One-dimensional Gorenstein local rings with decreasing Hilbert function

0.28 MB·

by  Anna Oneto

ONE-DIMENSIONAL GORENSTEIN LOCAL RINGS WITH
DECREASING HILBERT FUNCTION
ANNA ONETO, FRANCESCO STRAZZANTI, AND GRAZIA TAMONE
6
1
0
Abstract. In this paper we solve a problem posed by M.E. Rossi: Is the Hilbert function
2
of a Gorenstein local ring of dimension one not decreasing? Moreprecisely,foranyinteger

b h > 1, h ∈/ {14+22k, 35+46k | k ∈ N}, we construct infinitely many one-dimensional
e
Gorenstein local rings, included integral domains, reduced and non-reduced rings, whose
F
Hilbert function decreases at level h; moreover we prove that there are no bounds to the
0
decrease of the Hilbert function. The key tools are numerical semigroup theory, especially
1
some necessary conditions to obtain decreasing Hilbert functions found by the first and

] the third author, and a construction developed by V. Barucci, M. D’Anna and the second
C
author, that gives a family of quotients of the Rees algebra. Many examples are included.
A
.
h
t
a
Introduction
m
[ Given a one-dimensional Cohen-Macaulay local ring (R,m,k), let G be its associated

2 graded ring G = ⊕ mh/mh+1 and H be the Hilbert function of R, defined as H (h) =
h≥0 R R
v
4 H (h) = dim mh/mh(cid:16)+1 . The (cid:17)Cohen-Macaulayness of G and the behaviour of the Hilbert
G k
3
function are cl(cid:16)assic topic(cid:17)s in local algebra. Starting from 1970s with the basic results of
3
0 J.D. Sally [26], [27], [28], many authors have contributed to these themes; for instance we
0 recall J. Elias [16], M.E. Rossi and G. Valla [25] and Rossi’s survey [24]. It is well-known
.
2 that if G is Cohen-Macaulay, the function H is non-decreasing. On the other hand, when
R
0
depth(G) = 0, H can decrease, i.e. H (h − 1) > H (h) for some h; in this case we
6 R R R
1 say that H decreases at level h and that R has decreasing Hilbert function. When R is
R
: Gorenstein, M.E. Rossi asked in [24, Problem 4.9] if H is always non-decreasing and in the
v R
i last decade several authors found partial positive answers to this problem, especially in the
X
case of numerical semigroup rings:
r
a • In [2] F. Arslan and P. Mete, for large families of complete intersection rings and the
Gorenstein numerical semigroup rings with embedding dimension 4, under some arithmetical
conditions;
• In [3] F. Arslan, P. Mete and M. Şahin, for infinitely many families of Gorenstein rings
obtained by introducing the notion of nice gluing of numerical semigroups;
• In [22] D.P. Patil and the third author, for the rings associated with balanced numerical
semigroups with embedding dimension 4;
• In [4] F. Arslan, N. Sipahi and N. Şahin, for other 4-generated Gorenstein numerical
Date: February 11, 2016.
2010 Mathematics Subject Classification. Primary 13H10; 13A30; Secondary 20M14.
Key words and phrases. Hilbertfunction,Gorensteinring,AlmostGorensteinring,Numericalsemigroup,
Numerical duplication, Almost symmetric semigroup.
1
2 ANNAONETO, FRANCESCO STRAZZANTI,ANDGRAZIA TAMONE
semigroup rings constructed by non-nice gluing;
• In [19] R. Jafari and S. Zarzuela Armengou, for some families of numerical Gorenstein
semigroup rings through the concept of extension;
• In [1] F. Arslan, A. Katsabekis, and M. Nalbandiyan, for other families of Gorenstein
4-generated numerical semigroup rings;
• In [21] the first and the third author, for numerical semigroup rings such that ν ≥ e−4,
where ν and e denote respectively the embedding dimension and the multiplicity of R.
In this paper we show that Rossi’s problem has negative answer, by constructing, among
others, explicit examples of Gorenstein numerical semigroup rings with decreasing Hilbert
function. They are particular rings of a family introduced and studied by V. Barucci, M.
D’Anna and the second author in [6] and [7] to provide a unified approach to Nagata’s
idealization and amalgamated duplication. Given a commutative ring R and an ideal I, for
anya,b ∈ Rtherings R(I) aredefined assuitable quotients oftheRees algebraofI. These
a,b
have many good properties, in particular, if R is a one-dimensional local ring, so is R(I) .
a,b
If R is Cohen-Macaulay, another important fact is that R(I) is Gorenstein if and only if I
a,b
is a canonical ideal of R; in this case, when R is almost Gorenstein, we prove that the Hilbert
function of R(I) depends only on the Cohen-Macaulay type and the Hilbert function of
a,b
R. The crucial result is that if R is an almost Gorenstein ring with H (h−2) > H (h) for
R R
some h ≥ 3 and I is a canonical ideal of R, then R(I) is a one-dimensional Gorenstein
a,b
local ring with Hilbert function decreasing at level h.
We find such rings through numerical semigroup theory. A numerical semigroup S is
a submonoid of the natural numbers that has finite complement in N; if S is generated
by s0,...,sν−1 and k is a field, then k[[S]] := k[[ts0,...,tsν−1]] is the numerical semigroup
ring associated with S and many properties of k[[S]], such as the Hilbert function, are still
contained in S. In this context k[[S]] is almost Gorenstein if and only if S is a so-called
almost symmetric semigroup.
Hence to achieve our results we look for almost symmetric semigroups with decreasing
Hilbert function. If e and ν are the multiplicity and embedding dimension of S (or equiva-
lently of k[[S]]) we first show that we need e−v ≥ 4. By using a result of [21] and a theorem
of H. Nari [20] we give an explicit construction of a family of almost symmetric semigroups
with the required properties. We also show other examples with the above properties. In
conclusion we prove that for any integers m ≥ 1 and h > 1,h ∈/ {14+22k, 35+46k | k ∈ N},
there exist infinitely many non-isomorphic one-dimensional Gorenstein local rings R such
that H (h − 1) − H (h) > m; this class always contains numerical semigroup rings, non-
R R
reduced rings, and reduced rings that are not integral domains.
We include several examples in the numerical semigroup case. In fact if R is a numerical
semigroup ring and b = tm, with m odd, the ring R(I) is isomorphic to the numerical
0,−b
semigroup ring associated with the numerical duplication, a construction introduced and
studied by M. D’Anna and the second author in [14].
The structure of the paper is the following. In the first section we introduce the family
R(I) and show how to reduce the problem to find a suitable almost Gorenstein ring, see
a,b
Corollary 1.5. In Section 2 we describe a procedure that gives infinitely many almost Goren-
stein semigroup rings satisfying the desired properties, see Construction 2.6 and Theorem
ONE-DIMENSIONAL GORENSTEIN LOCAL RINGS WITH DECREASING HILBERT FUNCTION 3
2.9. In Section 3 we prove the main result, see Theorem 3.3; moreover we give explicit
examples of one-dimensional Gorenstein local semigroup rings with decreasing Hilbert func-
tion and other interesting examples based on the above constructions; see e.g. Example
3.4 with Hilbert function [1,53,54,54,53,53,56,59,61,63,64→], Example 3.9 with Hilbert
function decreasing at many levels, and Example 3.10 for a ring with smaller multiplicity
and embedding dimension. Finally the appendix contains the technical results needed to
prove Theorem 2.9.
1. Reduction to the almost Gorenstein case
Let R be a commutative ring with identity and let I be a proper ideal of R. The Rees
algebra of I is the ring R := ⊕ Intn ⊆ R[t]. Consider the ideal (t2 + at + b) of R[t],
+ n≥0
where a and b are elements of R, and let I2(t2 +at+b) denote its contraction to the Rees
algebra. In [6] it is introduced and studied the following family of rings
R
+
R(I) := .
a,b
I2(t2 +at+b)
One aim of this construction was to provide a unified approach to Nagata’s idealization (see
[5]) and amalgamated duplication (see [11] and [13]), which are isomorphic to R(I) and
0,0
R(I) respectively. In fact in [6] and [7] it is proved that several properties of the family
−1,0
are independent of a and b. In particular R is local if and only if R(I) is local, R and
a,b
R(I) have the same dimension and R(I) is Cohen-Macaulay if and only if R is Cohen-
a,b a,b
Macaulay and I is a maximal Cohen-Macaulay module; further, in the last case if I contains
a regular element, R(I) is Gorenstein if and only if I is a canonical ideal of R.
a,b
We are interested in the Hilbert function of these rings. In [6, Proposition 2.3] it is stated
that their Hilbert functions do not depend on a and b, but actually in the proof it is shown
more and then we restate that proposition in the version we need:
Proposition 1.1. If (R,m) is a local ring and I is an ideal of R, then for any h ≥ 2
H (h) = H (h) + ℓ Imh−1/Imh , where ℓ denotes the length as R-module. In par-
R(I)a,b R R R
ticular it is independent o(cid:16)f a and b. (cid:17)
We are interested in the case in which R is a one-dimensional Cohen-Macaulay local ring
and I is a canonical ideal of R. In this case we can easily compute ℓ (Imh−1/Imh), under an
R
extra hypothesis on R: almost Gorensteinness. Following [8] and [18] we recall the definition
in the one-dimensional case:
Definition 1.2. Let(R,m)beaone-dimensionalCohen-Macaulaylocalringwithacanonical
module ω such that R ⊆ ω ⊆ R. Then R is said to be almost Gorenstein if mω = m.
R R R
FromnowonweassumethatRisone-dimensional. Inthesettingofthepreviousdefinition,
chosen a regular element a ∈ R such that aω ⊂ R, the ideal I = aω is a canonical ideal of
R R
R and all canonical ideals of R can be obtained in this way (see e.g. [18, Corollary 2.8]).
If R is an almost Gorenstein ring and I is a canonical ideal of R, for any h ≥ 2 we have
Imh−1 aω mh−1 amh−1 mh−1
R
(1.3) ℓ = ℓ = ℓ = ℓ = H (h−1).
R Imh ! R aωRmh ! R amh ! R mh ! R
Therefore we get the following:
4 ANNAONETO, FRANCESCO STRAZZANTI,ANDGRAZIA TAMONE
Proposition 1.4. Let R be an almost Gorenstein ring and let I be a canonical ideal of
R. Let ν(R) and t(R) denote the embedding dimension and the Cohen-Macaulay type of R
respectively. Then the Hilbert function of R(I) is:
a,b
H (0) = 1
R(I)a,b
H (1) = ν(R)+t(R)
R(I)a,b
H (h) = H (h)+H (h−1) if h ≥ 2.
R(I)a,b R R
Proof. By Proposition 1.1 and (1.3) we only need to show the statement about H (1):
R(I)a,b
actually this equality is true even if R is not almost Gorenstein . In fact, since the minimal
number of generators of a canonical module is the Cohen-Macaulay type of the ring (see e.g.
(cid:3)
[9, Proposition 3.3.11]), it is easy to deduce the thesis from Proposition 1.1.
Corollary 1.5. Let R be an almost Gorenstein ring and let I be a canonical ideal of R. Then
R(I) is a Gorenstein ring for any choice of a,b ∈ R. Further H (1)−H (2) =
a,b R(I)a,b R(I)a,b
t(R)−H (2) and H (h−1)−H (h) = H (h−2)−H (h) for any h ≥ 3.
R R(I)a,b R(I)a,b R R
Proof. SinceI isacanonicalideal,R(I) isaGorensteinringby[6,Corollary3.3]. Moreover,
a,b
by the previous proposition, for any h ≥ 3 we have
H (h−1)−H (h) = H (h−1)+H (h−2)−H (h)−H (h−1) = H (h−2)−H (h).
R(I)a,b R(I)a,b R R R R R R
The first formula can be found in the same way, since H (1) = ν(R). (cid:3)
R
Remark 1.6. In this paper we are interested in negative results, anyway it is clear that
Proposition 1.1 can be also used to get positive results. A local ring S is said to have
minimal multiplicity if its multiplicity is 1+codimS. In [23, Theorem 1.1] it is proved that
if S is a two-dimensional Cohen-Macaulay local ring with minimal multiplicity and (R,m) is
a one-dimensional Cohen-Macaulay local ring which is a quotient of S, then every maximal
Cohen-Macaulay R-module M has non-decreasing Hilbert function. Here the h-th value of
the Hilbert function of M is defined as the length of Mmh/Mmh+1. Therefore, Proposition
1.1 implies that, under the above hypothesis, R(I) has non-decreasing Hilbert function
a,b
for any maximal Cohen-Macaulay ideal I; in particular, if I is a canonical ideal, R(I) is a
a,b
one-dimensional Gorenstein local ring with non-decreasing Hilbert function for all a,b ∈ R.
See [23] for several explicit cases in which the above hypothesis hold.
2. Construction of almost symmetric semigroups
In order to obtain Gorenstein local rings with decreasing Hilbert function, by Corollary
1.5 it is enough to find almost Gorenstein rings R such that H (h − 2) > H (h); in this
R R
section we construct infinitely many semigroup rings verifying these conditions.
First, we briefly recall some definitions and properties about numerical semigroup theory
that we need. A numerical semigroup S is a submonoid of the natural numbers such that
|N\S| < ∞. The maximum element of N\S is called Frobenius number of S and it will be
denoted by f(S). If S is generated by n ≤ n ≤ ··· ≤ n , we write S = hn ,...,n i.
0 1 ν−1 0 ν−1
It is well-known that a numerical semigroup has an unique minimal system of generators
and its cardinality is the embedding dimension ν of S. The smallest non-zero element of
S is n ; it is called multiplicity of S and we will denote it by e(S) or simply e, if the
0
ONE-DIMENSIONAL GORENSTEIN LOCAL RINGS WITH DECREASING HILBERT FUNCTION 5
semigroup is clear from the context. A numerical semigroup ring is a local ring of the form
k[[S]] := k[[tn0,tn1,...,tnν−1]], where S = hn0,n1...,nν−1i is a numerical semigroup and k
a field. A relative ideal E of a numerical semigroup is a subset of Z such that there exists
x ∈ N for which x + E ⊆ S and E + S ⊆ E; if E is contained in S we say that E is a
(proper) ideal of S. An example of ideal is the maximal ideal M = M(S) := S\{0} = v(m),
where m = (tn0,tn1,...,tnν−1) and v : k((t)) −→ Z∪{∞} is the usual valuation. An example
of a relative ideal is the standard canonical ideal K(S) := {x ∈ N | f(S) − x ∈/ S}; more
generally, we call canonical ideals all the relative ideals K(S)+z for any z ∈ Z.
The properties of a semigroup ring are strictly related to those of the associated numerical
semigroup. In particular: H (h) = |v(mh)\v(mh+1)| = |hM \(h+1)M| for each h ≥ 1.
K[[S]]
We shall denote this function by H and its values by[1,ν(S),H (2),...,e,→]. It is well-
S S
known that:
• k[[S]] is Gorenstein if and only if K(S) = S; in this case S is said to be symmetric;
• k[[S]] is almost Gorenstein if and only if M + K(S) = M; in this case S is said to be
almost symmetric.
Definition 2.1. Let S be a numerical semigroup.
i. If s is an element of S, the order of s is ord(s) := max{i | s ∈ iM}.
ii. The Apéry set of S is Ap(S) := {s ∈ S | s−e ∈/ S}, shortly Ap; it has cardinality e.
iii. Ap := {s ∈ Ap | ord(s) = k}.
k
iv. D := {s ∈ S | ord(s) = h−1 and ord(s+e) > h}, Dt := {s ∈ D | ord(s+e) = t}.
h h h
v. C := {s ∈ S | ord(s) = k and s−e ∈/ (k −1)M}.
k
WenoticethatC = Ap {∪ (Dk+e)|2 ≤ h ≤ k−1}andH (k−1)−H (k) = |D |−|C |,
k k h h S S k k
see for instance [10] and [22].
S
The following theorem of H. Nari characterizes the almost symmetric numerical semigroups
by means of their Apéry sets. First we recall that a pseudo-Frobenius number of S is an
integer x ∈ Z\S such that x+s ∈ S for any s ∈ M. We denote the set of pseudo-Frobenius
number of S by PF(S); it is straightforward to see that f(S) ∈ PF(S). The cardinality of
PF(S) is the type of S and it will be denoted by t(S); it is well-known that t(S) = t(k[[S]]).
Theorem 2.2. [20, Theorem 2.4] Let S be a numerical semigroup. Set Ap = A ∪ B,
where A := {0 < α < ··· < α }, B := {β < ··· < β }, with m = e−t(S), and
1 m 1 t(S)−1
PF(S) = {β −e | 1 ≤ i ≤ t(S)−1} ∪ {α −e = f(S)}. The following conditions are
i m
equivalent:
i. S is almost symmetric;
ii. α + α = α for all i ∈ {1,2,...,m − 1} and β +β = α +e for all
i m−i m j t(S)−j m
j ∈ {1,2,...,t(S)−1}.
Since we are looking for a semigroup with decreasing Hilbert function, we need that
|Ap | ≥ 3, by [12, Corollary 3.11]; then we focus on the simpler case, |Ap | = 3.
2 2
Proposition 2.3. Assume |Ap | = 3, Ap = ∅ for all k ≥ 3 and H decreasing. Then S
2 k S
cannot be almost symmetric.
Proof. The assumptions on the Apéry set imply that ν = e − 3. Since H decreases, by
S
[21, Theorem 4.2.3] there exist n 6= n ∈ Ap such that Ap = {2n ,n +n ,2n } and we
1 2 1 2 1 1 2 2
6 ANNAONETO, FRANCESCO STRAZZANTI,ANDGRAZIA TAMONE
assume n < n . The element n −e is not a pseudo-Frobenius number, because 2n −e ∈/ S;
1 2 1 1
therefore if S is almost symmetric, with the notation of the previous theorem, A is non-
empty. It follows that ord(α ) > 1 and so α = 2n . On the other hand by [21, Proposition
m m 2
4.3.1] we have 3n −e ∈ Ap, that is a contradiction because 3n −e > 2n . (cid:3)
2 2 2
According to the above proposition, we consider the next case |Ap | = 1. In this context
3
the following proposition holds:
Proposition 2.4. [21, Proposition 3.4] Assume |Ap | = 3, |Ap | = 1 and H decreasing.
2 3 S
Let ℓ = min{h | H decreases at level h} and let d = max{ord(σ) | σ ∈ Ap}. Then ℓ ≤ d
S
and there exist n ,n ∈ Ap such that for 2 ≤ h ≤ ℓ
1 2 1
C = {hn ,(h−1)n +n ,...,n +(h−1)n ,hn } = Ap ∪(D +e)
h 1 1 2 1 2 2 h h−1
D +e = {(d+1)n ,ℓn +n ,(ℓ−1)n +2n ,...,(ℓ+1)n }.
ℓ 1 1 2 1 2 2
Further, if (ℓ,d) 6= (3,3), then Ap = kn , for 3 ≤ k ≤ d.
k 1
The next proposition shows that, in the setting of the previous one, we only need to find
analmost symmetric semigroup with decreasing Hilbert function. This is not true ingeneral,
for instance the numerical semigroup S = h30,35,42,47,108,110,113,118,122,127,134,139i
is almost symmetric and its Hilbert function is H = [1,12,17,16,25,30 →]. Therefore H
S S
decreases, but H (h−2) ≤ H (h) for any h ≥ 2.
S S
Proposition 2.5. Assume that |Ap | = 3,|Ap | = 1, and H is decreasing. Let ℓ be the
2 3 S
minimum level in which the Hilbert function of S decreases:
i. If ℓ ≥ 3, then H (h) = H (ℓ−1) for all h ∈ [1,ℓ−1]. Further, H (ℓ−2)−H (ℓ) = 1.
S S S S
ii. If S is almost symmetric then α = dn and kn ∈ A for any k ≤ d (see Theorem 2.2
m 1 1
and Proposition 2.4 for the notation).
iii. If S is almost symmetric, then ℓ ≥ 3.
Proof. i. By Proposition 2.4, if 2 < h < ℓ we have C = (D + e) ∪ Ap and then
h h−1 h
|D | = |C | − 1, since |Ap | = 1. Moreover, in this range, we have |C | = |C | + 1,
h−1 h h h h−1
because of the previous proposition. Hence for any h = 2,...,ℓ−1 we have
H (h−1)−H (h) = |D |−|C | = |C |−1−(|C |−1) = 0.
S S h h h+1 h+1
Consequently we get H (1) = H (2) = ··· = H (ℓ−1). As for the last part of the statement
S S S
it is enough to note that, by the previous proposition, we have
H (ℓ−1)−H (ℓ) = |D |−|C | = ℓ+2−(ℓ+1) = 1.
S S ℓ ℓ
ii. Of course α is the greatest element of the Apéry set and in our case it can be either dn
m 1
or 2n . If α = 2n , then there would exist n ∈ Ap, such that (d−1)n +n = 2n , but it is
2 m 2 1 2
impossible, sinced ≥ 3impliesord(2n ) ≥ 3. Clearlykn ∈ A, becausekn +(d−k)n = dn .
2 1 1 1 1
iii. Assume ℓ = 2. By Proposition 2.4, (d+1)n −e ∈ D and so ord((d+1)n −e) = 1;
1 2 1
consequently, if S is almost symmetric, there exists n ∈ Ap such that (d+ 1)n −e+n =
1
dn +ke, with either k = 0 or k = 1. Hence n +n = (1+k)e ≤ 2e, impossible. (cid:3)
1 1
The next construction is based on Proposition 2.4 and indeed we will prove that it defines
almost symmetric numerical semigroups with decreasing Hilbert functions satisfying the
assumptions of Proposition 2.4.
ONE-DIMENSIONAL GORENSTEIN LOCAL RINGS WITH DECREASING HILBERT FUNCTION 7
Construction 2.6. Let ℓ ∈ N,ℓ ≥ 4, ℓ ∈/ {14+22k, 35+46k | k ∈ N}, let
e := ℓ2 +3ℓ+4
n := ℓ2 +5ℓ+3 = e+(2ℓ−1), n := 2ℓ2 +3ℓ−2 = e+(ℓ2 −6), if ℓ odd
1 2
" n1 := ℓ2 +4ℓ+1 = e+(ℓ−3), n2 := 2ℓ2 +2ℓ−2 = e+(ℓ2 −ℓ−6), if ℓ even
and let S be the semigroup generated by the subset Γ ⊆ N
Γ = {e,n ,n } ∪{t ,t }∪{s }∪{r }\{n +n ,2n }
1 2 1 2 p,q p,q 1 2 2
where:
{s } = {pn +qn −(p+q−2)e | 0 ≤ p ≤ ℓ, 1 ≤ q ≤ ℓ+1, 2 ≤ p+q ≤ ℓ+1}
p,q 1 2
{r } = {ℓn +e−s | 2 ≤ p+q ≤ ℓ+1, p ≥ 1, q ≥ 1}
p,q 1 p,q
t = (ℓ+1)n −(ℓ−1)e,
1 1
t = ℓn +e−t = (ℓ−1)e−n
2 1 1 1
|{s }| = (ℓ2 +3ℓ)/2 := u, |{r }| = (ℓ2 +ℓ)/2, |Γ| = |{s }|+|{r }|+3 = e−ℓ−1
p,q p,q p,q p,q
(ℓ−1)e = −ℓn +(ℓ+2)n
1 2
We note that the elements {e,n ,n ,t } ∪ {s } are given to obtain the structure of S
1 2 1 p,q
required in Proposition 2.4, with d = ℓ, while the elements {r }∪{t }, impose the almost
p,q 2
symmetry of S following Theorem 2.2 (with α = ℓn , and B ⊇ {t }∪{s }\{s }). In
m 1 1 p,q 0,ℓ+1
this construction, for s ∈ {qn −(q−2)e,2 ≤ q ≤ ℓ+1}∪{n }∪{pn ,1 ≤ p ≤ ℓ}, we don’t
2 2 1
need to add the corresponding ℓn +e−s, or ℓn −s, because such element is already inside
1 1
S (see Lemma 4.2.i).
Remark 2.7. Looking foranalmost symmetric semigroup S satisfying Proposition2.4, with
d = ℓ, it is natural to impose e ≥ ℓ2+3ℓ+4. In fact the first idea to construct this semigroup
is to impose that a system of generators of S contains the set
{e,n ,n }∪{ℓn −n }∪{t ,t }∪{s }∪ α +e−s,s ∈ {s } \{n +n ,2n }.
1 2 1 2 1 2 p,q m p,q 1 2 2
By counting the number of conditions for a givenn ℓ, we get o
ℓ2 +3ℓ
e(S) ≥ 4+2 = ℓ2 +3ℓ+4.
2
Further, following Proposition 2.4, we need
Ap = Ap ∪{2n ,n +n ,2n }∪{kn ,3 ≤ k ≤ ℓ}
1 1 1 2 2 1
thustheembedding dimensionofS must beν = e(S)−ℓ−1. If|{r }| = |{s }|−ℓandℓn −
p,q p,q 1
n ∈ {s }, we could fix the minimal value e = ℓ2+3ℓ+4, for the multiplicity. This happens
2 p,q
if we define e,n , and n as in Construction 2.6. In fact this choice gives the basic relation
1 2
ℓn = (ℓ+2)n −(ℓ−1)e
1 2
which assures that, for 2 ≤ q ≤ ℓ, ℓn + e −s ∈ {s } and so it reduces the number of
1 0,q p,q
independent conditions to 2|{s }|−ℓ+3 (see Lemma 4.2).
p,q
We give some examples before to prove of the exactness of the construction.
Example 2.8. In this example we show the almost symmetric semigroups constructed by
means of the above algorithm for ℓ = 4,5 and the Hilbert function H of R = k[[S]].
S
ℓ=4. The semigroup S is minimally generated by {32, 33= n , 38= n , 69, 72, 73, 74, 75,
1 2
77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95}. Moreover
8 ANNAONETO, FRANCESCO STRAZZANTI,ANDGRAZIA TAMONE
Ap = {66,71,76}, Ap = {99 = 3n }, Ap = {132 = ℓn }, PF(S) = {37, 39, 40,
2 3 1 4 1
41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58,59, 60, 61,63, 100}.
H = [1,27,27,27,26,27,29,30,31,32→] .
S
ℓ=5. The minimal generating system of S is {44, 53= n , 63= n , 117, 125, 127, 134, 135,
1 2
136, 137, 142, 143, 144, 145, 146, 147, 152, 153, 154, 155, 156, 157, 162, 163, 164,
165, 166, 167, 172, 173, 174, 175, 182, 183, 184, 192, 193, 202}.
Ap = {106,116,126}, Ap = {159}, Ap = {212}, Ap = {265} PF(S) = {72, 73,
2 3 4 5
81, 82, 83, 90, 91, 92, 93, 98, 99, 100, 101, 102, 103, 108, 109, 110, 111, 112, 113,
118, 119, 120, 121, 122, 123, 128, 129, 130, 131, 138, 139, 140, 148, 149, 221 }.
H = [1,38,38,38,38,37,44→]
S
To validate Construction 2.6, we need some technical lemmas proved in the appendix.
Theorem 2.9. With the assumptions of Construction 2.6:
i. The ring R = k[[S]] is almost Gorenstein with Hilbert function decreasing at level ℓ:
H = [1,ν,ν,...,ν,ν −1,H (ℓ+1)...].
R R
ii. The embedding dimension of R is ν(R) = e−(ℓ+1) = ℓ2 +2ℓ+3.
iii. The Cohen-Macaulay type of R is t(R) = ν(R)−1 = ℓ2 +2ℓ+2.
Proof. i. By Proposition 4.5, we know the Apéry set of S and the subsets A,B of Theorem
2.2. Then S is almost symmetric, since the elements of A and B verify the conditions of
Theorem 2.2, respectively by Lemma 4.2.i and by the definition of {r }.
p,q
Now we show that the Hilbert function of k[[S]] decreases at level ℓ: we shall prove that
• for each k ∈ [2,ℓ+1], C = {kn ,(k −1)n +n ,(k −2)n +2n ,...,kn }, |C | = k +1
k 1 1 2 1 2 2 k
•foreachk ∈ [2,ℓ−1],D +e = {kn +n ,(k−1)n +2n ,...,(k+1)n } = C \{(k+1)n },
k 1 2 1 2 2 k+1 1
|D | = k +1
k
• D +e = {(ℓ+1)n ,ℓn +n ,(ℓ−1)n +2n ,...,(ℓ+1)n }, |D | = ℓ+2.
ℓ 1 1 2 1 2 2 ℓ
Hence the thesis follows, recalling that for each k ≥ 2 : H(k) = H(k −1)+|C |−|D |.
k k
Let s = an +bn , with a+b = k ∈ [3,ℓ+1] and a < k, if k 6= ℓ+1. First we prove that
1 2
if an +bn = d+e, with d ∈ D , then ord(an +bn )=h+1 (1)
1 2 h 1 2
In fact, if ord(an + bn ) = k′ = h + p, with p ≥ 2, we know, by [21, Proposition 2.2.1],
1 2
that given a maximal expression an + bn = a n , with n ∈ Ap , a = k′, for any
1 2 i i i 1 i
y = b n , 0 ≤ b ≤ a , b = q ≤ p+1, then y ∈ Ap . Since k′ > 3 and Ap = {3n }, this
i i i i i P q P 3 1
wouldimplyan +bn = k′n , impossiblebyLemma 4.4.ii. Hencean +bn ∈ C ∩(D +e).
P 1 2 P 1 1 2 h+1 h
By definition, C = Ap and by Proposition 4.5, D ⊇ {2n +n −e,n +2n −e,3n −e}.
2 2 2 1 2 1 2 2
Then D = {2n +n −e,n +2n −e,3n −e}, otherwise the Hilbert function decreases at
2 1 2 1 2 2
level 2, impossible by Proposition 2.5.iii. Hence C = (D +e)∪{3n }.
3 2 1
Now we proceed by induction on k. First we recall that, if x ∈ C has maximal rep-
k
resentation x = a n , a = k,n ∈ Ap and y = b n ,0 ≤ b ≤ a with b = h,
i i i i 1 i i i i i
then y ∈ C by [21, Proposition 1.4.1]. In our case C = {2n ,n + n ,2n }, hence
h P P P 2 1 1 2 P2
C ⊆ {an +bn | a+b = k}. Assume 3 ≤ k ≤ ℓ and the thesis true for k−1. Therefore we
k 1 2
know the structures of C ,...C , D ,...,D . Let an +bn with a+b = k +1 ∈ [4,ℓ+1]
2 k 2 k−1 1 2
and a < k+1, if k 6= ℓ. Then, by Lemma 2.6, s = an +bn −(k−1)e ∈ Ap is such that
a,b 1 2 1
ord(s +(k−1)e) ≥ k+1. Moreover, since s +e ∈/ D , we know that its order is 2; hence
a,b a,b 2
ONE-DIMENSIONAL GORENSTEIN LOCAL RINGS WITH DECREASING HILBERT FUNCTION 9
there exists r ∈ [1,k −2] such that ord(s + re) = r +1 and ord(s +(r +1)e) > r +2
a,b a,b
i.e. s + re ∈ D , with r + 2 ≤ k. If r + 2 < k, by induction there would exist
a,b r+2
a′n1 + b′n2 − e ∈ Dr+2 such that sa,b + re = a′n1 + b′n2 − e = sa′,b′ + (a′ + b′ − 3)e:
impossible because sa,b and sa′,b′ have distinct residues (mode), by Lemma 4.4.ii. Hence
r = k −2, an +bn −e ∈ D and an +bn ∈ C by (1). This proves i.
1 2 k 1 2 k+1
ii. Since there are ℓ + 1 elements of the Apéry set with order greater than 1, we have
ν(R) = |Ap |+1 = e−1−(ℓ+1)+1 = ℓ2 +2ℓ+3.
1
iii. The Cohen-Macaulay type of k[[S]] is the cardinality of the Pseudo-Frobenius set of S:
ℓ2 +3ℓ ℓ2 +ℓ
t(R) = |B|+1 = −1+ +3 = ℓ2 +2ℓ+2. (cid:3)
2 2
Without using Construction 2.6, but by similar techniques, it is possible to construct other
almost symmetric semigroups such that H (h−1) > H (h), even if h = 2,3, as shown in the
S S
first two semigroups of the next example. Moreover the last one is another almost symmetric
semigroup with |Ap | = 3,|Ap | = 1 and decreasing Hilbert function.
2 3
Example 2.10. i. The numerical semigroup S = h33,41,42,46,86,90,91,95,96,97,98,
100,101,103,104,105,106,109,110,111,113,114,118,122i is almost symmetric and its
Hilbert function [1,24,23,23,31,33→] decreases at level 2. Moreover
Ap = {82,83,84,87,88,92,127}, Ap = {126}, Ap = {168}, Ap = ∅ if k ≥ 5.
2 3 4 k
ii. The numerical semigroup S = h32,33,38,58,59,60,61,62,63,67,68,69,72,73,74,75,
77,78,79,80,81,82,83,84,85,86,87,88i is almost symmetric with Hilbert function
[1,28,28,27,27,29,30,31,32→] decreasing at level 3 and
Ap = {66,71,76,121}, Ap = ∅ if k ≥ 3.
2 k
iii. The numerical semigroup S = h30,33,37,64,68,71,73,75,76,77,78,79,80,81,82,83,
84,85,86,87,88,89,91,92,94,95,98,101i is almost symmetric with Hilbert function
[1,25,25,25,24,27,28,29,30→]
Ap = {66,70,74}, Ap = {99}, Ap (S) = {132}, Ap = ∅ if k ≥ 5.
2 3 4 k
3. The Gorenstein case
In this section we give explicit examples of local one-dimensional Gorenstein rings with
decreasing Hilbert function and other interesting examples. Several computations are per-
formed by using the GAP system [17] and, in particular, the NumericalSgps package [15].
Luckily, if R is a numerical semigroup ring and b = tm ∈ R, with m odd, then R(I) is a
0,−b
numerical semigroup ring and it is exactly the ring associated with the so-called numerical
duplication. Anyway we note that in general, for other choices of a and b, the ring R(I)
a,b
is not a numerical semigroup. For example if a = −1 and b = 0 it is isomorphic to the
amalgamated duplication that, in this case, is reduced but not a domain; while R(I) is
0,0
isomorphic to the idealization and then it is not reduced. In this section we describe the
particular case of the numerical duplication, that is probably the easiest case; we show the
most notable and simple examples among the various we have constructed.
Let S be a numerical semigroup, b ∈ S be an odd integer and E be a proper ideal of S.
The numerical duplication of S with respect to E and b, introduced in [14], is the numerical
10 ANNAONETO, FRANCESCO STRAZZANTI,ANDGRAZIA TAMONE
semigroup
S✶bE := {2·S}∪{2·E +b},
where 2·X = {2x | x ∈ X} for any set X; we note that 2·X is different from 2X = X +X.
As mentioned above, if R = k[[S]] is a numerical semigroup ring and b = tm ∈ R with m
odd, it is proved in [6, Theorem 3.4] that
R(I) is isomorphic to k[[S✶mE]],
0,−b
where E := v(I) is the valuation of I, see [6] for more detail. We recall that I is a canonical
ideal of R if and only if v(I) is a proper canonical ideal of S; hence S✶bE is symmetric if
and only if E is a canonical ideal, see also [14, Proposition 3.1] for a simpler proof.
It is easy to compute the generators of S✶bE, in fact if G(S) = {n ,...,n } is the set of
1 r
the minimal generators of S and E is generated, as ideal, by {m ,...,m }, then
1 s
S✶bE = h2n ,...,2n ,2m +b,...,2m +bi.
1 r 1 s
In particular, we recall that K(S) is minimally generated by the elements f(S)−x, where
x ∈ PF(S), and therefore, if E = K(S)+z, the semigroup S✶bE is generated by
{2n , 2(f(S)−x +z)+b | n ∈ G(S), x ∈ PF(S)};
i j i j
moreover if S is almost symmetric, it follows from Theorem 2.2 that S ✶b (K(S) + z) is
minimally generated by {2n ,2z + b,2x + 2z + b | n ∈ G(S),x ∈ PF(S) \ {f(S)}}.
i j i j
Finally we remember that if S = hs ,...,s i is a symmetric numerical semigroup then
1 ν
k[[S]] := k[[ts1,...,tsν]] is a one-dimensional Gorenstein local ring for any field k.
For each h ≥ 4, h ∈/ {14 + 22k, 35 + 46k | k ∈ N}, Construction 2.6 allows to produce
Gorenstein rings whose Hilbert function decreases at level h, while for h = 3 we can use
Example 2.10.ii. The next example is useful to complete the case h = 2.
Example 3.1. Consider the numerical semigroup
S = h68,72,78,82,107,111,117,121,158,162,166,168,170,172,174,176,178,180,182,184,
186,188,190,192,194,196,197,198,200,201,202,205,206,207,209,210,211,213,215,
217,219,221,223,225,227,229,231,233,235,237,239,241,245,249i.
It is almost symmetric, has type 53, and its Hilbert function is [1,54,52,50,54,64,68 →].
Consequently by Proposition 1.4, k[[S✶bK]] is a Gorenstein ring and has Hilbert function
[1,107,106,102,104,118,132,136→] decreasing at level 2 for any canonical ideal K and for
any odd b ∈ S.
The next lemma will allow us to show that in general, even if R is Gorenstein, there are
no bounds for H (h−1)−H (h).
R R
Lemma 3.2. Let R(0) m be a local ring. The following hold:
i. Consider the ring R(i+1) := R(i)(m(i)) , where m(i) is the maximal ideal of R(i) and
a(i),b(i)
a(i),b(i) are two elements of R(i). Then
H (h) = 2H (h) = ··· = 2iH (h) for any h > 0.
R(i) R(i−1) R(0)