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.