Multigraded Structures and the Depth of Blow-up Algebras
[eng] 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 as...
| Autor: | |
|---|---|
| Tipo de recurso: | tesis doctoral |
| Estado: | Versión publicada |
| Fecha de publicación: | 2008 |
| País: | España |
| Institución: | Universidad de Barcelona |
| Repositorio: | Dipòsit Digital de la UB |
| OAI Identifier: | oai:diposit.ub.edu:2445/35140 |
| Acceso en línea: | https://hdl.handle.net/2445/35140 http://www.tdx.cat/TDX-1014108-120203 http://hdl.handle.net/10803/663 |
| Access Level: | acceso abierto |
| Palabra clave: | Àlgebra homològica Àlgebra commutativa Funcions característiques Geometria algebraica Esclatament (Matemàtica) Blowing up (Algebraic geometry) Algebra, Homological Commutative algebra Characteristic functions Algebraic geometry |
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Multigraded Structures and the Depth of Blow-up Algebras |
| title |
Multigraded Structures and the Depth of Blow-up Algebras |
| spellingShingle |
Multigraded Structures and the Depth of Blow-up Algebras Colomé Nin, Gemma Àlgebra homològica Àlgebra commutativa Funcions característiques Geometria algebraica Esclatament (Matemàtica) Blowing up (Algebraic geometry) Algebra, Homological Commutative algebra Characteristic functions Algebraic geometry |
| title_short |
Multigraded Structures and the Depth of Blow-up Algebras |
| title_full |
Multigraded Structures and the Depth of Blow-up Algebras |
| title_fullStr |
Multigraded Structures and the Depth of Blow-up Algebras |
| title_full_unstemmed |
Multigraded Structures and the Depth of Blow-up Algebras |
| title_sort |
Multigraded Structures and the Depth of Blow-up Algebras |
| dc.creator.none.fl_str_mv |
Colomé Nin, Gemma |
| author |
Colomé Nin, Gemma |
| author_facet |
Colomé Nin, Gemma |
| author_role |
author |
| dc.contributor.none.fl_str_mv |
Universitat de Barcelona. Departament d'Àlgebra i Geometria |
| dc.subject.none.fl_str_mv |
Àlgebra homològica Àlgebra commutativa Funcions característiques Geometria algebraica Esclatament (Matemàtica) Blowing up (Algebraic geometry) Algebra, Homological Commutative algebra Characteristic functions Algebraic geometry |
| topic |
Àlgebra homològica Àlgebra commutativa Funcions característiques Geometria algebraica Esclatament (Matemàtica) Blowing up (Algebraic geometry) Algebra, Homological Commutative algebra Characteristic functions Algebraic geometry |
| description |
[eng] 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. 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. 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. 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 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. 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. 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. |
| publishDate |
2008 |
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2008 |
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info:eu-repo/semantics/doctoralThesis info:eu-repo/semantics/publishedVersion |
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doctoralThesis |
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publishedVersion |
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https://hdl.handle.net/2445/35140 http://www.tdx.cat/TDX-1014108-120203 http://hdl.handle.net/10803/663 |
| url |
https://hdl.handle.net/2445/35140 http://www.tdx.cat/TDX-1014108-120203 http://hdl.handle.net/10803/663 |
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Inglés |
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Inglés |
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(c) Colomé Nin, 2008 info:eu-repo/semantics/openAccess |
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(c) Colomé Nin, 2008 |
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openAccess |
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application/pdf |
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Universitat de Barcelona |
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Universitat de Barcelona |
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Tesis Doctorals - Departament - Algebra i Geometria reponame:Dipòsit Digital de la UB instname:Universidad de Barcelona |
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Universidad de Barcelona |
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Dipòsit Digital de la UB |
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Dipòsit Digital de la UB |
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1869404328636186624 |
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Multigraded Structures and the Depth of Blow-up AlgebrasColomé Nin, GemmaÀlgebra homològicaÀlgebra commutativaFuncions característiquesGeometria algebraicaEsclatament (Matemàtica)Blowing up (Algebraic geometry)Algebra, HomologicalCommutative algebraCharacteristic functionsAlgebraic geometry[eng] 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. 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. 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. 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 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. 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. 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.[cat] TÍTOL DE LA TESI: "Estructures Multigraduades i la Profunditat d'Àlgebres de Blow-up" TEXT DEL RESUM: Un primer objectiu d'aquesta tesi és contribuir al coneixement de propietats cohomològiques de mòduls multigraduats no-estàndard. En particular estudiem la funció de Hilbert d'un mòdul multigraduat no-estàndard, la profunditat asimptòtica de les components homogènies d'un mòdul multigraduat i la profunditat asimptòtica dels mòduls de Veronese. Per a això, generalitzem alguns invariants cohomològics en el cas multigraduat no-estàndard i estudiem propietats d'anul·lació de mòduls de cohomologia local. En particular estudiem la profunditat generalitzada d'un mòdul multigraduat. En els capítols 2, 3 i 4, considerem anells multigraduats S finitament generats sobre l'anell local S0 per elements de graus g1,...,gr amb gi=(g1i,...,gii,...,0) vectors enters no-negatius i gii no nul per a i=1,...,r. Al Capítol 2, demostrem que la funció de Hilbert d'un S-mòdul multigraduat és quasi-polinòmica en un con de N^r. A més es satisfà la fórmula de Grothendieck-Serre en la nostra situació. Al Capítol 3, utilitzant el comportament quasi-polinòmic de la funció de Hilbert dels mòduls d'homologia de Koszul d'un S-mòdul M multigraduat respecte d'un sistema de generadors de l'ideal maximal de S0, podem demostrar que la profunditat de les components homogènies de M és constant per a graus en una subxarxa d'un con de N^r definit per g1,...,gr. En alguns casos es pot assegurar profunditat constant en tot un con. Considerant els anells de blow-up multigraduats associats a ideals I1,...,Ir en un anell local Noetherià (R,m), podem demostrar que la profunditat de R/I1^n1...Ir^nr és constant per a n1,...,nr prou grans. Al Capítol 4, estudiem la profunditat dels mòduls de (a,b)-Veronese per a a,b prou grans. En particular demostrem que en el cas quasi-estàndard (i.e. amb generadors de graus múltiples positius de la base canònica) amb S0 quocient d'un anell local regular, aquesta profunditat és constant per a a,b en certes regions de N^r. Per arribar a aquest resultat ens cal un estudi previ dels mòduls de Veronese i de l'anul·lació de mòduls de cohomologia local. En particular demostrem que, en el cas més general, si S0 és quocient d'un anell local regular, la profunditat generalitzada és invariant per transformacions Veronese. A més en el cas quasi-estàndard la profunditat generalitzada coincideix amb l'índex de graduació finita dels mòduls de cohomologia local respecte de l'ideal homogeni maximal. Un segon objectiu de la tesi és l'estudi de la profunditat de les àlgebres de blow-up associades a un ideal. Al Capítol 5 s'obtenen versions refinades de conjectures sobre la profunditat de l'anell graduat associat a un ideal. Utilitzant algunes estructures bigraduades no-estàndard, es poden interpretar els enters que apareixen a la Conjectura de Guerrieri i a la Conjectura de Wang com a multiplicitats de mòduls bigraduats. En particular hem pogut donar resposta a una pregunta formulada per A. Guerrieri i C. Huneke al 1993. Hem demostrat que donat un ideal I m-primari en un anell local (R,m) Cohen-Macaulay de dimensió d>0 amb reducció minimal J, suposant que les longituds de les components homogènies del mòdul de Valabrega-Valla de I i J siguin menors o iguals que 1, aleshores la profunditat de l'anell graduat associat a I és major o igual que d-2. Finalment, al Capítol 6, l'estudi de la funció de Hilbert de certs submòduls dels mòduls bigraduats estudiats anteriorment, permet provar alguns casos en què la funció de Hilbert d'un ideal m-primari en un anell local Cohen-Macaulay de dimensió 1, és no decreixent.Universitat de BarcelonaUniversitat de Barcelona. Departament d'Àlgebra i Geometria2008info:eu-repo/semantics/doctoralThesisinfo:eu-repo/semantics/publishedVersionapplication/pdfhttps://hdl.handle.net/2445/35140http://www.tdx.cat/TDX-1014108-120203http://hdl.handle.net/10803/663Tesis Doctorals - Departament - Algebra i Geometriareponame:Dipòsit Digital de la UBinstname:Universidad de BarcelonaInglés(c) Colomé Nin, 2008info:eu-repo/semantics/openAccessoai:diposit.ub.edu:2445/351402026-05-27T06:46:51Z |
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15,300719 |