# On Certain Gluing and indispensable resolution of semigroup rings

Sanjay Kumar Singh and Pranjal Srivastava  Discipline of Mathematics, IISER
Bhopal, Madhya Pradesh, India<EMAIL_ADDRESS>Discipline of
Mathematics, IISER Bhopal, Madhya Pradesh, India.
<EMAIL_ADDRESS>
###### Abstract.

In this paper, our aim is twofold: First, by using the technique of gluing
semigroups, we give infinitely many families of a projective closure with the
Cohen-Macaulay (Gorenstein) property. Also, we give an effective technique for
constructing large families of $1$-dimensional Gorenstein local rings
associated to monomial curves, which supports Rossi’s question, saying that
every Gorenstein local ring has a non-decreasing Hilbert function. In the
second part, we study strong indispensable minimal free resolutions of
semigroup rings, focusing on the operation of the join of affine semigroups,
which provide class of examples supporting Charalambous and Thoma’s question
on the class of lattice ideal which has a strong indispensable free
resolution..

###### Key words and phrases:

Affine Semigroups, Gröbner bases, Associated graded rings, Betti numbers,
Cohen-Macaulay.

###### 2020 Mathematics Subject Classification:

Primary 13H10, 13P10, 20M25.

The second author thanks IISER Bhopal for the Institute post-doc fellowship
IISERB/DoFA/PDF/2023/80.

##  1\. Introduction

Let $\mathbb{N}$ denote the set of non-negative integers and $\mathbb{K}$
denote a field. Let $e\geq 3$ and
$\mathbf{\underline{n}}=(n_{1},\ldots,n_{e})$ be a sequence of $e$ distinct
positive integers with $\gcd(\mathbf{\underline{n}})=1$. Let us assume that
the numbers $n_{1},\ldots,n_{e}$ generate the numerical semigroup
$\Gamma(n_{1},\ldots,n_{e})=\langle
n_{1},\ldots,n_{e}\rangle=\\{\sum_{j=1}^{e}z_{j}n_{j}\mid
z_{j}\in\mathbb{N}\\}$ minimally, that is, if $n_{i}=\sum_{j=1}^{e}z_{j}n_{j}$
for some non-negative integers $z_{j}$, then $z_{j}=0$ for all $j\neq i$ and
$z_{i}=1$. We often write $\Gamma$ in place of $\Gamma(n_{1},\ldots,n_{e})$,
when there is no confusion regarding the defining sequence
$n_{1},\ldots,n_{e}$. Let
$\eta:R=\mathbb{K}[x_{1},\,\ldots,\,x_{e}]\rightarrow\mathbb{K}[t]$ be the
mapping defined by $\eta(x_{i})=t^{n_{i}},\,1\leq i\leq e$. The ideal
$\ker(\eta)=\mathfrak{p}(n_{1},\ldots,n_{e})$ (or simply
$\mathfrak{p}(\Gamma)$) is called the defining ideal of
$\Gamma(n_{1},\ldots,n_{e})$ and it defines the affine monomial curve
$\\{(u^{n_{1}},\ldots,u^{n_{e}})\in\mathbb{A}^{e}_{\mathbb{K}}\mid
u\in\mathbb{K}\\}=:C(n_{1},\ldots,n_{e})$ (or simply $C(\Gamma)$). We write
$\mathbb{K}[x_{1},\ldots,x_{e}]/\mathfrak{p}(n_{1},\ldots,n_{e})=:\mathbb{K}[\Gamma(n_{1},\ldots,n_{e})]$
(or simply $\mathbb{K}[\Gamma]$), which is called the semigroup ring for the
semigroup $\Gamma(n_{1},\ldots,n_{e})$. It is known that
$\mathfrak{p}(\Gamma)$ is generated by the binomials $x^{a}-x^{b}$, where $a$
and $b$ are $e$-tuples of non-negative integers with
$\eta(x^{a})=\eta(x^{b})$.

The theory of numerical semigroups has been developed mainly in connection
with the study of curve singularities. Numerical semigroup rings are one-
dimensional domains and hence Cohen-Macaulay; these are the coordinate rings
of affine monomial curves. A lot of interesting studies have been undertaken
by several authors from the viewpoints of singularities, homology and also
from purely semigroup theoretic aspects. Our main object in this paper is to
explore the gluing of numerical semigroup rings and produce projective closure
of a monomial curve and tangent cone of the monomial curve with the Cohen-
Macaulay and Gorenstein properties. The concept of gluing was introduced by
Rosales in [12] and used by several authors to produce new examples of
numerical semigroup with nice property such as set-theoretic and ideal-
theoretic complete intersection, Cohen-Macaulay, Gorenstein etc. In this
paper, we particularly study the projective closure of gluing of numerical
semigroup rings and then the tangent cone of gluing of numerical semigroup
rings.

_Projective Closure of Numerical Semigroup rings._ Let $n_{e}>n_{i}$ for all
$i<e$, and $n_{0}=0$. We define the semigroup $\overline{\Gamma(n_{1},\ldots
n_{e})}=\langle\\{(n_{i},n_{e}-n_{i})\mid 0\leq i\leq
e\\}\rangle=\\{\sum_{i=0}^{e}z_{i}(n_{i},n_{e}-n_{i})\mid
z_{i}\in\mathbb{N}\\}$, often written as $\overline{\Gamma}$. Let
$\eta^{h}:S=\mathbb{K}[x_{0},\ldots,x_{e}]\longrightarrow\mathbb{K}[s,t]$ be
the $\mathbb{K}$-algebra map defined as
$\eta^{h}(x_{i})=t^{n_{i}}s^{n_{e}-n_{i}},0\leq i\leq e$ and
$\ker(\eta^{h})=\overline{\mathfrak{p}(n_{1},\ldots,n_{e})}$ (or simply
$\overline{\mathfrak{p}(\Gamma)}$). The homogenization of
$\mathfrak{p}(n_{1},\ldots,n_{e})$ with respect to the variable $x_{0}$ is
$\overline{\mathfrak{p}(n_{1},\ldots n_{e})}$, which defines
$\\{[(v^{n_{e}}:v^{n_{e}-n_{1}}u^{n_{1}}:\cdots:u^{n_{e}})]\in\mathbb{P}^{e}_{\mathbb{K}}\mid
u,v\in\mathbb{K}\\}=:\overline{C(n_{1},\ldots,n_{e})}$ (or simply
$\overline{C(\Gamma)}$), the projective closure of the affine monomial curve
$C(n_{1},\ldots,n_{e})$. The $\mathbb{K}$-algebra
$\mathbb{K}[x_{0},\ldots,x_{e}]/\overline{\mathfrak{p}(n_{1},\ldots,n_{e})}=:\mathbb{K}[\overline{\Gamma(n_{1},\ldots,n_{e})}]$
(or simply $\mathbb{K}[\overline{\Gamma}]$) denotes the coordinate ring. It
can be proved that $\overline{C(n_{1},\ldots,n_{e})}$ is a projective curve,
which is said to be arithmetically Cohen-Macaulay if the vanishing ideal
$\overline{\mathfrak{p}(n_{1},\ldots,n_{e})}$ is a Cohen-Macaulay ideal. The
projective closure of a numerical semigroup ring is defined by an affine
semigroup in $\mathbb{N}^{2}$. The Cohen-Macaulay property and many other
properties, such as Betti numbers, Goresntein, etc., of numerical semigroup
rings are not preserved under the operation of projective closure. It is not
easy to find examples where the projective closure retains the Cohen-Macaulay
property; see [11]. Gluing of affine semigroup can be an effective technique
for producing projective closure with the Cohen-Macaulay property, and from
gluing perspective, Saha et al. in [14] asked the following question.

_Question 1_([14, Question 1]). Suppose the projective closures of two affine
monomial curves are arithmetically Cohen-Macaulay (respectively Gorenstein).
Which conditions on gluing of numerical semigroups ensure that the
arithmetically Cohen-Macaulay (respectively Gorenstein) properties are
preserved for the projective closure?

An answer to the above Question would help us create a large family of affine
monomial curves with arithmetically Cohen-Macaulay (respectively Gorenstein)
projective closure. Saha et al. in [14] answer this question for _star gluing_
and _simple gluing_. In particular, they proved that the Cohen-Macaulay
(respectively Gorenstein) property preserves for the projective closure of
star gluing of numerical semigroup rings provided projective closures of
original numerical semigroups are Cohen-Macaulay (respectively Gorenstein).
Feza Arslan et al. in [1] introduced the concept of nice gluing to give
infinitely many families of $1$-dimensional local rings with Cohen Macaulay
tangent cone. We prove that, the projective closure of a nice gluing of
numerical semigroup rings is Cohen-Macaulay (respectively Gorenstein) under
some conditions.

_Associated graded ring of numerical semigroup rings._ For a numerical
semigroup $\Gamma$, $\mathbb{K}[\Gamma]$ is the coordinate ring of the affine
monomial curve $C(\Gamma)$. The tangent cone of $C(\Gamma)$ is the associated
graded ring
$\mathrm{gr}_{\mathfrak{m}}(\mathbb{K}[\Gamma])=\oplus_{i=0}^{\infty}\mathfrak{m}^{i}/\mathfrak{m}^{i+1}$,
with respect to the maximal ideal
$\mathfrak{m}=(t^{s}:s\in\Gamma\setminus\\{0\\})\mathbb{K}[\Gamma]$ at the
origin. The algebraic properties of
$\mathrm{gr}_{\mathfrak{m}}(\mathbb{K}[\Gamma])$ is a focal point of recent
research, with substantial progress happening in the directions of finding its
defining equations, Hilbert functions, Cohen-Macaulayness and Betti numbers.
In this section, we study the Hilbert functions of local rings associated to
affine monomial curves obtained by using the technique of gluing numerical
semigroups. Arslan et al. [1] give large families of local rings with non-
decreasing Hilbert functions using nice gluing, which are, in fact, special
types of gluings.

In this article we answer the following question for _star gluing_ : If
$C(\Gamma_{1})$ and $C(\Gamma_{2})$ have Cohen-Macaulay tangent cones, is the
tangent cone of the monomial curve $C(\Gamma)$ obtained by gluing these two
monomial curves necessarily Cohen-Macaulay? The following example shows that
the answer is no.

######  Example 1.1 ([1], Example 2.2).

Let $C(\Gamma_{1})$ and $C(\Gamma_{2})$ be the monomial curves
$C(\Gamma_{1})=C(5,12)$ and $C(\Gamma_{2})$ = $C(7,8)$. Obviously, they have
Cohen-Macaulay tangent cones. By a gluing of $\Gamma_{1}$ and $\Gamma_{2}$, we
obtain the monomial curve $C=C(21\times 5,21\times 12,17\times 7,17\times 8)$.
$C$ has a non Cohen-Macaulay tangent cone.

We demonstrate that _star gluing_ is also an effective technique for
constructing large families of $1$-dimensional Gorenstein local rings
associated to monomial curves, which support the question due to Rossi saying
that every Gorenstein local ring has non-decreasing Hilbert function [2].

_Extension of affine semigroups._ In the numerical case, the finiteness of
$\mathbb{N}\setminus\Gamma$ implies that there exists at least a positive
integer $a\in\mathbb{N}\setminus\Gamma$ such that
$a+\Gamma\setminus\\{0\\}\subseteq\Gamma\,(\text{provided that
}\Gamma\neq\mathbb{N})$. These integers are called pseudo-Frobenius numbers
and the biggest one is the so-called Frobenius number. But for affine
semigroups in $\mathbb{N}^{d}$, the existence of such elements is not always
guaranteed. The study of pseudo-Frobenius elements in affine semigroups over
$\mathbb{N}^{d}$ is done in [5], where the authors consider the complement of
the affine semigroup in its rational polyhedral cone. In [5], the authors
prove that an affine semigroup $\Gamma$ has pseudo-Frobenius elements if and
only if the length of the graded minimal free resolution of the corresponding
semigroup ring is maximal. Affine semigroups having pseudo-Frobenius elements
are called maximal projective dimension (MPD for short) semigroups. Projective
closure which are non Cohen-Macaulay are comes under the class of maximal
projective dimension semigroups. Let $\Gamma$ be an MPD-semigroup and let
$\mathrm{Cone}(\Gamma)$ denote the rational polyhedral cone of $\Gamma$. Set
$\mathcal{H}(\Gamma):=(\mathrm{Cone}(\Gamma)\setminus\Gamma)\cap\mathbb{N}^{d}$.
For a fixed term order $\prec$ on $\mathbb{N}^{d}$, Bhardwaj et al. in [3]
introduced the $\prec$-symmetric semigroup, provided
$\mathrm{F}(\Gamma)_{\prec}=\rm{max}_{\prec}\mathcal{H}(\Gamma)$ exists.

One of the widely studied class of numerical semigroups is symmetric
semigroups. The motivation to study these semigroups comes from the work E.
Kunz, who proved that a one-dimensional analytically irreducible Noetherian
local ring is Gorenstein if and only if its value semigroup is symmetric (See
[10]). In other words, a numerical semigroup is symmetric if and only if the
associated semigroup ring is Gorenstein. In [18], Watanabe gives a special
technique of producing symmetric numerical semigroup. Later, Stamate in [17]
called this technique to _simple gluing_ and used this to produce Cohen-
Macaulay tangent cone of numerical semigroup rings. In [14, Theorem 4.11] Saha
et al. proved that the projective closure of numerical semigroup ring obtained
from the simple gluing is Cohen-Macaulay (respectively Gorenstein) provided
projective closure of original numerical semigroup ring is Cohen-Macaulay
(respectively Gorenstein). In this paper, we study the simple gluing of MPD
semigroups and prove that simple gluing preserves the $\prec$-symmetric MPD
semigroups provided original MPD semigroup should be $\prec$-symmetric.

_Strong Indispensable Free Resolution._ Indispensable binomials are those that
appear in every minimal binomial generating set up to a constant multiple.
Strongly indispensable binomials are those appearing in every minimal
generating set, up to a constant multiple. In the same vein, as introduced for
the first time by Charalambous and Thoma in [4], strongly indispensable higher
syzygies are those appearing in every minimal free resolution. Semigroups all
of whose higher syzygy modules are generated minimally by strongly
indispensable elements are said to have a strongly indispensable minimal free
resolution, SIFRE for short. The statistical models having SIFREs or
equivalently having uniquely generated higher syzygy modules are a subclass of
those having a unique Markov basis and therefore have a better potential
statistical behaviour. Motivated by the third question stated by Charalambous
and Thoma at the end of [4], our main aim in this article is to identify some
semigroups having SIFREs. It is difficult to construct examples having SIFREs.
Şahin and Stella in [15] produces SIFR using the gluing of affine semigroups.
Saha et al. in [13], introduced the notion of _Join of affine semigroups_
which has some nice property. In Section 3, we prove that $\Gamma_{1}$ and
$\Gamma_{2}$ have SIFRs if and only if their join is also have SIFR (See
Theorem 3.4 )

##  2\. Gluing of monomial curves

We start this section with the example.

######  Example 2.1.

We consider monomial curves $\overline{C(3,5)}$ and $\overline{C(7,12)}$. They
are Cohen-Macaulay (Gorenstein), but their gluing with respect to the elements
$p=8,q=19$ is $\overline{C(57,95,56,96)}$, which is not even arithmetically
Cohen-Macaulay (verified with the help of MACAULAY 2 [6]).

This motivates authors in [14] to ask Question 1, described in the
Introduction. We summarise two results which will be useful for our purpose.

######  Lemma 2.2 ([9], Lemma 2.1).

Let $I$ be an ideal in $R=\mathbb{K}[x_{1},\ldots,x_{e}]$ and $I^{h}\subset
R[x_{0}]$ its homogenization, with respect to the variable $x_{0}$. Let $<$ be
any reverse lexicographic monomial order on $R$ and $<_{0}$ the reverse
lexicographic monomial order on $R[x_{0}]$, extended from $R$, such that
$x_{i}>x_{0}$. If $\\{f_{1},\ldots,f_{d}\\}$ is the reduced Gröbner basis for
$I$, with respect to $<$, then $\\{f_{1}^{h},\ldots,f_{d}^{h}\\}$ is the
reduced Gröbner basis for $I^{h}$, with respect to $<_{0}$, and
$\mathrm{in}_{<_{0}}(I^{h})=(\mathrm{in}_{<}(I))R[x_{0}]$.

The other one that is useful for checking the arithmetically Cohen-Macaulay
property of the projective closure. It states the following:

######  Theorem 2.3 ([9], Theorem 2.2).

Let $\mathbf{n}=(n_{1},\ldots,n_{e})$ be a sequence of positive integers with
$n_{e}>n_{i}$ for all $i<n$. Let $<$ be any reverse lexicographic order on
$R=\mathbb{K}[x_{1},\ldots,x_{e}]$, such that $x_{i}>x_{e}$, for all $1\leq
i<e$. Let $<_{0}$ be the induced reverse lexicographic order on $R[x_{0}]$,
where $x_{e}>x_{0}$. Then the following conditions are equivalent:

  1. (i)

The projective monomial curve $\overline{C(n_{1},\ldots,n_{e})}$ is
arithmetically Cohen-Macaulay.

  2. (ii)

$\mathrm{in}_{<_{0}}((\mathfrak{p}(n_{1},\ldots,n_{e}))^{h})$ (homogenization
w.r.t. $x_{0}$) is a Cohen-Macaulay ideal.

  3. (iii)

$\mathrm{in}_{<}(\mathfrak{p}(n_{1},\ldots,n_{e}))$ is a Cohen-Macaulay ideal.

  4. (iv)

$x_{e}$ does not divide any element of
$G(\mathrm{in}_{<}(\mathfrak{p}(n_{1},\ldots,n_{e})))$.

######  Definition 2.4 ([12]).

Let $\Gamma_{1}=\Gamma(m_{1},\dots,m_{l})$ and
$\Gamma_{2}=\Gamma(n_{1},\dots,n_{k})$ be two numerical semigroups, with
$m_{1}<\cdots<m_{l}$ and $n_{1}<\cdots<n_{k}$. Let
$p=b_{1}m_{1}+\cdots+b_{l}m_{l}\in\Gamma_{1}$ and
$q=a_{1}n_{1}+\cdots+a_{k}n_{k}\in\Gamma_{2}$ be two positive integers
satisfying $\gcd(p,q)=1$, with $p\notin\\{m_{1},\dots,m_{l}\\}$,
$q\notin\\{n_{1},\ldots,n_{k}\\}$ and
$\\{qm_{1},\ldots,qm_{l}\\}\cap\\{pn_{1},\ldots,pn_{k}\\}=\emptyset$. The
numerical semigroup $\Gamma_{1}\\#_{p,q}\Gamma_{2}=\langle
qm_{1},\ldots,qm_{l},pn_{1},\ldots,pn_{k}\rangle$ is called a gluing of the
semigroups $\Gamma_{1}$ and $\Gamma_{2}$ with respect to $p$ and $q$.

######  Remark 2.5 ([7], Lemma 2.2).

If $\Gamma$ be obtained by gluing $\Gamma_{1}=\Gamma(m_{1},\dots,m_{l})$ and
$\Gamma_{2}=\Gamma(n_{1},\dots,n_{k})$ with respect to
$p=\sum_{i=1}^{l}b_{i}m_{i}$ and $q=\sum_{i=1}^{k}a_{i}n_{i}$, and if the
defining ideals
$\mathfrak{p}(\Gamma_{1})\subset\mathbb{K}[x_{1},\ldots,x_{l}]$ and
$\mathfrak{p}(\Gamma_{2})\subset\mathbb{K}[y_{1},\ldots,y_{k}]$ are generated
by the sets $G_{1}=\\{f_{1},\ldots,f_{d}\\}$ and
$G_{2}=\\{g_{1},\ldots,g_{r}\\}$ respectively, then the defining ideal
$\mathfrak{p}(\Gamma)\subset
R=\mathbb{K}[x_{1},\ldots,x_{l},y_{1},\ldots,y_{k}]$ is generated by the set
$G=G_{1}\cup G_{2}\cup\\{\rho\\}$, where $\rho=x_{1}^{b_{1}}\dots
x_{l}^{b_{l}}-y_{1}^{a_{1}}\dots y_{k}^{a_{k}}$.

######  Definition 2.6 ([1], Definiton 2.3).

The numerical semigroup $\Gamma_{1}\\#_{p,q}\Gamma_{2}$, obtained by gluing of
$\Gamma_{1}=\Gamma(m_{1},\dots,m_{l})$ and
$\Gamma_{2}=\Gamma(n_{1},\dots,n_{k})$, with respect to the positive integers
$p$ and $q$, is called a nice gluing if
$p=b_{1}m_{1}+\dots+b_{l}m_{l}\in\Gamma_{1}$ and $q=a_{1}n_{1}\in\Gamma_{2}$,
with $b_{1}+b_{2}+\dots+b_{l}\geq a_{1}$. If we take
$q=a_{1}n_{1}+\dots+a_{k}n_{k}$ with
$b_{1}+b_{2}+\dots+b_{l}>a_{1}+\dots+a_{k}$. Then the numerical semigroup
minimally generated by $\langle
qm_{1},\dots,qm_{l},pn_{1},\dots,pn_{k}\rangle$ is called generalized nice
gluing of $\Gamma_{1}$ and $\Gamma_{2}$ with respect to $p,q$. We use the
notation $\Gamma_{1}\\#^{n}_{p,q}\Gamma_{2}$ to denote the generalized nice
gluing of $\Gamma_{1}$ and $\Gamma_{2}$, with respect to the positive integers
$p$ and $q$.

######  Remark 2.7.

For the Cohen-Macaulayness of projective closure of the numerical semigroup
$\Gamma=\Gamma_{1}\\#^{n}_{p,q}\Gamma_{2}$, obtained by nice gluing of
$\Gamma_{1}$ and $\Gamma_{2}$ with respect to $p,q$, it is important to find
the largest integer from the generator of the numerical semigroup
$\Gamma_{1}\\#^{n}_{p,q}\Gamma_{2}$. Choices for the largest integer $\Gamma$
is either $qm_{l}$ or $pn_{k}$. If the largest integer is $qm_{l}$ then
$\overline{C(\Gamma)}$ need not be Cohen-Macaulay (See Example 2.8).

######  Example 2.8.

Let $\Gamma_{1}=\langle 5,7,11\rangle$ and $\Gamma_{2}=\langle 25,28\rangle$.
Let $p=2\cdot 5+7=17,\,q=2\cdot 25=50$. Then
$\Gamma=\Gamma_{1}\\#^{n}_{17,50}\Gamma_{2}=\langle
250,350,550,425,476\rangle$ and $\overline{C(\Gamma)}$ is not arithmetically
Cohen-Macaulay (verified with the help of MACAULAY 2 [6]).

Condition A: Let $\mathrm{lcm}(b_{i},\alpha_{i})\neq b_{i}$ for all $1\leq
i\leq l$, where $x_{1}^{\alpha_{1}}\dots x_{l}^{\alpha_{l}}=\mathrm{LM}(f)$
and $f$ is an arbitrary element of a reduced Gröbner basis of
$\mathfrak{p}(\Gamma_{1})$ and $b_{i}$ is according to Definition 2.6. We
deonte $\mathrm{lcm}(m,n)$ by $[m,n]$. This condition has been defined in such
a way that $S$-polynomial of $\rho$ and $f$ reduces to zero after a
appropriate multidivison by $G$ (defined in Lemma 2.9).

######  Lemma 2.9.

Let $\Gamma_{1}=\Gamma(m_{1},\dots,m_{l})$ and
$\Gamma_{2}=\Gamma(n_{1},\dots,n_{k})$ be two numerical semigroups with
$m_{1}<\cdots<m_{l}$ and $n_{1}<\cdots<n_{k}$. Let
$p=b_{1}m_{1}+\dots+p_{l}m_{l}\in\Gamma_{1}$ and
$q=a_{1}n_{1}+\cdots+a_{k}n_{k}\in\Gamma_{2}$ be two positive integers
satisfying $\gcd(p,q)=1$, such that $\Gamma=\Gamma_{1}\\#^{n}_{p,q}\Gamma_{2}$
is the generalized nice gluing of $\Gamma_{1}$ and $\Gamma_{2}$ with
$b_{1},\dots,b_{l}$ satisfy Condition $A$. Suppose that $G_{1}$ and $G_{2}$
are Gröbner bases of $\mathfrak{p}(\Gamma_{1})$ and
$\mathfrak{p}(\Gamma_{2})$, with respect to the degree reverse lexicographic
ordering induced by $x_{1}>\dots>x_{l}$ and $y_{1}>\dots>y_{k}$ respectively.
Then

$G_{1}^{h}\cup G_{2}^{h}\cup\\{x^{b_{1}}_{1}\dots
x_{l}^{b_{l}}-x_{0}^{b_{1}\dots+b_{l}-(a_{1}+\cdots+a_{k})}y^{a_{1}}_{1}y^{a_{2}}_{2}\dots
y^{a_{k}}_{k}\\}$

is a Gröbner bases of $\overline{\mathfrak{p}(\Gamma)}$ with respect to the
degree reverse lexicographic ordering induced by
$x_{1}>\dots>x_{l}>y_{1}>\dots>y_{k}>x_{0}$ or
$y_{1}>\dots>y_{k}>x_{1}>\dots>x_{l}>x_{0}$, where $G_{1}^{h}$, $G_{2}^{h}$
denotes homogenization of $G_{1}$ and $G_{2}$ with respect to $x_{0}$.

###### Proof.

Let $G_{1}=\\{f_{1},\ldots,f_{d}\\}$ be a reduced Gröbner basis of the ideal
$\mathfrak{p}(\Gamma_{1})\subset\mathbb{K}[x_{1},\ldots,x_{l}]$, with respect
to the degree reverse lexicographical ordering induced by $x_{1}>\cdots>x_{l}$
and $G_{2}=\\{g_{1},\ldots,g_{r}\\}$ be a reduced Gröbner basis of the ideal
$\mathfrak{p}(\Gamma_{2})\subset\mathbb{K}[y_{1},\ldots,y_{k}]$, with respect
to the degree reverse lexicographical ordering induced by
$y_{1}>\ldots>y_{k}$. We show that
$G=\\{f_{1},\dots,f_{d},g_{1},\dots,g_{r},\rho=x_{1}^{b_{1}}\dots
x_{l}^{b_{l}}-y^{a_{1}}_{1}y^{a_{2}}_{2}\dots y^{a_{k}}_{k}\\}$ is a Gröbner
basis of $\mathfrak{p}(\Gamma)$, with respect to the degree reverse
lexicographical ordering induced by $x_{1}>\cdots>x_{l}>y_{1}>\cdots>y_{k}$.
By Lemma 2.2 in [7], the defining ideal $\mathfrak{p}(\Gamma)$ of the affine
curve $C(\Gamma)$, obtained by gluing, is generated by the set $G$. Let
$S(f,g)$ denote the $S$-polynomial of polynomials $f$ and $g$ in
$\mathbb{K}[x_{1},\ldots,x_{l},y_{1},\ldots,y_{k}]$. With respect to the said
monomial order, the leading monomial of any element of $G_{1}$ is the product
of monomials of the form $x_{i}^{\alpha_{i}}$ only, and $G_{2}$ is the product
of monomials of the form $y_{j}^{\beta_{j}}$ only, for some non-negative
integers $\alpha_{i}$, $\beta_{j}$ and $1\leq i\leq l$, $1\leq j\leq k$.
Therefore $S(f_{i},g_{j})\rightarrow_{G}0$. It is clear from the condition of
generalized nice gluing that $\mathrm{LM}(x_{1}^{b_{1}}\dots
x_{l}^{b_{l}}-y^{a_{1}}_{1}y^{a_{2}}_{2}\dots
y^{a_{k}}_{k})=x_{1}^{b_{1}}\dots x_{l}^{b_{l}}$. Let
$f_{i}=x_{1}^{\alpha_{1}}\dots x_{l}^{\alpha_{l}}-x_{1}^{\beta_{1}}\dots
x_{l}^{\beta_{l}}$ be an arbitrary element of $G_{1}$ with
$\mathrm{LM}(f_{i})=x_{1}^{\alpha_{1}}\dots x_{l}^{\alpha_{l}}$. Now
$S(\rho,f_{i})=-x_{1}^{[b_{1},\alpha_{1}]-b_{1}}\dots
x_{l}^{[b_{l},\alpha_{l}]-b_{l}}(y_{1}^{a_{1}}\dots
y_{k}^{a_{k}})+x_{1}^{[b_{1},\alpha_{1}]-\alpha_{1}}\dots
x_{l}^{[b_{l},\alpha_{l}]-\alpha_{l}}x_{1}^{\beta_{1}}\dots
x_{l}^{\beta_{l}}$. Since
$\sum_{s=1}^{l}([b_{s},\alpha_{s}]-b_{s})+a_{1}+\dots+a_{k}\geq\sum_{s=1}^{l}([b_{s},\alpha_{s}]-\alpha_{s})+\beta_{1}+\dots+\beta_{k}$,
we have $\mathrm{LM}(S(\rho,f_{i}))=x_{1}^{[b_{1},\alpha_{1}]-b_{1}}\dots
x_{l}^{[b_{l},\alpha_{l}]-b_{l}}(y_{1}^{a_{1}}\dots y_{k}^{a_{k}})$ and using
multi variable division algorithm $S(\rho,f_{i})$ reduced to zero after
division by $G_{1}\cup G_{2}\cup\\{\rho\\}$ due to condition A. Since for any
$g_{i}\in G_{2}$, $\mathrm{gcd}(\mathrm{LM}(\rho)=x_{1}^{b_{1}}\dots
x_{l}^{b_{l}},\mathrm{LM}(g_{i})=\text{some monomial in }y_{j}^{\prime}s)=1$,
we have $S(\rho,g_{i})$ reduces to zero after division by $G_{1}\cup
G_{2}\cup\\{\rho\\}$. By the Buchberger’s criterion, $G=G_{1}\cup
G_{2}\cup\\{\rho\\}$ is a Gröbner basis of $\mathfrak{p}(\Gamma)$ and by Lemma
2.2 , $G^{h}=G_{1}^{h}\cup
G_{2}^{h}\cup\\{x^{b_{l}}_{l}-x_{0}^{b_{l}-(a_{1}+\cdots+a_{k})}y^{a_{1}}_{1}y^{a_{2}}_{2}\dots
y^{a_{k}}_{k}\\}$ is a Gröbner basis of $\overline{\mathfrak{p}(\Gamma)}$. ∎

######  Theorem 2.10.

Assuming the hypothesis of Theorem 2.9, if the associated projective closure
$\overline{C(\Gamma_{1})}$ and $\overline{C(\Gamma_{2})}$ are arithmetically
Cohen-Macaulay, then

  * •

If $pn_{k}$ is the largest integer among the generators of
$\Gamma=\Gamma_{1}\\#^{n}_{p,q}\Gamma_{2}$ then the projective closure
$\overline{C(\Gamma)}$ associated to $\Gamma$ is arithmetically Cohen-
Macaulay.

  * •

If $qm_{l}$ is the largest integer among the generators of
$\Gamma=\Gamma_{1}\\#^{n}_{p,q}\Gamma_{2}$ then the projective closure
$\overline{C(\Gamma)}$ associated to $\Gamma$ is never arithmetically Cohen-
Macaulay.

###### Proof.

  1. (1)

Assume that $pn_{k}$ is the largest integer among the generators of
$\Gamma=\Gamma_{1}\\#^{n}_{p,q}\Gamma_{2}$ then $y_{k}$ will be the deciding
variable for the Cohen-Macaulayness of $\overline{C(\Gamma)}$. From Lemma 2.9,
$G^{h}=G_{1}^{h}\cup
G_{2}^{h}\cup\\{x^{b_{l}}_{l}-x_{0}^{b_{l}-(a_{1}+\cdots+a_{k})}y^{a_{1}}_{1}y^{a_{2}}_{2}\dots
y^{a_{k}}_{k}\\}$ is a Gröbner bases of $\overline{\mathfrak{p}(\Gamma)}$,
with respect to the degree reverse lexicographic ordering
$x_{1}>\dots>x_{l}>y_{1}>\dots>y_{k}>x_{0}$, where $G_{1}$ and $G_{2}$ are
Gröbner bases of $\mathfrak{p}(\Gamma_{1})$ and $\mathfrak{p}(\Gamma_{2})$,
with respect to the degree reverse lexicographic ordering $x_{1}>\dots>x_{l}$
and $y_{1}>\dots>y_{k}$ respectively. Since $\overline{C(\Gamma_{2})}$ is
arithmetically Cohen-Macaulay, by Theorem 2.3, $y_{k}$ does not divide the
leading monomial of any element in $G_{1}^{h}$ and $G_{2}^{h}$, and also
$\mathrm{LM}(x^{b_{1}}_{1}\dots
x_{l}^{b_{l}}-x_{0}^{b_{l}-(a_{1}+\cdots+a_{k})}y^{a_{1}}_{1}y^{a_{2}}_{2}\dots
y^{a_{k}}_{k})=x^{b_{1}}_{1}\dots x_{l}^{b_{l}}$. Thus, $y_{k}$ does not
divide the leading monomial of any element in $G^{h}$, which is a Gröbner
basis with respect to to the degree reverse lexicographic ordering induced by
$x_{1}>...>x_{l}>y_{1}>...>y_{k}>x_{0}$. Thus, by Theorem 2.3,
$\overline{C(\Gamma)}$ is arithmetically Cohen-Macaulay.

  2. (2)

If $qm_{l}$ is the largest integer among the generators of
$\Gamma=\Gamma_{1}\\#^{n}_{p,q}\Gamma_{2}$ then $x_{l}$ is the deciding
variable and $x_{l}$ divides $\mathrm{LM}(\rho)$, therefore by Theorem 2.3,
$\overline{C(\Gamma)}$ is not arithmetically Cohen-Macaulay.

∎

######  Corollary 2.11.

Under the hypothesis of generalized nice gluing of $\Gamma_{1}$ and
$\Gamma_{2}$ with respect to $p,q$, let
$\Gamma=\Gamma_{1}\\#^{n}_{p,q}\Gamma_{2}$. If $\overline{C(\Gamma_{1})}$ and
$\overline{C(\Gamma_{2})}$ are Gorenstein then $\overline{C(\Gamma)}$ is
Gorenstein.

###### Proof.

Proof is directly follows from Theorem 2.9, [7, Proposition 3.3] and [14,
Theorem 3.13]. ∎

######  Example 2.12.

Let $\Gamma_{1}=\langle 3,5,7\rangle$ and $\Gamma_{2}=\langle 9,11\rangle$
with $p=3\cdot 3+5=14$, $q=9\cdot 2+11=29$. Then
$\Gamma=\Gamma_{1}\\#^{n}_{p,q}\Gamma_{2}=\langle 87,145,203,126,154\rangle$.
Note that $x_{3}$ is the variable corresponding to the largest integer of the
minimal generator of $\Gamma$. Thus $\overline{C(\Gamma)}$ is not
arithmetically Cohen-Macaulay (verified with the help of MACAULAY 2 [6]). But
take $p=2\cdot 3+3\cdot 5=21,q=29$, then
$\Gamma=\Gamma_{1}\\#^{n}_{p,q}\Gamma_{2}=\langle 87,145,203,189,231\rangle$.
Here $y_{2}$ is the variable corresponding to the largest integer of the
minimal generator of $\Gamma$. Thus $\overline{C(\Gamma)}$ is arithmetically
Cohen-Macaulay

###  2.1. Gluing of tangent Cone

We recall that an affine monomial curve $C(\Gamma)$ is a curve with generic
zero $(t^{n_{1}},\dots,t^{n_{e}})$ in the affine $e$-space $\mathbb{A}^{e}$
over an algebraically closed field $\mathbb{K}$, where $\Gamma=\langle
n_{1},\dots,n_{e}\rangle$. The semigroup ring associated to the monomial curve
$C(\Gamma)$ is $\mathbb{K}[[t^{n_{1}},\dots,t^{n_{e}}]])$, and the Hilbert
function of this local ring is the Hilbert function of its associated graded
ring $\mathrm{gr}_{\mathfrak{m}}(\mathbb{K}[[t^{n_{1}},\dots,t^{n_{e}}]])$,
which is isomorphic to the ring
$\mathbb{K}[x_{1},\dots,x_{e}]/\mathfrak{p}^{\star}(\Gamma)$, where
$\mathfrak{p}(\Gamma)$ is the defining ideal of $C(\Gamma)$ and
$\mathfrak{p}^{\star}(\Gamma)$ is the ideal generated by the polynomials
$f^{\star}$, with $f$ in $\mathfrak{p}(\Gamma)$ and $f^{\star}$ being the
homogeneous summand of $f$ of least degree. In other words,
$\mathfrak{p}^{\star}(\Gamma)$ is the defining ideal of the tangent cone of
$C(\Gamma)$ at $0$.

Let us recall the definition of _star gluing_ of numerical semigroups.

######  Definition 2.13.

The numerical semigroup $\Gamma_{1}\\#_{p,q}\Gamma_{2}$, obtained by gluing of
$\Gamma_{1}=\Gamma(m_{1},\dots,m_{l})$ and
$\Gamma_{2}=\Gamma(n_{1},\dots,n_{k})$, with respect to the positive integers
$p$ and $q$, is called a generalized star gluing if
$p=b_{1}m_{1}+\dots+b_{l}m_{l}\in\Gamma_{1}$ and
$q=a_{1}n_{1}+a_{2}n_{2}+\dots+a_{k}n_{k}\in\Gamma_{2}$, with
$a_{1}+a_{2}+\dots+a_{k}<b_{l}+\dots+b_{l}.$ We use the notation
$\Gamma_{1}\star_{p,q}\Gamma_{2}$ to denote the generalized star gluing of
$\Gamma_{1}$ and $\Gamma_{2}$, with respect to the positive integers $p$ and
$q$.

To prove the Cohen-Macaulayness of tangent cone of monomial curve obtained by
star gluing, we first write the criterion for checking the Cohen-Macaulayness
of the tangent cone of a monomial curve

######  Lemma 2.14.

Let $\Gamma=\langle n_{1},\dots,n_{k}\rangle$ be a numerical semigroup
minimally generated by $n_{1}<\dots<n_{k}$, let $C=C(\Gamma)$ be the
associated monomial curve and let $G=\\{f_{1},\dots,f_{s}\\}$ be a minimal
Gröbner basis of the ideal
$\mathfrak{p}(\Gamma)\subset\mathbb{K}[x_{1},\dots,x_{k}]$ with respect to the
negative degree lexicographic ordering that makes $x_{1}$ the lowest variable.
$C$ has Cohen-Macaulay tangent cone at the origin if and only $x_{1}$ does not
divide $\mathrm{LM}(f_{i})$ for $1\leq i\leq s$.

###### Proof.

See [1, Lemma 2.7]. ∎

######  Remark 2.15.

For the Cohen-Macaulayness of tangent cone of affine monomial curve, it is
important to find the smallest integer among the generator of associated
numerical semigroup. In case of numerical semigroup
$\Gamma_{1}\star_{p,q}\Gamma_{2}$, the possibility of smallest generator is
either $qm_{1}$ or $pn_{1}$.

Condition B: Let $\mathrm{lcm}(a_{i},\alpha_{i})\neq a_{i}$ for all $1\leq
i\leq k$, where $y_{1}^{\alpha_{1}}\dots y_{k}^{\alpha_{k}}=\mathrm{LM}(g)$
and $g$ is an arbitrary element of a reduced Gröbner basis of
$\mathfrak{p}(\Gamma_{2})$ and $a_{i}$ is according to Definition 2.13 . This
condition has been defined in such a way that $S$-polynomial of $\rho$ and $g$
reduces to zero after a appropriate multidivison by $G$ (defined in Theorem
2.16).

######  Theorem 2.16.

Let $\Gamma_{1}=\Gamma(m_{1},\dots,m_{l})$ and
$\Gamma_{2}=\Gamma(n_{1},\dots,n_{k})$ be two numerical semigroups with
$m_{1}<\cdots<m_{l}$ and $n_{1}<\cdots<n_{k}$. Let
$p=b_{1}m_{1}+\dots+p_{l}m_{l}\in\Gamma_{1}$ and
$q=a_{1}n_{1}+\cdots+a_{k}n_{k}\in\Gamma_{2}$ be two positive integers
satisfying $\gcd(p,q)=1$, such that $\Gamma_{1}\star_{p,q}\Gamma_{2}$ is the
generalized star gluing of $\Gamma_{1}$ and $\Gamma_{2}$ with
$b_{1},\dots,b_{l}$ satisfy condition $B$. If the associated monomial curves
$C(\Gamma_{1})$ and $C(\Gamma_{2})$ have Cohen-Macaulay tangent cone at the
origin.

  * •

If $qm_{1}$ is the smallest integer among the generators of
$\Gamma=\Gamma_{1}\star_{p,q}\Gamma_{2}$ then $C=C(\Gamma)$ also has a Cohen-
Macaulay tangent cone at the origin.

  * •

If $pn_{1}$ is the smallest integer among the generators of
$\Gamma=\Gamma_{1}\star_{p,q}\Gamma_{2}$ then $C=C(\Gamma)$ has never a Cohen-
Macaulay tangent cone at the origin.

###### Proof.

Let $G_{1}=\\{f_{1},\ldots,f_{d}\\}$ be a reduced Gröbner basis of the ideal
$\mathfrak{p}(\Gamma_{1})\subset\mathbb{K}[x_{1},\ldots,x_{l}]$, with respect
to the negative degree reverse lexicographical ordering induced by
$x_{l}>\cdots>x_{1}$ and $G_{2}=\\{g_{1},\ldots,g_{r}\\}$ be a reduced Gröbner
basis of the ideal
$\mathfrak{p}(\Gamma_{2})\subset\mathbb{K}[y_{1},\ldots,y_{k}]$, with respect
to the negative degree reverse lexicographical ordering induced by
$y_{l}>\ldots>y_{1}$. We show that
$G=\\{f_{1},\dots,f_{d},g_{1},\dots,g_{r},\rho=x_{1}^{b_{1}}\dots
x_{l}^{b_{l}}-y^{a_{1}}_{1}y^{a_{2}}_{2}\dots y^{a_{k}}_{k}\\}$ is a Gröbner
basis of $\mathfrak{p}(\Gamma)$, with respect to the negative degree reverse
lexicographical ordering induced by $x_{l}>\cdots>x_{1}>y_{k}>\cdots>y_{1}$.
Let $S(f,g)$ denote the $S$-polynomial of polynomials $f$ and $g$ in
$\mathbb{K}[x_{1},\ldots,x_{l},y_{1},\ldots,y_{k}]$. With respect to the said
monomial order, the leading monomial of any element of $G_{1}$ is the product
of monomials of the form $x_{i}^{\alpha_{i}}$ only, and $G_{2}$ is the product
of monomials of the form $y_{j}^{\beta_{j}}$ only, for some non-negative
integers $\alpha_{i}$, $\beta_{j}$ and $1\leq i\leq l$, $1\leq j\leq k$.
Therefore $S(f_{i},g_{j})\rightarrow_{G}0$. It is clear from the condition of
generalized star gluing and definition of negative degree reverse
lexicographic ordering that $\mathrm{LM}(x_{1}^{b_{1}}\dots
x_{l}^{b_{l}}-y^{a_{1}}_{1}y^{a_{2}}_{2}\dots
y^{a_{k}}_{k})=y_{1}^{a_{1}}\dots y_{k}^{a_{k}}$. Let
$g_{i}=y_{1}^{\alpha_{1}}\dots x_{l}^{\alpha_{l}}-y_{1}^{\beta_{1}}\dots
y_{l}^{\beta_{l}}$ be an arbitrary element of $G_{2}$ with
$\mathrm{LM}(g_{i})=y_{1}^{\alpha_{1}}\dots y_{l}^{\alpha_{l}}$. Now the
spolynomial of $\rho$ and $g_{i}$ is given by
$S(\rho,g_{i})=-x_{1}^{b_{1}}x_{2}^{b_{2}}\dots
x_{l}^{b_{l}}y_{1}^{[\alpha_{1},a_{1}]-a_{1}}\dots
y_{k}^{[\alpha_{k},a_{k}]-a_{k}}-y_{1}^{[\alpha_{1},a_{1}]-\alpha_{1}+\beta_{1}}\dots
y^{[\alpha_{k},a_{k}]-\alpha_{k}+\beta_{k}}$. Since
$\sum_{s=1}^{k}([a_{s},\alpha_{s}]-a_{s})+b_{1}+\dots+b_{l}>\sum_{s=1}^{k}([a_{s},\alpha_{s}]-\alpha_{s})+\beta_{1}+\dots+\beta_{k}$,
we have $\mathrm{LM}(S(\rho,f_{i}))=-x_{1}^{b_{1}}x_{2}^{b_{2}}\dots
x_{l}^{b_{l}}y_{1}^{[\alpha_{1},a_{1}]-a_{1}}y_{k}^{[\alpha_{k},a_{k}]-a_{k}}$,
and using multi variable division algorithm $S(\rho,f_{i})$ reduced to zero
after division by $G_{1}\cup G_{2}\cup\\{\rho\\}$ due to condition B. Since
for any $g_{i}\in G_{2}$, $\mathrm{gcd}(\mathrm{LM}(\rho)=x_{1}^{b_{1}}\dots
x_{l}^{b_{l}},\mathrm{LM}(g_{i})=\text{some monomial in }y_{j}^{\prime}s)=1$,
we have $S(\rho,g_{i})$ reduces to zero after division by $G_{1}\cup
G_{2}\cup\\{\rho\\}$. By the Buchberger’s criterion, $G=G_{1}\cup
G_{2}\cup\\{\rho\\}$ is a minimal Gröbner basis of $\mathfrak{p}(\Gamma)$.

If $qm_{1}$ is the smallest integer among the generators of $\Gamma$ and it is
straightforward to check that $y_{1}$ do not divide the leading monomial of
any elements of $G$ as $\Gamma_{2}$ has Cohen-Macaulay tangent cone, hence by
Lemma 2.14, $C(\Gamma)$ has Cohen-Macaulay tangent cone at origin. If $pn_{1}$
is the smallest integer then we can see that $x_{1}$ divide
$\mathrm{LM}(\rho)$, therefore again by Lemma 2.14, $C=C(\Gamma)$ has never a
Cohen-Macaulay tangent cone at the origin. ∎

######  Remark 2.17.

If we take two Gorenstein monomial curves with Cohen-Macaulay tangent cone
then the monomial curve obtained by star gluing is also Gorenstein (See [7]),
and has a Cohen-Macaulay tangent cone. Since monomial curves with Cohen-
Macaulay tangent cone has a non-decreasing Hilbert function, hence we can
produce infinitely many $1$-dimensional Gorenstein monomial curve with non-
decreasing Hilbert function, which gives affirmative answer to Rossi’s
question.

######  Example 2.18.

Let $\Gamma_{1}=\langle 3,5,7\rangle$ and $\Gamma_{2}=\langle 9,11\rangle$
with $p=4.7=28$, $q=2.9+11=29$. Then
$\Gamma=\Gamma_{1}\star_{p,q}\Gamma_{2}=\langle 87,145,203,189,231\rangle$.
Then
$G=\\{x_{2}^{2}-x_{1}x_{3},x_{3}^{2}-x_{1}^{3}x_{2},x_{2}x_{3}-x_{1}^{4},y_{2}^{9}-y_{1}^{11},y_{1}^{2}y_{2}-x_{3}^{4}\\}$
is a reduced Gröbner basis with respect to the negative degree reverse
lexicographic ordering induced by $y_{2}>y_{1}>x_{3}>x_{2}>x_{1}$ of
$\mathfrak{p}(\Gamma)$. Note that $x_{1}$ is the variable corresponding to the
smallest integer of the minimal generator of $\Gamma$ and $x_{1}$ does not
divide leading monomial of any element of $G$. Thus the tangent cone of
$C(\Gamma)$ is Cohen-Macaulay.

###  2.2. Extension of MPD semigroup

Let
$\mathcal{A}=\\{\mathbf{a}_{1},\dots,\mathbf{a}_{n}\\}\subset\mathbb{N}^{d}$
and let $\Gamma$ is a submonoid of $\mathbb{N}^{d}$ generated by
$\mathcal{A}$. An affine semigroup $\Gamma$ is said to be maximal projective
dimension semigroup (MPD-semigorup) if
$\mathrm{pd}_{R}(\mathbb{K}[\Gamma])=n-1$. Equivalently,
$\mathrm{depth}_{R}(\mathbb{K}[\Gamma])=1$. Consider the cone of $\Gamma$ in
$\mathbb{N}^{d}$,

$\mathrm{Cone}(\Gamma):=\Big{\\{}\sum_{i=1}^{n}\lambda_{i}\mathbf{a_{i}}:\lambda_{i}\in\mathbb{Q}_{\geq
0},i=1,\dots,n\Big{\\}}$

and set
$\mathcal{H}(\Gamma):=(\mathrm{Cone}(\Gamma)\setminus\Gamma)\cap\mathbb{N}^{d}$.
An element $f\in\mathcal{H}(\Gamma)$ is called a pseudo-Frobenius element of
$\Gamma$ if $f+\gamma\in\Gamma$ for all
$\gamma\in\Gamma\setminus\\{\mathbf{0}\\}$. The set of pseudo-Frobenius
elements of $\Gamma$ is denoted by $\mathrm{PF}(\Gamma)$. In [3, Theorem 6],
the authors proved that $\Gamma$ is a MPD-semigroup if and only if
$\mathrm{PF}(\Gamma)\neq\emptyset$. In particular, they prove that if $\Gamma$
is a MPD-semigroup then $b\in\Gamma$ is the $\Gamma$-degree of the
$(n-2)^{th}$-minimal syzygy of $\mathbb{K}[\Gamma]$ if and only if
$b\in\\{\mathbf{a}+\sum_{i=1}^{n}\mathbf{a}_{i}:\mathbf{a}\in\mathrm{PF}(\Gamma)\\}$.

Let $(\mathbf{F},\phi)$ be a graded minimal free $R$\- resolution of
$\mathbb{K}[\Gamma]$, where

$\mathbf{F}:0\rightarrow
R^{\beta_{k}}\xrightarrow{\phi_{k}}R^{\beta_{k-1}}\xrightarrow{\phi_{k-1}}\cdots\xrightarrow{\phi_{2}}R^{\beta_{1}}\xrightarrow{\phi_{0}}\rightarrow\mathbb{K}[\Gamma]\rightarrow
0.$

The elements $s_{i,j}\in\Gamma$ for which
$R^{\beta_{i}}$=$\oplus_{j=1}^{\beta_{i}}R[-s_{i,j}]$ are called $i$-Betti
$\Gamma$-degrees. Denote by $\mathcal{B}_{i}(\Gamma)$ the set of these
$i$-Betti $\Gamma$-degrees for $1\leq i\leq\mathrm{pd}(\Gamma)$ and let
$\mathcal{B}_{i}(\Gamma)=0$ otherwise, where $\mathrm{pd}(\Gamma)$ is the
projective dimension of $\mathbb{K}[\Gamma]$. Note that we allow
$\mathcal{B}_{i}(\Gamma)$ to contain repeating elements in a nonstandard way
for convenience.

######  Definition 2.19 ([16], Definition 2.1).

Given an affine semigroup $\Gamma$ generated minimally by the set
$\\{\mathbf{a}_{1},\dots,\mathbf{a}_{n}\\}$, recall that an extension of
$\Gamma$ is an affine semigroup denoted by $E$ and generated by
$l\mathbf{a}_{1},\dots,l\mathbf{a}_{n},\mathbf{a}$, where $l$ is a positive
integer coprime to a component of
$\mathbf{a}=u_{1}\mathbf{a}_{1}+\dots+u_{n}\mathbf{a}_{n}$ for some non-
negative integers $u_{1},\dots,u_{n}$.

######  Definition 2.20 ([3],Definition 3.1).

Let $\Gamma$ be an affine MPD-semigroup with $\mathcal{H}(\Gamma)$ is finite.
For a fixed term order $\prec$ on $\mathbb{N}^{d}$, where
$\rm{F}(\Gamma)\neq\emptyset$ such that
$\rm{F}(\Gamma)_{\prec}=\rm{max_{\prec}}\mathcal{H}(\Gamma)\in\rm{F}(\Gamma)$.
If $|\mathrm{PF}(\Gamma)|=1$ and
$\mathrm{PF}(\Gamma)=\\{\rm{F}(\Gamma)_{\prec}\\}$, then $\Gamma$ is called a
$\prec$-symmetric semigroup.

######  Theorem 2.21.

Let $\Gamma$ be an affine MPD-semigroup with $\mathcal{H}(\Gamma)$ is finite
then extension $E$ of affine semigroup $\Gamma$ is MPD-semigroup and
$\mathcal{H}(E)$ is finite. In particular,
$\mathrm{PF}(E)=\\{lf+(l-1)\mathbf{a}\,|f\in\mathrm{PF}(\Gamma)\\}$. Moreover,
if $\Gamma$ is $\prec$-symmetric then $E$ is also $\prec$-symmetric.

###### Proof.

Let $\mathbf{F}:0\rightarrow
R^{\beta_{k}}\xrightarrow{\phi_{k}}R^{\beta_{k-1}}\xrightarrow{\phi_{k-1}}\cdots\xrightarrow{\phi_{2}}R^{\beta_{1}}\xrightarrow{\phi_{0}}\rightarrow\mathbb{K}[\Gamma]\rightarrow
0$ be a graded minimal free $R$-resolution of $\mathbb{K}[\Gamma]$. It is
clear from the proof of [15, Lemma 3.4] that
$\mathcal{B}_{i}(E)=l\mathcal{B}_{i}(\Gamma)\cup
l[\mathcal{B}_{i-1}(\Gamma)+\mathbf{a}]$ for $1\leq
i\leq\mathrm{pd}(\Gamma)+1$ and $\mathrm{pd}(E)=\mathrm{pd}(\Gamma)+1$. Since
$\Gamma$ is MPD-semigroup, therefore $\mathrm{pd}(E)=n-1+1=n$, and so $E$ is
MPD semigroup. Now
$\mathcal{B}_{n-1}(E)=l\mathcal{B}_{n-1}(\Gamma)\cup[l\mathcal{B}_{n-2}(\Gamma)+\mathbf{a}]=l\mathcal{B}_{n-2}(\Gamma)+l\mathbf{a}$.
Therefore, for any $b_{n-2}\in\mathcal{B}_{n-2}(\Gamma)$, we have
$lb_{n-2}(\Gamma)+l\mathbf{a}-\sum_{i=1}^{n}l\mathbf{a}_{i}-\mathbf{a}=lb_{n-2}(\Gamma)-\sum_{i=1}^{n}l\mathbf{a}_{i}+l\mathbf{a}-\mathbf{a}=lf+(l-1)\mathbf{a}\in\mathrm{PF}(E)$.
Hence, $\mathrm{PF}(E)=\\{lf+(l-1)\mathbf{a}\,|f\in\mathrm{PF}(\Gamma)\\}$.
Let $\mathbf{x}\in\mathcal{H}(E)$ then
$\mathbf{x}=y_{1}\mathbf{a}_{1}+\dots+y_{n}\mathbf{a}_{n}+z\mathbf{a}$, for
some $y_{i},z\in\mathbb{Q}_{\geq 0}$ but since $\mathbf{a}\in\Gamma$ and one
of $y_{j}$ or $z$ is in $\mathbb{Q}\setminus\mathbb{N}$, we have
$\mathbf{x}\in\mathcal{H}(\Gamma)$. Therefore,
$\mathcal{H}(E)\subseteq\mathcal{H}(\Gamma)$ is finite. Since $\mathcal{H}(E)$
is finite and $\Gamma$ is $\prec$-symmetric, we have $|\mathrm{PF}(E)|=1$ and
$\mathrm{PF}(E)=\\{l\mathrm{F}(\Gamma)_{\prec}+(l-1)\mathbf{a}=\mathrm{F}(E)_{\prec}\\}$.
Therefore, $E$ is also $\prec$-symmetric. ∎

######  Example 2.22 ([5]).

Let $\Gamma$ be the submonoid of $\mathbb{N}^{2}$ generated by the columns of
the following matrix

$A=\begin{pmatrix}3&5&0&1&2\\\ 0&0&1&3&3\end{pmatrix}$

The minimal free resolution $\mathbb{K}[\Gamma]$, as a module over
$R=\mathbb{K}[x_{1},x_{2},x_{3},x_{4},x_{5}]$ is given by

$0\rightarrow R(18,9)\rightarrow R^{6}\rightarrow R^{11}\rightarrow
R^{7}\rightarrow R\rightarrow\mathbb{K}[\Gamma]\rightarrow 0.$

In particular, the degrees of minimal generators of the fourth syzygy modules
is $(18,9)$. Therefore, $\Gamma$ has one pseudo-Frobenius element $(7,2)$.
Consider the extension $E$ of $\Gamma$, which is the submonoid of
$\mathbb{N}^{2}$ generated by the columns of the following matrix

$B=\begin{pmatrix}6&10&0&2&4&6\\\ 0&0&2&6&6&9\end{pmatrix}$

The minimal free resolution $\mathbb{K}[\Gamma]$, as a module over
$R=\mathbb{K}[x_{1},x_{2},x_{3},x_{4},x_{5}]$ is given by

$0\rightarrow R(48,36)\rightarrow R^{8}\rightarrow R^{18}\rightarrow
R^{17}\rightarrow R^{7}\rightarrow R^{1}\rightarrow\mathbb{K}[E]\rightarrow
0.$

The degrees of minimal generators of the fifth syzygy modules is $(48,36)$.
Therefore, $E$ has one pseudo-Frobenius element $(48,36)-(28,23)=(20,13)$.
Note that $\mathcal{H}(\Gamma)$ is finite, so $\Gamma$ is $\prec$-symmetric
and also as $\mathcal{H}(E)\subseteq\mathcal{H}(\Gamma)$, hence $E$ is
$\prec$-symmetric.

##  3\. Strong Indispensable Minimal Free Resolution

Let $\Gamma$ be an affine semigroup, i.e, a finitely generated semigroup which
for some $r$ is isomorphic to a subsemigroup of $\mathbb{Z}^{r}$ containing
zero. Suppose that $\Gamma$ is a simplicial affine semigroup, fully embedded
in $\mathbb{N}^{d}$, minimally generated by
$\\{\mathbf{a}_{1}.\dots,\mathbf{a}_{d},\mathbf{a}_{d+1},\dots,\mathbf{a}_{d+r}\\}$
with the set of extremal rays $E=\\{\mathbf{a}_{1},\dots,\mathbf{a}_{d}\\}$.
The semigroup algebra $\mathbb{K}[\Gamma]$ over a field $\mathbb{K}$ is
generated by the monomials $\mathbf{x}^{\mathbf{a}}$, where
$\mathbf{a}\in\Gamma$, with maximal ideal
$\mathfrak{m}=(\mathbf{x}^{\mathbf{a}_{1}},\dots,\mathbf{x}^{\mathbf{a}_{d+r}})$.

Let $I(\Gamma)$ denote the defining ideal of $\mathbb{K}[\Gamma]$, which is
the kernel of the $\mathbb{K}-$algebra homomorphism
$\phi:R=\mathbb{K}[z_{1},\dots,z_{d+r}]\rightarrow\mathbb{K}[\mathbf{x}^{\mathbf{a}_{1}},\dots,\mathbf{x}^{\mathbf{a}_{d+r}}]$,
such that $\phi(z_{i})=\mathbf{x}^{\mathbf{a}_{i}}$, $i=1,\dots,d+r$. Let us
write $\mathbb{K}[\Gamma]\cong A/I(\Gamma)$. The defining ideal $I(\Gamma)$ is
a binomial prime ideal ([8], Proposition 1.4).

Let $(\mathbf{F},\phi)$ be a graded minimal free $R$\- resolution of
$\mathbb{K}[\Gamma]$, where

$\mathbf{F}:0\rightarrow
R^{\beta_{k}}\xrightarrow{\phi_{k}}R^{\beta_{k-1}}\xrightarrow{\phi_{k-1}}\cdots\xrightarrow{\phi_{2}}R^{\beta_{1}}\xrightarrow{\phi_{0}}\rightarrow\mathbb{K}[\Gamma]\rightarrow
0.$

The elements $s_{i,j}\in\Gamma$ for which
$R^{\beta_{i}}$=$\oplus_{j=1}^{\beta_{i}}R[-s_{i,j}]$ are calledd $i$-Betti
$\Gamma$-degrees. Denote by $\mathcal{B}_{i}(\Gamma)$ the set of these
$i$-Betti $\Gamma$-degrees for $1\leq i\leq\mathrm{pd}(\Gamma)$ and let
$\mathcal{B}_{i}(\Gamma)=0$ otherwise, where $\mathrm{pd}(\Gamma)$ is the
projective dimension of $\mathbb{K}[\Gamma]$. Note that we allow
$\mathcal{B}_{i}(\Gamma)$ to contain repeating elements in a nonstandard way
for convenience.

The resolution $(\mathbf{F},\phi)$ is called strongly indispensable if for any
graded minimal resolution $(\mathbf{G},\theta)$, we have an injective complex
map $i:(\mathbf{F},\phi)\rightarrow(\mathbf{G},\theta)$. When
$(\mathbf{F},\phi)$ is strongly indispensable $\Gamma$ or $\mathbb{K}[\Gamma]$
is said to have a SIFR for short.

###  3.1. Join of Affine Semigroups

We start this section with the definition of join of affine semigroups.

######  Definition 3.1 ([13]).

Consider affine semigroups $\Gamma_{1},\Gamma_{2}\subset\mathbb{N}^{r}$ of
dimensions $r_{1}$ and $r_{2}$, respectively such that they are minimally
generated by the disjoint sets
$\mathcal{G}_{1}=\\{\mathbf{a}_{1},\dots,\mathbf{a}_{r_{1}},\dots,\mathbf{a}_{r_{1}+n_{1}}\\}$
and
$\mathcal{G}_{2}=\\{\mathbf{b}_{1},\dots,\mathbf{b}_{r_{2}},\dots,\mathbf{b}_{r_{2}+n_{2}}\\}$,
with extremal rays
$E_{\Gamma_{1}}=\\{\mathbf{a}_{1},\dots,\mathbf{a}_{r_{1}}\\}$,
$E_{\Gamma_{2}}=\\{\mathbf{b}_{1},\dots,\mathbf{b}_{r_{2}}\\}$ respectively.
Assuming that the set
$\\{\mathbf{a}_{1},\dots,\mathbf{a}_{r_{1}},\mathbf{b}_{1},\dots,\mathbf{b}_{r_{2}}\\}$
is linearly independent over $\mathbb{Q}$, the semigroup
$\Gamma_{1}\sqcup\Gamma_{2}=\langle\mathcal{G}_{1}\cup\mathcal{G}_{2}\rangle$
is an affine semigroup with the set of extremal rays
$E_{\Gamma_{1}\sqcup\Gamma_{2}}=E_{\Gamma_{1}}\cup E_{\Gamma_{2}}$. Moreover,
the set $\mathcal{G}_{1}\cup\mathcal{G}_{2}$ is a minimal generating set of
the semigroup $\Gamma_{1}\sqcup\Gamma_{2}$. We call
$\Gamma_{1}\sqcup\Gamma_{2}$ the join of affine semigroups $\Gamma_{1}$ and
$\Gamma_{2}$. Clearly,
$\mathrm{dim}(\Gamma_{1}\sqcup\Gamma_{2})=\mathrm{dim}(\Gamma_{1})+\mathrm{dim}(\Gamma_{2})$.
Let $\Gamma_{1},\dots,\Gamma_{s}$ be affine semigroups in $\mathbb{N}^{r}$, of
dimensions $r_{1},\dots,r_{s}$, respectively. If $E_{\Gamma_{1}}\cup\dots\cup
E_{\Gamma_{s}}$ is linearly independent set of cardinality
$r_{1}+\dots+r_{s}=r$, then $\Gamma=\Gamma_{1}\sqcup\dots\sqcup\Gamma_{s}$ is
simplicial. Note that a simplicial affine semigroup in $\mathbb{N}^{r}$ is a
join of $r$ semigroups if and only if all elements in its minimal generating
set are located on the extremal rays of $\rm{cone}(\Gamma)$.

In an equivalent statement, one can say that an affine semigroup $\Gamma$ with
the minimal generating set $G$ and the set of extremal rays $E$, is a join of
two of its sub-semigroups if there exist $G_{1},G_{2},E_{1},E_{2}$ such that
$G=G_{1}\cup G_{2},\,E=E_{1}\cup E_{2},\,E_{1}\cap E_{2}=\phi$ and $E_{i}$ is
the set of extremal rays of $\Gamma_{i}=\langle G_{i}\rangle$, for $i=1,2$. We
call $\Gamma_{1}\sqcup\Gamma_{2}$ the join of affine semigroups $\Gamma_{1}$
and $\Gamma_{2}$, denoted by $\Gamma_{c}$.

Let $A_{1}=\mathbb{K}[x_{1},\ldots,x_{r_{1}+n_{1}}]$ and
$A_{2}=\mathbb{K}[y_{1},\ldots,y_{r_{2}+n_{2}}]$ be polynomial rings with
disjoint set of indeterminates $\\{x_{1},\ldots,x_{r_{1}+n_{1}}\\}$ and
$\\{y_{1},\dots,y_{r_{2}+n_{2}}\\}$. Let $\mathbb{K}[\mathbf{t}]$ be another
polynomial ring, where $\mathbf{t}=t_{1},\ldots,t_{r}$. We consider maps
$\phi_{\Gamma_{1}}:A_{1}\rightarrow\mathbb{K}[\mathbf{t}]$, defined by
$\phi_{\Gamma_{1}}(x_{i})=\mathbf{t}^{\mathbf{a_{i}}}$, $1\leq i\leq
r_{1}+n_{1}$, and $\phi_{\Gamma_{2}}:A_{2}\rightarrow\mathbb{K}[\mathbf{t}]$,
defined by $\phi_{\Gamma_{2}}(y_{j})=\mathbf{t}^{\mathbf{b_{j}}}$, $1\leq
j\leq r_{2}+n_{2}$. We write
$A_{12}=\mathbb{K}[x_{1},\dots,x_{r_{1}+n_{1}},y_{1},\dots,y_{r_{2}+n_{2}}]$
and consider
$\phi_{\Gamma_{1}\sqcup\Gamma_{2}}:A_{12}\rightarrow\mathbb{K}[\mathbf{t}]$,
defined by
$\phi_{\Gamma_{1}\sqcup\Gamma_{2}}(x_{i})=\mathbf{t}^{\mathbf{a_{i}}}$, $1\leq
i\leq r_{1}+n_{1}$, and
$\phi_{\Gamma_{1}\sqcup\Gamma_{2}}(y_{j})=\mathbf{t}^{\mathbf{b_{j}}}$, $1\leq
j\leq r_{2}+n_{2}$. For the sake of simplicity assume $l_{1}=r_{1}+n_{1}$ and
$l_{2}=r_{2}+n_{2}$.

######  Theorem 3.2.

Let $I_{\Gamma_{1}},I_{\Gamma_{2}}$ be the defining ideals of
$\mathbb{K}[\Gamma_{1}]$ and $\mathbb{K}[\Gamma_{2}]$ respectively. Then the
defining ideal of $\mathbb{K}[\Gamma_{1}\sqcup\Gamma_{2}]$ is
$I_{\Gamma_{1}\sqcup\Gamma_{2}}=I_{\Gamma_{1}}A_{12}+I_{\Gamma_{2}}A_{12}$.
The tensor product of minimal graded free resolutions of
$\mathbb{K}[\Gamma_{1}]$ and $\mathbb{K}[\Gamma_{2}]$ over $\mathbb{K}$ is a
minimal graded free resolution of $\mathbb{K}[\Gamma_{1}\sqcup\Gamma_{2}]$
over $\mathbb{K}$.

###### Proof.

See [13, Theorem 4.2]. ∎

######  Lemma 3.3.

A minimal graded free resolution of $\mathbb{K}[\Gamma]$ is strongly
indispensable if and only if
$\pm(\mathbf{b}_{i}-\mathbf{b}^{\prime})\notin\Gamma$ for all
$\mathbf{b}_{i},\mathbf{b}^{\prime}\in\mathcal{B}_{i}(\Gamma)$ and for each
$1\leq i\leq\mathrm{pd}(\Gamma)$.

###### Proof.

See [15, Lemma 2.1]. ∎

######  Theorem 3.4.

Let $\Gamma_{1}$ and $\Gamma_{2}$ are two affine semigroups in
$\mathbb{N}^{r}$. Let $\Gamma=\Gamma_{1}\sqcup\Gamma_{2}$ be a join of affine
semigroups $\Gamma_{1},\Gamma_{2}$. Then $\Gamma$ has a SIFR if and only if
$\Gamma_{1}$ and $\Gamma_{2}$ have SIFRs.

###### Proof.

Let $\mathbf{F}_{i}$ be a graded minimal free resolution of
$\mathbb{K}[{\Gamma_{i}}]$ for $i=1,2$. By Theorem 3.2,
$\mathbf{F}_{1}\otimes\mathbf{F}_{2}$ gives the graded minimal free resolution
of $\mathbb{K}[{\Gamma}]$. Hence, the proof follows from the following

$\big{(}\mathbf{F}_{1}\otimes\mathbf{F}_{2}\big{)}_{i}=\bigoplus_{p_{1}+p_{2}=i}F_{p_{1}}\otimes
F_{p_{2}}$

since $\Gamma$-degrees of elements in $F_{p_{1}}\otimes F_{p_{2}}$ constitutes
the set $\mathcal{B}_{p_{1}}(\Gamma_{1})+\mathcal{B}_{p_{2}}(\Gamma_{2})$.
Hence, we have
$\mathcal{B}_{i}(\Gamma)=\bigcup_{p_{1}+p_{2}=i}\big{(}\mathcal{B}_{p_{1}}(\Gamma_{1})\cup\mathcal{B}_{p_{2}}(\Gamma_{2})\big{)}$.

claim(i): Fix $j\in\\{1,2\\}$, and
$\mathbf{b},\mathbf{b^{\prime}}\in\Gamma_{j}$. Then
$\mathbf{b}-\mathbf{b^{\prime}}\in\Gamma$ if and only if
$\mathbf{b}-\mathbf{b^{\prime}}\in\Gamma_{1}$ or
$\mathbf{b}-\mathbf{b^{\prime}}\in\Gamma_{2}$.
Without loss of generality assume that $j=1$. As $\Gamma_{1}\subseteq\Gamma$,
$\mathbf{b}-\mathbf{b^{\prime}}\in\Gamma_{1}$ implies
$\mathbf{b}-\mathbf{b^{\prime}}\in\Gamma$. Conversely, take
$\mathbf{b},\mathbf{b^{\prime}}\in\Gamma_{1}$ such that
$\mathbf{b}-\mathbf{b^{\prime}}\in\Gamma$. Then
$\mathbf{b}-\mathbf{b^{\prime}}=\gamma_{1}+\gamma_{2}$ for some
$\gamma_{1}\in\Gamma_{1}$ and $\gamma_{2}\in\Gamma_{2}$. Assume that
$\gamma_{2}\neq 0$. Since $E_{1}$ and $E_{2}$ are the set of extremal rays of
$\Gamma_{1}$ and $\Gamma_{2}$ respectively, hence
$\mathbf{b},\,\gamma_{1},\,\mathbf{b^{\prime}},\,\gamma_{2}$ can be written as
rational linear combination of elements of $E_{1}$ and $E_{2}$. But in that
case we get a non-zero linear combination of elements of $E_{1}$ and $E_{2}$
over $\mathbb{Q}$, which is a contradiction of the fact that
$\\{\mathbf{a}_{1},\dots,\mathbf{a}_{r_{1}},\mathbf{b}_{1},\dots,\mathbf{b}_{r_{2}}\\}$
is linearly independent over $\mathbb{Q}$. Therefore
$\mathbf{b}-\mathbf{b^{\prime}}\in\Gamma_{1}$.

By Lemma 3.3, the differences between the elements in
$\mathcal{B}_{i}(\Gamma)$ do not belong to $\Gamma$. Let
$\mathbf{b}_{i,j},\mathbf{b}^{{}^{\prime}}_{i,j}\in\mathcal{B}_{i}(\Gamma_{j})$,
for $j=1,2$. Since
$\mathcal{B}_{i}(\Gamma_{j})\subset\mathcal{B}_{i}(\Gamma)$, and
$\mathbf{b}_{i,j}-\mathbf{b}^{{}^{\prime}}_{i,j}\notin\Gamma$. Hence by claim
(i) $\mathbf{b}_{i,j}-\mathbf{b}^{{}^{\prime}}_{i,j}\notin\Gamma_{j}$, which
means $I_{\Gamma_{1}}$ and $I_{\Gamma_{2}}$ have SIFR by Lemma 3.3. Now, if
$\mathbf{b},\mathbf{b}^{{}^{\prime}}\in\mathcal{B}_{i}(\Gamma)$ then
$\mathbf{b},\mathbf{b}^{{}^{\prime}}\in\mathcal{B}_{p}(\Gamma_{1})+\mathcal{B}_{q}(\Gamma_{2})$,
for $p+q=i$. Let $\mathbf{b}=\mathbf{b}_{p,1}+\mathbf{b}_{q,2}$ and
$\mathbf{b}^{{}^{\prime}}=\mathbf{b}^{{}^{\prime}}_{r,1}+\mathbf{b}^{{}^{\prime}}_{s,2}$
with $p+q=i=r+s$. Suppose $\mathbf{b}-\mathbf{b}^{{}^{\prime}}\in\Gamma$.
Then,
$\mathbf{b}-\mathbf{b}^{{}^{\prime}}=\mathbf{b}_{p,1}+\mathbf{b}_{q,2}-\mathbf{b}^{{}^{\prime}}_{r,1}-\mathbf{b}^{{}^{\prime}}_{s,2}=\gamma_{1}+\gamma_{2}$,
for some $\gamma_{1}\in\Gamma_{1}$ and $\gamma_{2}\in\Gamma_{2}$. Thus,
$\mathbf{b}_{p,1}-\mathbf{b}^{\prime}_{r,1}-\gamma_{1}=\mathbf{b}_{q,2}-\mathbf{b}^{\prime}_{s,2}-\gamma_{2}$,
which leads to the contradiction of the fact
$\\{\mathbf{a}_{1},\dots,\mathbf{a}_{r_{1}},\mathbf{b}_{1},\dots,\mathbf{b}_{r_{2}}\\}$
is linearly independent over $\mathbb{Q}$, unless
$\mathbf{b}_{p,1}-\mathbf{b}^{\prime}_{r,1}-\gamma_{1}=\mathbf{b}_{q,2}-\mathbf{b}^{\prime}_{s,2}-\gamma_{2}=0$.
Hence, $\mathbf{b}_{p,1}-\mathbf{b}^{\prime}_{r,1}\in\Gamma_{1}$ and
$\mathbf{b}_{q,2}-\mathbf{b}^{\prime}_{s,2}\in\Gamma_{2}$, which is not
possible as $\Gamma_{1}$ and $\Gamma_{2}$ have SIFRs. Hence,
$\mathbf{b}-\mathbf{b}^{{}^{\prime}}\notin\Gamma$, and by Lemma 3.3, $\Gamma$
has a strongly indispensable free resolution. ∎

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