theorem 2.1] which characterizes regularity of rings

of prime characteristic in terms of Frobenius (see Theorem 5.9).

Throughout this article we adopt the following notation.

Notation 1.1. Let

be a ﬁeld of any characteristic and

a positive integer. Let

M⊆Zq

⩾0

be an

afﬁne monoid, i.e., a ﬁnitely generated submonoid of

. We ﬁx

{γ1,...,γu}

a minimal set of

generators of

. Let

denote the group generated by

and

C(M) = R⩾0M

the cone generated

by M.

2. BACKGROUND

In this section we include some preliminary information that is needed in the rest of the paper.

Afﬁne monoids and afﬁne semigroup rings. For proofs of the claims in this subsection and

further information about afﬁne monoids we refer the reader to Bruns and Gubeladze’s book [6].

Let

M⊆Zq

⩾0

be an afﬁne monoid. A subset

U⊆Qq

is an

-module if

U+M⊆U

. An

-module

is an ideal if it is contained in

. An ideal

U⊆M

is prime if whenever

a+b∈U

with

a,b∈M

we must have a∈Uor b∈U. The rank of Mis the dimension of the Q-vector space Q⊗ZZM.

Let

R=k[M]⊆k[

]:=k[x1,...,xq]

be the afﬁne semigroup associated to

. As a

-vector

space,

is generated by the monomials

{xα|α∈M}

. We note that the monomial ideals of

are precisely those generated by {xα|α∈U}for some ideal U⊆M. Under this correspondence,

prime monomial ideals of

correspond to prime ideals of

. For every

-module

U⊆Qq

we have

a corresponding

-module

RU :={xα+η|α∈M,η∈U}

in the algebraic closure of

k(x1,...,xq)

Moreover, we have dim(R) = rank(M).

4 A. DE STEFANI, J. MONTAÑO, AND L. NÚÑEZ-BETANCOURT

Graded algebras and modules. A non-negatively graded algebra

is a ring that admits a direct

sum decomposition

A=Lj⩾0Aj

of Abelian groups such that

Ai·Aj⊆Ai+j

. It follows from this

that

is a ring, and each

is an

-module. If we let

A+=Lj>0Aj

, then

A+

is an ideal of

called the irrelevant ideal.

Throughout this manuscript we will make the assumption that

is Noetherian or, equivalently,

that there exist ﬁnitely many elements

a1,...,an∈A+

such that

A=A0[a1,...,an]

, which can

be assumed to be homogeneous of degrees

d1,...,dn

. In this case, note that

is a quotient of a

polynomial ring A0[x1,...,xn]by a homogeneous ideal.

-graded

-module is an

-module

that admits a direct sum decomposition

N=Lj∈ZNj

of Abelian groups, and such that

Ai·Nj⊆Ni+j

. As a consequence, each

is an

-module.

Moreover, if

is Noetherian there exists

i0∈Z

such that

Ni=0

for all

i<i0

; on the other hand, if

Nis Artinian there exists j0∈Zsuch that Nj=0 for all j>j0.

Given a

-graded

-module

, and an integer

j∈Z

, we deﬁne the shift

N(j)

as the

-graded

-module whose

-th graded component is

N(j)i=Ni+j

. In particular,

A(−j)

is a free graded

A-module of rank one with generator in degree j.

Graded local cohomology and Castelnuovo-Mumford regularity. In this subsection we recall

general properties of local cohomology. We refer the interested reader to Brodmann and Sharp’s

book on this subject [3]. Let

be a ﬁeld and

S=k[x1,...,xn]

, with

deg(xi) = di>0

. Let

be a

ﬁnitely generated

-graded

-module. If we let

m= (x1,...,xn)

, then the graded local cohomology

modules Hi

m(N)are Artinian and Z-graded.

Deﬁnition 2.1. Let

ai(N) = sup{j∈Z|Hi

m(N)j̸=0}

be the

-th

-invariant of

. If

N̸=0

deﬁne the Castelnuovo-Mumford regularity of

reg(N) = sup{ai(N) + i|i=0,...,n}

. On the

other hand, if N=0 we let reg(N) = −∞.

In the standard graded case,

reg(N)

has a well-known interpretation in terms of graded Betti

numbers of

. In our setup this is still the case, but the degrees of the algebra generators of

must

be taken into account. For

i∈Z⩾0

, let

βS

i(N) = a0(TorS

i(N,k)) ∈Z∪ {−∞}

. As another way to

see this, for a non-zero graded

-module

let

β(T)

be the maximum degree of an element in a

minimal homogeneous generating set of T. If

F•: 0 //Fc//Fc−1//..... . //F1//F0//N//0,

is a minimal graded free resolution of

where

c:=pd(N)

is the projective dimension of

, then

βS

i(N) = β(Fi).

Deﬁnition 2.2. For

N̸=0

we let

Reg(N) = sup{βS

i(N)−i|i∈Z}

, while for

N=0

we let

Reg(N) = −∞.

In the standard graded case, that is, when

d1=... =dn=1

, then

reg(N) = Reg(N)

. In our more

general scenario, we still have the following relation between the two notions of regularity.

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

PURITYOFMONOIDSANDCHARACTERISTIC-FREESPLITTINGSINSEMIGROUPRINGSALESSANDRODESTEFANI1,JONATHANMONTAÑO2,ANDLUISNÚÑEZ-BETANCOURT3ABSTRACT.Inspiredbymethodsinprimecharacteristicincommutativealgebra,weintroduceandstudycombinatorialinvariantsofseminormalmonoids.Werelatesuchnumberswiththesingularitiesandhom...

展开>> 收起<<

PURITY OF MONOIDS AND CHARACTERISTIC-FREE SPLITTINGS IN SEMIGROUP RINGS ALESSANDRO DE STEFANI1 JONATHAN MONTAÑO2 AND LUIS NÚÑEZ-BETANCOURT3.pdf

共20页,预览4页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

PURITY OF MONOIDS AND CHARACTERISTIC-FREE SPLITTINGS IN SEMIGROUP RINGS ALESSANDRO DE STEFANI1 JONATHAN MONTAÑO2 AND LUIS NÚÑEZ-BETANCOURT3

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: