Local cohomology of bigraded modules [Elektronische Ressource] / vorgelegt von Ahad Rahimi

Description

Local cohomology of bigraded modulesDem Fachbereich 6Mathematik und Informatikder Universit¨at Duisburg-Essenzur Erlangung desDoktorgrades(Dr. rer. nat.)vorgelegt vonAhad Rahimiaus dem IranOctober 2007Vorsitzender: Prof. Dr. Wolfgang M. RuessErster Gutachter: Prof. Dr. Ju¨rgen HerzogZweiter Gutachter: Prof. Dr. Marc ChardinDedicated to my parents, my wife and my son AliAcknowledgmentsI am very grateful to my advisor Professor Ju¨rgen Herzog for his guidance, con-stant encouragement and support throughout the preparation of this thesis.I also express my gratitude to the university of Essen for its hospitality and thefacilities provided during my Ph.D study and research.Finally, I would like to thank the ministry of Science, Research and Technologyof Islamic Republic of Iran for the financial support offered me during my study.ContentsPreface 11 Preliminaries 41.1 Graded rings and graded modules . . . . . . . . . . . . . . . . . . . . 41.2 Local cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Castelnuovo-Mumford regularity and Hilbert function . . . . . . . . 91.4 Stanley-Reisner rings . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.5 Spectral sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.6 Tameness and local cohomology . . . . . . . . . . . . . . . . . . . . . 161.7 Graded and bigraded local cohomology . . . . . . . . . . . . . . . . .

Sujets

• Mathematics

Informations

• Publié par: universitat_duisburg-essen
• Publié le: 01 janvier 2007
• Nombre de lectures: 20

Extrait

Local

cohomology of bigraded

Dem Fachbereich 6 Mathematik und Informatik derUniversita¨tDuisburg-Essen zur Erlangung des Doktorgrades(Dr. rer. nat.)

vorgelegt von

Ahad Rahimi

aus dem Iran

October 2007

modules

Vorsitzender: Erster Gutachter: Zweiter Gutachter:

Prof. Prof. Prof.

Dr. Dr. Dr.

Wolfgang M. Ruess J¨urgenHerzog Marc Chardin

D

edicated

to

my

parents,

my

wife

and

my

son

Ali

Acknowledgments

IamverygratefultomyadvisorProfessorJu¨rgenHerzogforhisguidance,con- stant encouragement and support throughout the preparation of this thesis. I also express my gratitude to the university of Essen for its hospitality and the facilities provided during my Ph.D study and research. Finally, I would like to thank the ministry of Science, Research and Technology of Islamic Republic of Iran for the financial support offered me during my study.

Contents

Preface 1 1 Preliminaries 4 1.1 Graded rings and graded modules . . . . . . . . . . . . . . . . . . . . 4 1.2 Local cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Castelnuovo-Mumford regularity and Hilbert function . . . . . . . . 9 1.4 Stanley-Reisner rings . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.5 Spectral sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.6 Tameness and local cohomology . . . . . . . . . . . . . . . . . . . . . 16 1.7 Graded and bigraded local cohomology . . . . . . . . . . . . . . . . . 17 2 Regularity of local cohomology of bigraded algebras 20 2.1 Regularity of the graded components of local cohomology for modules of small dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 Hilbert function of the components of the top local cohomology of a hypersurface ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3 Regularity of the graded components of local cohomology for a special class of hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.4 Linear bounds for the regularity of the graded components of local cohomology for hypersurface rings . . . . . . . . . . . . . . . . . . . . 37 3 Local duality for bigraded modules 41 3.1 Proof of the duality theorem . . . . . . . . . . . . . . . . . . . . . . . 42 3.2 Some applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4 Local duality for graded modules 53 4.1 Proof of the duality theorem . . . . . . . . . . . . . . . . . . . . . . . 53 5 Tameness of local cohomology of monomial ideals with respect to monomial prime ideals 57 5.1 Local cohomology of Stanley-Reisner rings with respect to monomial prime ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.2 Local cohomology of monomial ideals with respect to monomial prime ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 References 73

Preface

Local cohomology is an extremely useful technique in commutative algebra and algebraic geometry. In this thesis we study local cohomology in various graded situ- ations. LetP 0 be a Noetherian ring,P=P 0 [y 1      y n ] be the polynomial ring over P 0 with the standard grading andP + = (y 1      y n ) the irrelevant graded ideal ofP. Then for any finitely generated gradedP-moduleM, the local cohomology modules H Pi + (M) are naturally gradedP-modules and each graded componentH i P + (M) j is a finitely generated gradedP 0 -module. The main topic of this thesis is to study the structure of theP 0 -modulesH i P + (M) j and their asymptotic behavior forj0. This thesis is organized as follows: We recall in the first chapter some basic definitions and known facts about graded rings and graded modules, local cohomol- ogy, Castelnuovo-Mumford regularity and Hilbert function, Stanley-Reisner rings, spectral sequences, tameness local cohomology and, graded and bigraded local co- homology. In Chapter 2 we consider the case thatP 0 =K[x 1      x m ] is a polyno- mial ring, so that theK-algebraPis naturally bigraded with degx i = (10) and degy i = (01). In this situation, ifMis a finitely generated bigradedP-module, then each of the modulesH i P + (M) is a bigradedP-module, and each H P i + (M) j = M H i P + (M) (kj) k is a finitely generated gradedP 0 -module whose grading is given by (H Pi + (M) j ) k = H i P + (M) (kj) . Thus it makes sense to investigate the regularity and Hilbert function of the gradedP 0 -modulesH Pi + (M) j . We show that ifMis a finitely generated bigradedP-module such that the dimension ofMP + MoverP 0 is at most one, then there exists an integercsuch that,−c≤regH Pi + (M) j ≤cfor alliand allj. The rest of this chapter is devoted to study the local cohomology of a hypersurface ringR=P f Pwheref∈Pis a bihomogeneous polynomial. show that We dim K H n P + (R) (kj) ≤dim K H P n + (R) (kj−1) for allk j In other words, the Hilbert series of theP 0 -moduleH Pn + (R) j is a nonincreasing func- tion ofj. In the special case thatH Pn + (R) j has finite length the regularity ofH Pn + (R) j is also a nonincreasing functions inj we compute the regularity of. NextH i P + (R) j for a special class of hypersurfaces. For the computation we use in an essential way a result of Stanley and J. Watanabe. They showed that a monomial complete intersection has the strong Lefschetz property. Using these facts the regularity and the Hilbert function ofH Pi + (P f λr P) j  Herecan be computed explicitly.r∈Nand f λ = P i n =1 λ i x i y i withλ i ∈K a consequence we see that. AsH P n + −1 (P f r P) j has a linear resolution and its Betti numbers can be computed. Finally we show that for any bigraded hypersurface ringR=P f Pfor which the idealI(f) generated

2

Preface

by all coefficients offism-primary wheremis the graded maximal ideal ofP 0 , the regularity ofH Pi + (R) j is linearly bounded inj. In Chapter 3 we prove a duality theorem for local cohomology of bigraded mod- ules. We letRbe a standard bigradedK-algebra with bigraded irrelevant idealsP generated by all elements of degree (10), andQgenerated by all elements of degree (01) andMbe a bigraded finitely generatedR-module. We define the bigraded Matlis-dual ofMto beM ∨ where the (i j)th bigraded component ofM ∨ is given by Hom K (M (−i−j)  K) We establish the following duality theorem: Theorem.LetRbe a standard bigradedK-algebra with irrelevant bigraded ideals PandQ, and letMbe a finitely generated bigradedR there exists-module. Then a convergent spectral sequence j−m E i 2 j =H P m−j  H R i + (M) ∨  = j ⇒H i Q + (M) ∨ of bigradedR-modules, wheremis the minimal number of homogeneous generators ofPandR + is the unique graded maximal ideal ofR. We show that the above spectral sequence degenerates whenMis Cohen-Macaulay and obtain for allkthe following isomorphims of bigradedR-modules H k P  H s R + (M) ∨  ∼ = H Q s−k (M) ∨ wheres= dimM let We, see Corollary 3.12.R 0 be theK-subalgebra ofRwhich is generated by the elements of bidegree (10) and setN=H Rs + (M) ∨ . ThenNis again ans-dimensional Cohen-Macaulay module and by the above isomorphism we obtain for alljthe isomorphisms of gradedR 0 -modules H P k 0 (N j ) = ∼  H s Q −k (M) −j  ∨ (1) whereP 0 is the graded maximal ideal ofR 0 . Here we used, thatH k P (N) j = ∼ H k P 0 (N j ) for allkandj. Brodmann and Hellus [4] raised the question whether the modulesH k Q (M) are tame ifMis a finitely generated gradedR other words, whether for-module. In eachkthere exists an integerj 0 such that eitherH k Q (M) j = 0 for allj≤j 0 , or else H k Q (M) j 6= 0 for allj≤j 0 various cases this problem has been answered in the. In  affirmative, see [3], [4], [24], [18], [20] and [2] for a survey on this problem. In case Mis Cohen-Macaulay the tameness problem translates, due to (1), to the following question: Given a finitely generated bigradedR-moduleN. Does there exist an integerj 0 such thatH k P 0 (N j ) = 0 for allj≥j 0 , or elseH Pk 0 (N j )6= 0 for allj≥j 0 ? More generally, one would expect that for a finitely generated gradedR 0 -moduleW and a finitely generated bigradedR-moduleNthere exists for allkan integerj 0 such that Ext R k 0 (N j  W) = 0 for allj≥j 0 , or else Ext k R 0 (N j  W)6= 0 for allj≥j 0 . However this is not the case as has been recently shown by Cutkosky and Herzog, see [10]. Their example also provides a counterexample to the general tameness