One half of almost symmetric numerical semigroups

Let $S,T$ be two numerical semigroups. We study when $S$ is one half of $T$, with $T$ almost symmetric. If we assume that the type of $T$, $t(T)$, is odd, then for any $S$ there exist infinitely many such $T$ and we prove that $1 \leq t(T) \leq 2t(S)+1$. On the other hand, if $t(T)$ is even, there exists such $T$ if and only if $S$ is almost symmetric and different from $\mathbb{N}$; in this case the type of $S$ is the number of even pseudo-Frobenius numbers of $T$. Moreover, we construct these families of semigroups using the numerical duplication with respect to a relative ideal.Comment: 17 page

The numerical duplication of a numerical semigroup

In this paper we present and study the numerical duplication of a numerical semigroup, a construction that, starting with a numerical semigroup $S$ and a semigroup ideal $E\subseteq S$, produces a new numerical semigroup, denoted by S\Join^b\E (where $b$ is any odd integer belonging to $S$), such that S=(S\Join^b\E)/2. In particular, we characterize the ideals $E$ such that $S\Join^bE$ is almost symmetric and we determine its type.Comment: 17 pages. Accepted for publication on: Semigroup Foru

A family of quotients of the Rees algebra and rigidity properties of local cohomology modules

This thesis deals with several problems in Commutative Algebra and Numerical Semigroup Theory. Here a new family of rings is introduced and developed in order to give a unified approach to idealization and amalgamated duplication of a ring with respect to an ideal. We prove that a lot of properties are common to all the members of the family, like dimension, noetherianity, Cohen-Macaulayness, Gorensteinness and almost Gorensteinness and some other properties are studied, like spectra and localizations. Unlike idealization and amalgamated duplication, some other members of this family can be integral domains. In particular, if we start from a numerical semigroup ring, in this family there are infinitely many numerical semigroup rings. Using this new construction, we solve a problem posed by M.E. Rossi proving that there exist one-dimensional Gorenstein local rings with decreasing Hilbert function (at some level); moreover we prove that there is no bound to the decrease and construct infinitely many examples whose Hilbert function decreases at different levels. We also apply this construction in Numerical Semigroup Theory, where we introduce its counterpart: the numerical duplication. We use this to characterize all the almost symmetric doubles of a numerical semigroup, generalizing some results about symmetric and pseudo-symmetric doubles due to J.C. Rosales and P.A. García-Sánchez; we also prove the existence of some other almost symmetric multiples. Moreover, we solve a problem posed by A.M. Robles-Pérez, J.C. Rosales, and P. Vasco finding a formula for the minimal genus of the multiples of a given numerical semigroup and we do the same for the symmetric doubles. Finally, we find a formula for the Frobenius number of the quotients of some families of numerical semigroups. In the last chapter we prove a rigidity property of the Hilbert function of local cohomology modules with a support on the maximal ideal; more precisely, we prove that if the i-th local cohomology modules of an ideal of a polynomial ring and its lex-ideal have the same Hilbert functions, then the same happens for all the j-th local cohomology modules with j greater than i. Moreover, we introduce the notion of the i-partially sequentially Cohen-Macaulay modules in order to characterize the ideals for which their j-th local cohomology modules and those of their generic initial ideals have the same Hilbert functions for all j greater than i

A family of quotients of the Rees algebra

A family of quotient rings of the Rees algebra associated to a commutative ring is studied. This family generalizes both the classical concept of idealization by Nagata and a more recent concept, the amalgamated duplication of a ring. It is shown that several properties of the rings of this family do not depend on the particular member.Comment: 17 pages. To appear on "Communications in Algebra

One-dimensional Gorenstein local rings with decreasing Hilbert function

In this paper we solve a problem posed by M.E. Rossi: {\it Is the Hilbert function of a Gorenstein local ring of dimension one not decreasing? } More precisely, for any integer $h>1$, $h \notin\{14+22k, \, 35+46k \ | \ k\in\mathbb{N} \}$, we construct infinitely many one-dimensional Gorenstein local rings, included integral domains, reduced and non-reduced rings, whose Hilbert function decreases at level$h$; moreover we prove that there are no bounds to the decrease of the Hilbert function. The key tools are numerical semigroup theory, especially some necessary conditions to obtain decreasing Hilbert functions found by the first and the third author, and a construction developed by V. Barucci, M. D'Anna and the second author, that gives a family of quotients of the Rees algebra. Many examples are included