On the Apéry sets of monomial curves
Autor: | Raheleh Jafari, Teresa Cortadellas Benítez, Santiago Zarzuela Armengou |
---|---|
Rok vydání: | 2012 |
Předmět: |
Monomial
Pure mathematics Commutative rings Anells commutatius Commutative Algebra (math.AC) Àlgebra commutativa Mathematics - Algebraic Geometry symbols.namesake Numerical semigroup FOS: Mathematics Algebraic Geometry (math.AG) Mathematics Discrete mathematics Hilbert series and Hilbert polynomial Ring (mathematics) Algebra and Number Theory Mathematics::Commutative Algebra Tangent cone Local ring Tangent Mathematics - Commutative Algebra Anells locals Local rings symbols Embedding Commutative algebra |
Zdroj: | Semigroup Forum. 86:289-320 |
ISSN: | 1432-2137 0037-1912 |
DOI: | 10.1007/s00233-012-9445-8 |
Popis: | In this paper, we use the Ap\'ery table of the numerical semigroup associated to an affine monomial curve in order to characterize arithmetic properties and invariants of its tangent cone. In particular, we precise the shape of the Ap\'ery table of a numerical semigroup of embedding dimension 3, when the tangent cone of its monomial curve is Buchsbaum or 2-Buchsbaum, and give new proofs for two conjectures raised by V. Sapko (Commun. Algebra {29}:4759-4773, 2001) and Y. H. Shen (Commun. Algebra {39}:1922-1940, 2001). We also provide a new simple proof in the case of monomial curves for Sally's conjecture (Numbers of Generators of Ideals in Local Rings, 1978) that the Hilbert function of a one-dimensional Cohen-Macaulay ring with embedding dimension three is non-decreasing. Finally, we obtain that monomial curves of embedding dimension 4 whose tangent cones are Buchsbaum, and also monomial curves of any embedding dimensions whose numerical semigroups are balanced, have non-decreasing Hilbert functions. Numerous examples are provided to illustrate the results, most of them computed by using the NumericalSgps package of GAP (Delgado et al., NumericalSgps-a GAP package, 2006). Comment: To appear in Semigroup Forum |
Databáze: | OpenAIRE |
Externí odkaz: |