Multigraded Structures and the Depth of Blow-up Algebras
A first goal of this thesis is to contribute to the knowledge of cohomological properties of non-standard multigraded modules. In particular we study the Hilbert function of a non-standard multigraded module, the asymptotic depth of the homogeneous components of a multigraded module and the asymptot...
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| Tipo de recurso: | tesis doctoral |
| Estado: | Versión publicada |
| Fecha de publicación: | 2008 |
| País: | España |
| Institución: | CBUC, CESCA |
| Repositorio: | TDR. Tesis Doctorales en Red |
| OAI Identifier: | oai:www.tdx.cat:10803/663 |
| Acceso en línea: | http://www.tdx.cat/TDX-1014108-120203 http://hdl.handle.net/10803/663 |
| Access Level: | acceso abierto |
| Palabra clave: | Mòduls de Veronese Funció de Hilbert Cohomologia Àlgebra de Blow-Up Estructures multigraduades (Matemàtica) Ciències Experimentals i Matemàtiques 512 |
| Sumario: | A first goal of this thesis is to contribute to the knowledge of cohomological properties of non-standard multigraded modules. In particular we study the Hilbert function of a non-standard multigraded module, the asymptotic depth of the homogeneous components of a multigraded module and the asymptotic depth of the Veronese modules. To reach our purposes, we generalize some cohomological invariants to the non-standard multigraded case and we study properties on the vanishing of local cohomology modules. In particular we study the generalized depth of a multigraded module.<br/><br/>In chapters 2, 3 and 4, we consider multigraded rings S, finitely generated over the local ring S0 by elements of degrees g1,.,gr with gi=(g1i,.,gii,.,0) non-negative integral vectors and gii not zero for i=1,.,r. In Chapter 2, we prove that the Hilbert function of a multigraded S-module is quasi-polynomial in a cone of N^r. Moreover the Grothendieck-Serre formula is satisfied in our situation as well.<br/><br/>In Chapter 3, using the quasi-polynomial behavior of the Hilbert function of the Koszul homology modules of a multigraded S-module M with respect to a system of generators of the maximal ideal of S0, we can prove that the depth of the homogeneous components of M is constant for degrees in a subnet of a cone of N^r defined by g1,.,gr. In some cases we can assure constant depth in all the cone. By considering the multigraded blow-up algebras associated to ideals I1,.,Ir in a Noetherian local ring (R,m), we can prove that the depth of R/I1^n1.Ir^nr is constant for n1,.,nr large enough.<br/><br/>In Chapter 4, we study the depth of (a,b)-Veronese modules for a, b large enough. In particular we prove that in almost-standard case (i.e. the degrees of the generators are positive multiples of the canonical basis) with S0 a quotient of a regular local ring, this depth is constant for a, b in some regions of N^r. To reach this result we need a previous study about Veronese modules and about the vanishing of local cohomology modules. In particular we prove that, in the more<br/>general case, if S0 is a quotient of a regular local ring, the generalized depth is invariant by taking Veronese transforms. Moreover in the almost-standard case the generalized depth coincides with the index of finite graduation of the local cohomology modules with respect to the homogeneous maximal ideal.<br/><br/>A second goal of the thesis is the study of the depth of blow-up algebras associated to an ideal. In Chapter 5 we obtain refined versions of some conjectures on the depth of the associated graded ring of an ideal. By using certain non-standard bigraded structures, the integers that appear in Guerrieri's Conjecture and in Wang's Conjecture can be interpreted as a multiplicities of some bigraded modules. In particular we have given an answer to the question formulated by A. Guerrieri and C. Huneke in 1993. We have proved that given an m-primary ideal I in a Cohen-Macaulay local ring (R,m) of dimension d>0 with minimal reduction J, assuming that the lengths of the homogeneous components of the Valabrega-Valla module of I and J are less than or equal to 1, then the depth of the associated graded ring of I is greater than or equal to d-2.<br/><br/>Finally, in Chapter 6, the study of the Hilbert function of certain submodules of the bigraded modules studied before, allows us to prove some cases in which the Hilbert function of an m-primary ideal in a one-dimensional Cohen-Macaulay local ring is non-decreasing. |
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