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A variation of gluing of numerical semigroups

Takahiro Numata

Nihon University

9th September 2014

Takahiro Numata (Nihon University) A variation of gluing of numerical semigroups 9th September 2014 1 / 20

Introduction

Introduction

Takahiro Numata (Nihon University) A variation of gluing of numerical semigroups 9th September 2014 2 / 20

Introduction

Definition (Rosales)

Let S1 = a1, ..., an and S2 = b1, ..., bm be two numerical semigroups. Let d1 ∈ S2 \ {b1, ..., bm} and d2 ∈ S1 \ {a1, ..., an}, where gcd(d1, d2) = 1. Then we say that S = d1a1, ..., d1an, d2b1, ..., d2bm , is a gluing of S1 and S2.

Takahiro Numata (Nihon University) A variation of gluing of numerical semigroups 9th September 2014 3 / 20

Introduction

Definition (Rosales)

Let S1 = a1, ..., an and S2 = b1, ..., bm be two numerical semigroups. Let d1 ∈ S2 \ {b1, ..., bm} and d2 ∈ S1 \ {a1, ..., an}, where gcd(d1, d2) = 1. Then we say that S = d1a1, ..., d1an, d2b1, ..., d2bm , is a gluing of S1 and S2.

Theorem (Delorme),(Rosales)

Let S be a gluing of two numerical semigroups S1 and S2. Then S is symmetric (resp. a complete intersection) ⇐ ⇒ S1 and S2 are symmetric (resp. complete intersections)

Takahiro Numata (Nihon University) A variation of gluing of numerical semigroups 9th September 2014 3 / 20

Introduction

Our purpose is to study the relation between two numerical semigroups S = a1, ..., an and T = da1, ..., dan−1, an , where d > 1 and gcd(d, an) = 1.

Takahiro Numata (Nihon University) A variation of gluing of numerical semigroups 9th September 2014 4 / 20

Introduction

Our purpose is to study the relation between two numerical semigroups S = a1, ..., an and T = da1, ..., dan−1, an , where d > 1 and gcd(d, an) = 1. [Watanabe,1973] If S = a1, ..., an−1 and an ∈ S \ {a1, ..., an−1}, then T is symmetric (resp. a complete intersection) ⇐ ⇒ S is symmetric (resp. a complete intersection)

Takahiro Numata (Nihon University) A variation of gluing of numerical semigroups 9th September 2014 4 / 20

Introduction

Our purpose is to study the relation between two numerical semigroups S = a1, ..., an and T = da1, ..., dan−1, an , where d > 1 and gcd(d, an) = 1. [Watanabe,1973] If S = a1, ..., an−1 and an ∈ S \ {a1, ..., an−1}, then T is symmetric (resp. a complete intersection) ⇐ ⇒ S is symmetric (resp. a complete intersection) We consider the case an / ∈ a1, ..., an−1.

Takahiro Numata (Nihon University) A variation of gluing of numerical semigroups 9th September 2014 4 / 20

Preliminaries

Preliminaries

Takahiro Numata (Nihon University) A variation of gluing of numerical semigroups 9th September 2014 5 / 20

Preliminaries

Definition

For a numerical semigroup S = a1, ..., an, we define its semigroup ring: k[S] := k[ta1, ..., tan] ⊂ k[t], where k is any field and t is an indeterminate.

Takahiro Numata (Nihon University) A variation of gluing of numerical semigroups 9th September 2014 6 / 20

Preliminaries

A semigroup ring k[S] = k[ta1, ..., tan] is a Z-graded ring in the natural way, a one-dimensional Cohen-Macaulay ring with the unique homogeneous maximal ideal m = (ta1, ...tan), and k[S] ∼ = k[X1, ..., Xn]/IS, where IS is the kernel of the surjective k-algebra homomorphism k[X1, ..., Xn] → k[S] Xi − → tai where deg(Xi) = ai for any 1 ≤ i ≤ n. Then IS is called the defining ideal of k[S].

Takahiro Numata (Nihon University) A variation of gluing of numerical semigroups 9th September 2014 7 / 20

Preliminaries

Definition

Let S be a numerical semigroup.

1 F(S) := max(Z \ S), the Frobenius number of S. 2 PF(S) := {x ∈ Z \ S | x + s ∈ S for any 0 = s ∈ S}.

x ∈ PF(S) : a pseudo-Frobenius number of S.

3 t(S) := #PF(S): the type of S. Takahiro Numata (Nihon University) A variation of gluing of numerical semigroups 9th September 2014 8 / 20

Preliminaries

Definition

Let S be a numerical semigroup.

1 F(S) := max(Z \ S), the Frobenius number of S. 2 PF(S) := {x ∈ Z \ S | x + s ∈ S for any 0 = s ∈ S}.

x ∈ PF(S) : a pseudo-Frobenius number of S.

3 t(S) := #PF(S): the type of S.

t(S) = r(k[S]), the Cohen-Macaulay type of k[S]. F(S) = a(k[S]), the a-invariant of k[S].

Takahiro Numata (Nihon University) A variation of gluing of numerical semigroups 9th September 2014 8 / 20

Preliminaries

Definition

Let S be a numerical semigroup.

1

S is symmetric.

def

⇐ ⇒ ∀x ∈ Z, either x ∈ S or F(S) − x ∈ S.

iff

⇐ ⇒ k[S] is Gorenstein.

2 S is a complete intersection.

def

⇐ ⇒ k[S] is a complete intersection.

Takahiro Numata (Nihon University) A variation of gluing of numerical semigroups 9th September 2014 9 / 20

Preliminaries

Let S = a1, ..., an, R = k[S] and A = k[X1, ..., Xn]. Since R is a

• ne-dimensional Cohen-Macaulay ring, the minimal graded free resolution
• f R is length n − 1:

0 → ⊕

j

A(−mn−1,j)βn−1,j → · · · → ⊕

j

A(−m1j)β1j → A → R → 0, where βij > 0 for each i, j.

Takahiro Numata (Nihon University) A variation of gluing of numerical semigroups 9th September 2014 10 / 20

Preliminaries

Let S = a1, ..., an, R = k[S] and A = k[X1, ..., Xn]. Since R is a

• ne-dimensional Cohen-Macaulay ring, the minimal graded free resolution
• f R is length n − 1:

0 → ⊕

j

A(−mn−1,j)βn−1,j → · · · → ⊕

j

A(−m1j)β1j → A → R → 0, where βij > 0 for each i, j. Taking HomA(∗, KA) ∼ = HomA(∗, A(−N)), we have 0 → A(−N) → ⊕

j

A(m1j − N)β1j → · · · → ⊕

j

A(mn−1,j − N)βn−1,j → KR → 0, where N = ∑n

i=1 ai, KR ∼

= Extn−1

A

(R, KA).

Takahiro Numata (Nihon University) A variation of gluing of numerical semigroups 9th September 2014 10 / 20

Main Results

Main Results

Takahiro Numata (Nihon University) A variation of gluing of numerical semigroups 9th September 2014 11 / 20

Main Results

Notation S = a1, ..., an. T = da1, ..., dan−1, an, where d > 1 and gcd(d, an) = 1. A = k[X1, ..., Xn], where deg(Xi) = ai for each 1 ≤ i ≤ n. B = k[Y1, ..., Yn], where deg(Yi) = dai for each 1 ≤ i ≤ n − 1, and deg(Yn) = an. f : A ֒ → B, where Xi → Yi for any 1 ≤ i ≤ n − 1, and Xn → Ynd. We note that f is faithfully flat, and if a ∈ A is a homogeneous element of degree m, then f(a) ∈ B is of degree dm.

Takahiro Numata (Nihon University) A variation of gluing of numerical semigroups 9th September 2014 12 / 20

Main Results

The minimal graded free resolution of k[S] over A: F• : 0 → ⊕

j

A(−mn−1,j)βn−1,j → · · · → ⊕

j

A(−m1j)β1j → A → k[S] → 0

Takahiro Numata (Nihon University) A variation of gluing of numerical semigroups 9th September 2014 13 / 20

Main Results

The minimal graded free resolution of k[S] over A: F• : 0 → ⊕

j

A(−mn−1,j)βn−1,j → · · · → ⊕

j

A(−m1j)β1j → A → k[S] → 0 Since f is faithfully flat, F• ⊗A B is the minimal graded free resolution of k[T] over B: F• ⊗A B : 0 → ⊕

j

B( − dmn−1,j)βn−1,j → · · · → ⊕

j

B(−dm1j)β1j → B → k[T] → 0

Takahiro Numata (Nihon University) A variation of gluing of numerical semigroups 9th September 2014 13 / 20

Main Results

Proposition

Under the above notations, the following statements hold true.

1 The Betti numbers of k[T] are equal to those of k[S]. In particular,

t(T) = t(S) and µ(IT ) = µ(IS), where µ(I) is the number of the minimal generators of I.

2 PF(T) = {d

f + (d − 1)an | f ∈ PF(S)}. In particular, F(T) = dF(S) + (d − 1)an.

Takahiro Numata (Nihon University) A variation of gluing of numerical semigroups 9th September 2014 14 / 20

Main Results

Proposition

Under the above notations, the following statements hold true.

1 The Betti numbers of k[T] are equal to those of k[S]. In particular,

t(T) = t(S) and µ(IT ) = µ(IS), where µ(I) is the number of the minimal generators of I.

2 PF(T) = {d

f + (d − 1)an | f ∈ PF(S)}. In particular, F(T) = dF(S) + (d − 1)an.

Corollary

T is symmetric (resp. a complete intersection) ⇐ ⇒ S is symmetric (resp. a complete intersection).

Takahiro Numata (Nihon University) A variation of gluing of numerical semigroups 9th September 2014 14 / 20

Main Results

Question

When is T = da1, ..., dan−1, an almost symmetric if it is not symmetric ?

Takahiro Numata (Nihon University) A variation of gluing of numerical semigroups 9th September 2014 15 / 20

Main Results

For a numerical semigroup S, put L(S) = {x ∈ Z \ S | F(S) − x / ∈ S}.

Definition (Barucci, Fr¨

• berg, 1997)

A numerical semigroup S is almost symmetric

def

⇐ ⇒ L(S) ⊂ PF(S).

Takahiro Numata (Nihon University) A variation of gluing of numerical semigroups 9th September 2014 16 / 20

Main Results

For a numerical semigroup S, put L(S) = {x ∈ Z \ S | F(S) − x / ∈ S}.

Definition (Barucci, Fr¨

• berg, 1997)

A numerical semigroup S is almost symmetric

def

⇐ ⇒ L(S) ⊂ PF(S). S is symmetric ⇐ ⇒ (S is almost symmetric and) t(S) = 1. S is pseudo-symmetric ⇐ ⇒ S is almost symmetric and t(S) = 2.

Takahiro Numata (Nihon University) A variation of gluing of numerical semigroups 9th September 2014 16 / 20

Main Results

Theorem (Nari, 2013)

Let S be a numerical semigroup with PF(S) = {f1 < f2 < · · · < ft = F(S)}. Then: S is almost symmetric ⇐ ⇒ fi + ft−i = F(S) for each 1 ≤ i ≤ t.

Takahiro Numata (Nihon University) A variation of gluing of numerical semigroups 9th September 2014 17 / 20

Main Results

Theorem (Nari, 2013)

Let S be a numerical semigroup with PF(S) = {f1 < f2 < · · · < ft = F(S)}. Then: S is almost symmetric ⇐ ⇒ fi + ft−i = F(S) for each 1 ≤ i ≤ t.

Proposition

T = da1, ..., dan−1, an is never almost symmetric if it is not symmetric.

Takahiro Numata (Nihon University) A variation of gluing of numerical semigroups 9th September 2014 17 / 20

Main Results

Theorem (Nari, 2013)

Let S is a numerical semigroup which is not symmetric. If S is a gluing of two numerical semigroups, then S is never almost symmetric.

Takahiro Numata (Nihon University) A variation of gluing of numerical semigroups 9th September 2014 18 / 20

Main Results

Theorem

Let S = a1, ..., an be a numerical semigroup which is not symmetric. If S is almost symmetric, then any (n − 1)-tuples of {a1, ..., an} are relatively coprime.

Takahiro Numata (Nihon University) A variation of gluing of numerical semigroups 9th September 2014 19 / 20

Main Results

Theorem

Let S = a1, ..., an be a numerical semigroup which is not symmetric. If S is almost symmetric, then any (n − 1)-tuples of {a1, ..., an} are relatively coprime.

Theorem (Rosales, Garc´ ıa-S´ anchez)

If S = a1, a2, a3 is pseudo-symmetric, then any pairs of {a1, a2, a3} are relatively coprime.

Takahiro Numata (Nihon University) A variation of gluing of numerical semigroups 9th September 2014 19 / 20

Thank you for your attention !

Takahiro Numata (Nihon University) A variation of gluing of numerical semigroups 9th September 2014 20 / 20