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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 S 1 = � a 1 , ..., a n � and S 2 = � b 1 , ..., b m � be two numerical semigroups. Let d 1 ∈ S 2 \ { b 1 , ..., b m } and d 2 ∈ S 1 \ { a 1 , ..., a n } , where gcd( d 1 , d 2 ) = 1 . Then we say that S = � d 1 a 1 , ..., d 1 a n , d 2 b 1 , ..., d 2 b m � , is a gluing of S 1 and S 2 . Takahiro Numata (Nihon University) A variation of gluing of numerical semigroups 9th September 2014 3 / 20

Introduction Definition (Rosales) Let S 1 = � a 1 , ..., a n � and S 2 = � b 1 , ..., b m � be two numerical semigroups. Let d 1 ∈ S 2 \ { b 1 , ..., b m } and d 2 ∈ S 1 \ { a 1 , ..., a n } , where gcd( d 1 , d 2 ) = 1 . Then we say that S = � d 1 a 1 , ..., d 1 a n , d 2 b 1 , ..., d 2 b m � , is a gluing of S 1 and S 2 . Theorem (Delorme),(Rosales) Let S be a gluing of two numerical semigroups S 1 and S 2 . Then S is symmetric (resp. a complete intersection) ⇐ ⇒ S 1 and S 2 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 = � a 1 , ..., a n � and T = � da 1 , ..., da n − 1 , a n � , where d > 1 and gcd( d, a n ) = 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 = � a 1 , ..., a n � and T = � da 1 , ..., da n − 1 , a n � , where d > 1 and gcd( d, a n ) = 1 . [Watanabe,1973] If S = � a 1 , ..., a n − 1 � and a n ∈ S \ { a 1 , ..., a n − 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 = � a 1 , ..., a n � and T = � da 1 , ..., da n − 1 , a n � , where d > 1 and gcd( d, a n ) = 1 . [Watanabe,1973] If S = � a 1 , ..., a n − 1 � and a n ∈ S \ { a 1 , ..., a n − 1 } , then T is symmetric (resp. a complete intersection) ⇐ ⇒ S is symmetric (resp. a complete intersection) We consider the case a n / ∈ � a 1 , ..., a n − 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 = � a 1 , ..., a n � , we define its semigroup ring : k [ S ] := k [ t a 1 , ..., t a n ] ⊂ 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 [ t a 1 , ..., t a n ] is a Z -graded ring in the natural way, a one-dimensional Cohen-Macaulay ring with the unique homogeneous maximal ideal m = ( t a 1 , ...t a n ) , and k [ S ] ∼ = k [ X 1 , ..., X n ] /I S , where I S is the kernel of the surjective k -algebra homomorphism k [ X 1 , ..., X n ] → k [ S ] t a i X i �− → where deg( X i ) = a i for any 1 ≤ i ≤ n . Then I S 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. S is symmetric . 1 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 = � a 1 , ..., a n � , R = k [ S ] and A = k [ X 1 , ..., X n ] . Since R is a one-dimensional Cohen-Macaulay ring, the minimal graded free resolution of R is length n − 1 : A ( − m n − 1 ,j ) β n − 1 ,j → · · · → A ( − m 1 j ) β 1 j → A → R → 0 , ⊕ ⊕ 0 → j j where β ij > 0 for each i, j . Takahiro Numata (Nihon University) A variation of gluing of numerical semigroups 9th September 2014 10 / 20

Preliminaries Let S = � a 1 , ..., a n � , R = k [ S ] and A = k [ X 1 , ..., X n ] . Since R is a one-dimensional Cohen-Macaulay ring, the minimal graded free resolution of R is length n − 1 : A ( − m n − 1 ,j ) β n − 1 ,j → · · · → A ( − m 1 j ) β 1 j → A → R → 0 , ⊕ ⊕ 0 → j j where β ij > 0 for each i, j . Taking Hom A ( ∗ , K A ) ∼ = Hom A ( ∗ , A ( − N )) , we have A ( m 1 j − N ) β 1 j → · · · ⊕ 0 → A ( − N ) → j A ( m n − 1 ,j − N ) β n − 1 ,j → K R → 0 , ⊕ → j i =1 a i , K R ∼ where N = ∑ n = Ext n − 1 ( R, K A ) . A 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 = � a 1 , ..., a n � . T = � da 1 , ..., da n − 1 , a n � , where d > 1 and gcd( d, a n ) = 1 . A = k [ X 1 , ..., X n ] , where deg( X i ) = a i for each 1 ≤ i ≤ n . B = k [ Y 1 , ..., Y n ] , where deg( Y i ) = da i for each 1 ≤ i ≤ n − 1 , and deg( Y n ) = a n . → B , where X i �→ Y i for any 1 ≤ i ≤ n − 1 , and X n �→ Y nd . f : A ֒ 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 : A ( − m n − 1 ,j ) β n − 1 ,j → · · · → A ( − m 1 j ) β 1 j → A → k [ S ] → 0 ⊕ ⊕ F • : 0 → j j 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 : A ( − m n − 1 ,j ) β n − 1 ,j → · · · → A ( − m 1 j ) β 1 j → A → k [ S ] → 0 ⊕ ⊕ F • : 0 → j j Since f is faithfully flat, F • ⊗ A B is the minimal graded free resolution of k [ T ] over B : B ( − dm n − 1 ,j ) β n − 1 ,j → · · · ⊕ F • ⊗ A B : 0 → j B ( − dm 1 j ) β 1 j → B → k [ T ] → 0 ⊕ → j 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 ( I T ) = µ ( I S ) , where µ ( I ) is the number of the minimal generators of I . 2 PF( T ) = { d f + ( d − 1) a n | f ∈ PF( S ) } . In particular, F( T ) = d F( S ) + ( d − 1) a n . 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 ( I T ) = µ ( I S ) , where µ ( I ) is the number of the minimal generators of I . 2 PF( T ) = { d f + ( d − 1) a n | f ∈ PF( S ) } . In particular, F( T ) = d F( S ) + ( d − 1) a n . 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 = � da 1 , ..., da n − 1 , a n � almost symmetric if it is not symmetric ? Takahiro Numata (Nihon University) A variation of gluing of numerical semigroups 9th September 2014 15 / 20