Numerical Semigroup Algebra Numerical Semigroup Algebra Joint with Kee, Mee-Kyoung International meeting on numerical semigroups with applications - Levico Terme 2016 1 / 63
Numerical Semigroup Algebra Introduction We are interested in properties of numerical semigroup rings, such as Cohen-Macaulyness, Gorensteiness and complete intersection. We emphasize morphisms rather than objects. Ap´ ery sets are relative notion. Gluing is an operation of adding radicals. 2 / 63
Numerical Semigroup Algebra Introduction We are interested in properties of numerical semigroup rings, such as Cohen-Macaulyness, Gorensteiness and complete intersection. We emphasize morphisms rather than objects. Ap´ ery sets are relative notion. Gluing is an operation of adding radicals. 3 / 63
Numerical Semigroup Algebra Introduction We are interested in properties of numerical semigroup rings, such as Cohen-Macaulyness, Gorensteiness and complete intersection. We emphasize morphisms rather than objects. Ap´ ery sets are relative notion. Gluing is an operation of adding radicals. 4 / 63
Numerical Semigroup Algebra Introduction We are interested in properties of numerical semigroup rings, such as Cohen-Macaulyness, Gorensteiness and complete intersection. We emphasize morphisms rather than objects. Ap´ ery sets are relative notion. Gluing is an operation of adding radicals. 5 / 63
Numerical Semigroup Algebra Numerical semigroup rings Let S be a numerical semigroup and κ be a field. Being a subring of the power series ring κ [[ u ]], the numerical semigroup ring κ [[ u S ]] consists of power series s ∈ S a s u s with coefficients a s ∈ κ . � Let t be the smallest non-zero number of S . The monoid S / t is called a normalized numerical semigroup. A numerical semigroup is a monoid generated by finitely many positive rational numbers. If S is generated by s 1 , · · · , s n , then we write κ [[ u S ]] = κ [[ u s 1 , · · · , u s n ]]. If S is normalized, we often write the numerical semigroup ring as κ [[ e S ]]. 6 / 63
Numerical Semigroup Algebra Numerical semigroup rings Let S be a numerical semigroup and κ be a field. Being a subring of the power series ring κ [[ u ]], the numerical semigroup ring κ [[ u S ]] consists of power series s ∈ S a s u s with coefficients a s ∈ κ . � Let t be the smallest non-zero number of S . The monoid S / t is called a normalized numerical semigroup. A numerical semigroup is a monoid generated by finitely many positive rational numbers. If S is generated by s 1 , · · · , s n , then we write κ [[ u S ]] = κ [[ u s 1 , · · · , u s n ]]. If S is normalized, we often write the numerical semigroup ring as κ [[ e S ]]. 7 / 63
Numerical Semigroup Algebra Numerical semigroup rings Let S be a numerical semigroup and κ be a field. Being a subring of the power series ring κ [[ u ]], the numerical semigroup ring κ [[ u S ]] consists of power series s ∈ S a s u s with coefficients a s ∈ κ . � Let t be the smallest non-zero number of S . The monoid S / t is called a normalized numerical semigroup. A numerical semigroup is a monoid generated by finitely many positive rational numbers. If S is generated by s 1 , · · · , s n , then we write κ [[ u S ]] = κ [[ u s 1 , · · · , u s n ]]. If S is normalized, we often write the numerical semigroup ring as κ [[ e S ]]. 8 / 63
Numerical Semigroup Algebra Numerical semigroup rings Let S be a numerical semigroup and κ be a field. Being a subring of the power series ring κ [[ u ]], the numerical semigroup ring κ [[ u S ]] consists of power series s ∈ S a s u s with coefficients a s ∈ κ . � Let t be the smallest non-zero number of S . The monoid S / t is called a normalized numerical semigroup. A numerical semigroup is a monoid generated by finitely many positive rational numbers. If S is generated by s 1 , · · · , s n , then we write κ [[ u S ]] = κ [[ u s 1 , · · · , u s n ]]. If S is normalized, we often write the numerical semigroup ring as κ [[ e S ]]. 9 / 63
Numerical Semigroup Algebra Numerical semigroup rings Let S be a numerical semigroup and κ be a field. Being a subring of the power series ring κ [[ u ]], the numerical semigroup ring κ [[ u S ]] consists of power series s ∈ S a s u s with coefficients a s ∈ κ . � Let t be the smallest non-zero number of S . The monoid S / t is called a normalized numerical semigroup. A numerical semigroup is a monoid generated by finitely many positive rational numbers. If S is generated by s 1 , · · · , s n , then we write κ [[ u S ]] = κ [[ u s 1 , · · · , u s n ]]. If S is normalized, we often write the numerical semigroup ring as κ [[ e S ]]. 10 / 63
Numerical Semigroup Algebra Numerical semigroup rings Let S be a numerical semigroup and κ be a field. Being a subring of the power series ring κ [[ u ]], the numerical semigroup ring κ [[ u S ]] consists of power series s ∈ S a s u s with coefficients a s ∈ κ . � Let t be the smallest non-zero number of S . The monoid S / t is called a normalized numerical semigroup. A numerical semigroup is a monoid generated by finitely many positive rational numbers. If S is generated by s 1 , · · · , s n , then we write κ [[ u S ]] = κ [[ u s 1 , · · · , u s n ]]. If S is normalized, we often write the numerical semigroup ring as κ [[ e S ]]. 11 / 63
Numerical Semigroup Algebra Numerical semigroup rings Let S be a numerical semigroup and κ be a field. Being a subring of the power series ring κ [[ u ]], the numerical semigroup ring κ [[ u S ]] consists of power series s ∈ S a s u s with coefficients a s ∈ κ . � Let t be the smallest non-zero number of S . The monoid S / t is called a normalized numerical semigroup. A numerical semigroup is a monoid generated by finitely many positive rational numbers. If S is generated by s 1 , · · · , s n , then we write κ [[ u S ]] = κ [[ u s 1 , · · · , u s n ]]. If S is normalized, we often write the numerical semigroup ring as κ [[ e S ]]. 12 / 63
Numerical Semigroup Algebra Numerical semigroup rings Let S be a numerical semigroup and κ be a field. Being a subring of the power series ring κ [[ u ]], the numerical semigroup ring κ [[ u S ]] consists of power series s ∈ S a s u s with coefficients a s ∈ κ . � Let t be the smallest non-zero number of S . The monoid S / t is called a normalized numerical semigroup. A numerical semigroup is a monoid generated by finitely many positive rational numbers. If S is generated by s 1 , · · · , s n , then we write κ [[ u S ]] = κ [[ u s 1 , · · · , u s n ]]. If S is normalized, we often write the numerical semigroup ring as κ [[ e S ]]. 13 / 63
Numerical Semigroup Algebra Numerical semigroup algebras A morphism of numerical semigroups S 1 → S 2 is a multiplication by a positive rational number t satisfying tS 1 ⊂ S 2 . For a morphism t : S 1 → S 2 of numerical semigroups, there is an inclusion κ [[ v S 1 ]] → κ [[ u S 2 ]] identifying v s with u ts for s ∈ S 1 . We call κ [[ u S 2 ]] /κ [[ v S 1 ]] a numerical semigroup algebra. Elements of κ [[ v S 1 ]] are called coefficients of the algebra. If t = 1, we use the notation κ [[ u S 2 ]] /κ [[ u S 1 ]] for the algebra. To study a ring κ [[ e S ]], it is the same thing to study the algebra κ [[ e S ]] /κ [[ e ]] with coefficients in the power series ring κ [[ e ]]. 14 / 63
Numerical Semigroup Algebra Numerical semigroup algebras A morphism of numerical semigroups S 1 → S 2 is a multiplication by a positive rational number t satisfying tS 1 ⊂ S 2 . For a morphism t : S 1 → S 2 of numerical semigroups, there is an inclusion κ [[ v S 1 ]] → κ [[ u S 2 ]] identifying v s with u ts for s ∈ S 1 . We call κ [[ u S 2 ]] /κ [[ v S 1 ]] a numerical semigroup algebra. Elements of κ [[ v S 1 ]] are called coefficients of the algebra. If t = 1, we use the notation κ [[ u S 2 ]] /κ [[ u S 1 ]] for the algebra. To study a ring κ [[ e S ]], it is the same thing to study the algebra κ [[ e S ]] /κ [[ e ]] with coefficients in the power series ring κ [[ e ]]. 15 / 63
Numerical Semigroup Algebra Numerical semigroup algebras A morphism of numerical semigroups S 1 → S 2 is a multiplication by a positive rational number t satisfying tS 1 ⊂ S 2 . For a morphism t : S 1 → S 2 of numerical semigroups, there is an inclusion κ [[ v S 1 ]] → κ [[ u S 2 ]] identifying v s with u ts for s ∈ S 1 . We call κ [[ u S 2 ]] /κ [[ v S 1 ]] a numerical semigroup algebra. Elements of κ [[ v S 1 ]] are called coefficients of the algebra. If t = 1, we use the notation κ [[ u S 2 ]] /κ [[ u S 1 ]] for the algebra. To study a ring κ [[ e S ]], it is the same thing to study the algebra κ [[ e S ]] /κ [[ e ]] with coefficients in the power series ring κ [[ e ]]. 16 / 63
Numerical Semigroup Algebra Numerical semigroup algebras A morphism of numerical semigroups S 1 → S 2 is a multiplication by a positive rational number t satisfying tS 1 ⊂ S 2 . For a morphism t : S 1 → S 2 of numerical semigroups, there is an inclusion κ [[ v S 1 ]] → κ [[ u S 2 ]] identifying v s with u ts for s ∈ S 1 . We call κ [[ u S 2 ]] /κ [[ v S 1 ]] a numerical semigroup algebra. Elements of κ [[ v S 1 ]] are called coefficients of the algebra. If t = 1, we use the notation κ [[ u S 2 ]] /κ [[ u S 1 ]] for the algebra. To study a ring κ [[ e S ]], it is the same thing to study the algebra κ [[ e S ]] /κ [[ e ]] with coefficients in the power series ring κ [[ e ]]. 17 / 63
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