Goto Numbers of a Numerical Semigroup Ring and the Gorensteiness of Associated Graded Rings

Abstract

The Goto number of a parameter ideal Q in a Noetherian local ring (R,m) is the largest integer q such that Q : m^q is integral over Q. The Goto numbers of the monomial parameter ideals of R = k[[x^{a_1}, x^{a_2},..., x_{a_{\nu}}]] are characterized using the semigroup of R. This helps in computing them for classes of numerical semigroup rings, as well as on a case-by-case basis. The minimal Goto number of R and its connection to other invariants is explored. Necessary and sufficient conditions for the associated graded rings of R and R/x^{a_1}R to be Gorenstein are also given, again using the semigroup of R. Comment: 36 pages, corrected typos and improved exposition throughout. To appear in Communications in Algebra

Full text

Page 1

arXiv:0809.0476v2 [math.AC] 26 Oct 2009
GOTO NUMBERS OF A NUMERICAL SEMIGROUP RING AND THE
GORENSTEINESS OF ASSOCIATED GRADED RINGS
LANCE BRYANT
Abstract. The Goto number of a parameter ideal Q in a Noetherian local ring (R,m)
is the largest integer q such that Q : mqis integral over Q. The Goto numbers of
the monomial parameter ideals of R = k[[xa1,xa2,...,xaν]] are characterized using the
semigroup of R. This helps in computing them for classes of numerical semigroup rings,
as well as on a case-by-case basis. The minimal Goto number of R and its connection to
other invariants is explored. Necessary and sufficient conditions for the associated graded
rings of R and R/xa1R to be Gorenstein are also given, again using the semigroup of R.
1. Introduction
Quasi-socle ideals are ideals of the form I = Q : mqin a Noetherian local ring (R,m)
with dimension d > 0, where Q = (a1,a2,...,ad) is a parameter ideal. In response to a
conjecture of Polini and Ulrich (1998) rooted in linkage theory, Wang (2007) proved the
following: Let R be Cohen-Macaulay with d ≥ 2, d > 2 when R is regular. If Q ⊂ mqand
I ?= R, then I ⊂ mq, mqI = mqQ, and I2= QI. Recently Goto, Kimura and Matsuoka
(2007) have inquired about a modification of this result for the one-dimensional case.
They began by studying Gorenstein numerical semigroup rings.
One component involved in making this modification is determining when Q : mqis
integral over Q. Heinzer and Swanson (2008) made the following definition.
Definition 1.1. Let g(Q) be the largest integer q such that Q : mqis integral over Q.
This is called the Goto number of Q.
In this paper the Goto numbers of the parameter ideals in a numerical semigroup ring
(R,m) = k[[S]] = k[[xa1,xa2,...,xaν]] are considered.
monomial parameter ideals.
Let e be the multiplicity of R. In section 2, Corollary 2.7 states that there is an e-
tuple of integers (σ(1),σ(2),...,σ(e)) called the Goto vector of S such that {g(u)|0 ?=
u ∈ S} ⊂ {σ(1),σ(2),...,σ(e)} where g(u) is the Goto number of xuR. In addition,
min{g(Q)|Q is a parameter ideal of R} = min{σ(1),σ(2),...,σ(e)}. If the Goto vector
of S is known, then the computation of g(u) is reduced to determining the set {α ∈
{1,2,...,e}|u−α ∈ S}. Using this set, several results about g(u) are obtained. Bounds
for the Goto numbers of R are given in Proposition 2.16.
Of particular interest are the
Date: October 26, 2009.
1

Page 2

2LANCE BRYANT
Section 3 introduces a class of numerical semigroups called M-pure semigroups. For
(R,m) = k[[S]], the M-purity of S is closely connected to the Gorenstein property of
the associated graded rings grm(R) =?
¯R = R/xa1R and ¯ m = m/xa1R. The definition of M-purity involves the Ap´ ery set of
the semigroup S and the m-adic order of R. The following equivalencies are proven
in Theorem 3.14: (i) gr¯ m(¯R) is Gorenstein if and only if S is M-pure symmetric, (ii)
grm(R) is Gorenstein if and only if S is M-pure symmetric and g(a1) = r, where g(a1)
is the Goto number of xa1R and r is the reduction number of m (with respect to xa1R).
This section is ended by showing that in general there are no implications among the
conditions symmetry, M-purity, and M-additivity. A semigroup S is M-additive if and
only if grm(R) is Cohen-Macaulay.
Section 4 is largely concerned with the inequalities δ ≤ γ ≤ ord(C) ≤ τ ≤ g(a1) ≤ r
where ord(C) is the m-adic order of the conductor C = R : k[[x]] and τ = min{g(Q)|Q
is a parameter ideal of R}. δ and γ are invariants that are derived from the Ap´ ery set
of S. Sufficient conditions are provided for many of these inequalities to be equalities.
A guiding question for this work has been: when is τ = g(a1)? Some partial results are
the following equivalencies stated in Theorem 4.4: (i) S is M-pure if and only δ = g(a1),
(ii) S is M-pure and grm(R) is Cohen-Macaulay if and only if δ = r, and (iii) grm(R)
is Gorenstein if and only if S is symmetric and δ = r. Furthermore τ = g(a1) for every
semigroup with multiplicity less than 5 with the exception of S =< 4,5,7 >, and for every
symmetric semigroup with multiplicity less than 7 with the exceptions S =< 5,6,9 > and
S =< 6,7,10,11 >.
In the last section Goto numbers are expressed in terms of the minimal generators of the
semigroup. This enables one to consider a class of semigroups (e.g., those with embedding
dimension 2) instead of considering semigroups on a case-by-case basis. The contents of
Theorems 5.8, 5.10, 5.11, 5.12, and 5.13 are the computations for the following classes:
(i) S is symmetric and generated by an arithmetic sequence, (ii) S is symmetric and of
almost maximal embedding dimension, (iii) S is of maximal embedding dimension, (iv)
S has multiplicity less than 5, (v) S is symmetric with multiplicity equal to 5. Moreover,
in Theorem 5.4 the Goto numbers of an arbitrary semigroup S are expressed in terms of
the Ap´ ery set of S.
i≥0mi/mi+1and gr¯ m(¯R) =?
i≥0¯ mi/¯ mi+1where
2. The Goto Numbers of a Numerical Semigroup
A numerical semigroup, or semigroup, S is a subsemigroup of N0that contains 0 and
has a finite complement in N0. For two elements u and u′in S, u ? u′if there exists an
s ∈ S such that u + s = u′. This defines a partial ordering on S. The minimal elements
in S \ {0} with respect to this ordering form the unique minimal set of generators for S,
which is denoted by {a1,a2,...,aν} where a1< a2< ··· < aν. S = {?ν
i=1ciai|ci≥ 0}

Page 3

GOTO NUMBERS OF A NUMERICAL SEMIGROUP RING3
is represented using the notation S = ?a1,...,aν?. Since the minimal generators of S are
distinct modulo a1, the set of minimal generators is finite. Furthermore, having finite
complement in N0is equivalent to gcd{ai|1 ≤ i ≤ ν} = 1.
S has the following invariants:
• f = f(S) = max{z ∈ Z\S} is the Frobenius number of S
• e = e(S) = min{u ∈ S |u ?= 0} is the multiplicity of S
• ν = ν(S) = #{a1,...,aν} is the embedding dimension of S
Notice that we always have e = a1and ν ≤ e.
The Ap´ ery set of S with respect to n ∈ N is Ap(S;n) = {w ∈ S |w −n ?∈ S}. This is a
finite set and it is immediate from the definition that if w ∈ Ap(S;n) and s ? w for some
s ∈ S, then s ∈ Ap(S;n). For u ∈ S, #Ap(S;u) = u and no two elements of Ap(S;u) are
congruent modulo u. Ap(S) = Ap(S;e) and is usually called the Ap´ ery set of S. There
are two natural ways to order the elements of Ap(S). One is Ap(S) = {w0,...,we−1}
where w0 < w1 < ··· < we−1, and the other is Ap(S) = {v0,...,ve−1} where vn ≡ n
mod e. Notice that we always have w0= v0= 0.
We will allow the subscript of vnto be any integer by agreeing that vn= vmif and only
if n ≡ m mod e. This will allow us to perform arithmetic operations on the subscripts of
the vi’s as in Lemma 2.1. Recall that the floor function ⌊x⌋ denotes the greatest integer
less than or equal to x, and that the ceiling function ⌈x⌉ denotes the least integer greater
than or equal to x.
Lemma 2.1. Let S = ?a1,...,aν? be a semigroup with Ap(S) = {v0,...,ve−1}. Then
(1) vn+ vm= vn+m+ ta1where t ≥ 0.
(2) vn− vm= vn−m− ta1where t ≥ 0.
(3) vn+ v−n= (1 +?vn
Proof. (1) Clearly vn+vm= vn+m+ta1for some t since vn+vm≡ vn+m mod e. Moreover,
vn+ vm∈ S so t ≥ 0.
(2) Again vn−vm= vn−m−ta1for some t since vn−vm≡ vn−m mod e. If t < 0, then
vn− a1∈ S, which is a contradiction.
(3) By (1), vn+ v−n= v0+ ta1= ta1where t ≥ 0. Thus vn− a1
is a positive multiple of a1when vn?= 0. Since both vn− a1
positive and strictly less than a1, we have vn− a1
a1
?+?v−n
a1
?)a1if vn?= 0.
?vn
a1
?+ v−n− a1
?= a1.
?v−n
a1
?
?vn
a1
?and v−n− a1
a1
?v−n
a1
?are
?vn
a1
?+ v−n− a1
?v−n
?
Statements (1) and (2) of Lemma 2.1 are also given by Madero-Craven and Herzinger
(2005, Lemma 2.6). The next definition facilitates the use of both notations for Ap(S).
Definition 2.2. For i ∈ {0,1,...,e−1}, let ? ı ∈ {1,2,...,e} denote the integer such that
wi= vb ı.

Page 4

4 LANCE BRYANT
Lemma 2.1 and Definition 2.2 will be used extensively in Section 5. However, they
are occasionally used throughout the paper and hence recorded here. Notice that ? ı ∈
σ(? ı) (see Definition 2.6).
k is a field. This is a one-dimensional local domain with maximal ideal m = (xa1,...,xaν).
The multiplicity and embedding dimension of S coincide with the ring-theoretic notions
for the ring R. Unless stated otherwise, it is assumed that R ?= k[[x]]. For the semigroup
S, this means that S ?= N0, or equivalently ν ≥ 2.
A parameter ideal Q of R is a principal ideal generated by a nonzero, nonunit element.
Although ultimately all of the parameter ideals of R are of interest (and some results
with this generality are provided), the focus is on monomial parameter ideals. These are
principal ideals generated by a monomial. The benefit is that one can work directly with
the numerical semigroup.
For the remainder of this paper g(u) denotes the Goto number of xuR and {g(u)|0 ?=
u ∈ S} is the set of Goto numbers of S. Moreover ord(n) = ord(xn) = k where xn∈
mk\mk+1if n ∈ S. This is the m-adic order of xn. To simplify statements, we will accept
the convention that ord(n) = −1 if n ?∈ S.
Theorem 2.4 and Corollary 2.7 are the main results of this section. The former charac-
terizes the Goto number of a nonzero element of S using the Ap´ ery sets Ap(S;α) where
1 ≤ α ≤ e, and the latter introduces the Goto vector of S. We begin by defining the
following sets.
{1,2,...,e} instead of {0,1,...,e−1}. This will be useful in Section 5 where we compute
The numerical semigroup ring corresponding to S is R = k[[S]] = k[[xa1,...,xaν]] where
Definition 2.3.
(1) T = {z ∈ Z\S |z + u ∈ S for all 0 ?= u ∈ S}
(2) A(u) = {α ∈ {1,2,...,e}|u − α ∈ S}
(3) A(S) = {1,2,...,e}
Results about T can be found in (Fr¨ oberg, Gottlieb and H¨ aggkvist, 1987) and (Barucci, Dobbs and Fontana
1997). A(u) is introduced here for its usefulness in studying Goto numbers. Both have
interesting connections to the Ap´ ery set of S as shown in Lemma 3.2 and Corollary 5.2.
Theorem 2.4. Let S = ?a1,...,aν? be a semigroup and 0 ?= u ∈ S. Then the following
integers are equal.
(1) g(u)
(2) minα∈A(u)
(3) minα∈A(u)
?max{ord(w)|w ∈ Ap(S,α)}?
?max{ord(p + α)|p ∈ T}?
Proof. Let R = k[[S]] be the ring corresponding to S. The integral closure of xuR is
xuk[[x]] ∩ R =?xs|s ∈ S and s ≥ u?R.

Page 5

GOTO NUMBERS OF A NUMERICAL SEMIGROUP RING5
(1) = (2) Let
N = min
α∈A(u)
?max{ord(w)|w ∈ Ap(S,α)?
We first show that g(u) ≤ N. For some α ∈ A(u), N = max{ord(w)|w ∈ Ap(S,α)}. We
have s −α ∈ S for all s ∈ S with ord(s) ≥ N + 1. But α = u −b where b ∈ S and b < u.
So s + b ∈ u + S for all s ∈ S with ord(s) ≥ N + 1. Therefore xb∈ xuR : mN+1and it
follows that xuR : mN+1is not contained in the integral closure of xuR. This proves that
g(u) ≤ N.
Next we show that N ≤ g(u). Let b ∈ S with b < u and choose k such that b + ka1<
u ≤ b+(k +1)a1. Then u−b−ka1∈ A(u). By the definition of N there exists an s ∈ S
with ord(s) ≥ N such that s − u + b + ka1?∈ S. Thus we have s + b + ka1?∈ u + S which
implies that s + b ?∈ u + S. So xb?∈ xuR : mN. Since b was an arbitrary element of S
strictly less than u, we conclude that xuR : mNis integral over xuR. This shows that
N ≤ g(u).
(2) = (3) For a fixed α we show that max{ord(w)|w ∈ Ap(S,α)} = max{ord(p +
α)|p ∈ T}. Recall that by convention, if p + α ?∈ Ap(S;α), then ord(p + α) = −1. Thus
we need only consider the case when p+α ∈ Ap(S;α), and the inequality “≥” holds. For
the reverse inequality “≤”, let w ∈ Ap(S;α) have the largest order among the elements in
Ap(S;α). Then w−α ?∈ S, but w−α+s ∈ S for all 0 ?= s ∈ S. Therefore w−α = p ∈ T
and w = p + α.
?
Remark 2.5. A nice way to think of Theorem 2.4(3) is that g(u) = ord(p + α) for some
α ∈ A(u) and p ∈ T satisfying the following conditions.
(1) For any q ∈ T, ord(p + α) ≥ ord(q + α)
(2) For any α′∈ A(u), there exists a p′∈ T such that ord(p + α) ≤ ord(p′+ α′)
Notice that every Goto number of S is determined by the order of p + α for some
p ∈ T and α ∈ A(S). Moreover, for a fixed α, we need only know the maximum order of
p +α as p ranges over the elements of T. If these values have been computed for S, then
determining g(u) is reduced to determining A(u). This motivates the following definition.
Definition 2.6. Let S = ?a1,...,aν? be a semigroup. For each α ∈ A(S) let σ(α) =
max{ord(w)|w ∈ Ap(S;α)} = max{ord(p + α)|p ∈ T}. Then gv(S) = (σ(1),...,σ(e))
will be called the Goto vector of S. Sometimes it is convenient to work with a set rather
than a vector, so let gs(S) = {σ(1),...,σ(e)} be the Goto set of S.
The next corollary follows immediately from Theorem 2.4.
Corollary 2.7. Let S be a semigroup, gv(S) = (σ(1),...,σ(e)) the Goto vector of S, and
0 ?= u ∈ S. Then g(u) = min{σ(α)|α ∈ A(u)}.
Example 2.8. This example demonstrates how the Goto vector of a semigroup can be
used to find the Goto numbers. Let S =< 4,5,7 >= {0,4,5,7,→} where the → indicates