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  • Koszul Algebras, Castelnuovo-Mumford Regularity, and Generic InitialIdeals

    by

    Giulio Caviglia

    Submitted to the Department of Mathematics

    and the Faculty of the Graduate School

    of the University of Kansas

    in partial fulfillment of the requirements

    for the degree of Doctor of Philosophy

    Thesis Committee:

    Professor Craig Huneke, Chair

    Professor Arvin Agah

    Professor Rober Brown

    Professor Daniel Katz

    Professor Bangere Purnaprajna

  • c Copyright 2004

    by

    Giulio Caviglia

    All Rights Reserved

    ii

  • Abstract

    KOSZUL ALGEBRAS, CASTELNUOVO-MUMFORD REGULARITY, AND

    GENERIC INITIAL IDEALS

    Giulio Caviglia

    The University of Kansas

    Advisor: Craig Huneke

    August, 2004

    The central topics of this dissertation are: Koszul Algebras, bounds for the Castel-

    nuovo Mumford regularity, and methods involving the use of generic changes

    of coordinates and generic hyperplane restrictions. We give an introduction to

    Koszul algebras and prove some criteria to show that an algebra is Koszul. We

    use these methods to show that the Pinched Veronese, i.e. the toric ring defined as

    K[X3,X2Y,XY 2,Y 3,X2Z,Y 2Z,XZ2,Y Z2,Z3], is Koszul.

    The middle chapters are devoted to the Castelnuovo-Mumford regularity. We

    give a collection of techniques and formulas to compute the regularity by using

    hyperplane sections. For example we obtain some variations of a criterion due to

    Bayer and Stillman and a formula for the regularity that involves the postulation

    numbers.

    We study the combinatorial properties of a special kind of monomial ideal that

    we call weakly stable. We employ them to give a uniform bound, depending on

    the degree of the generators, for the regularity of all the homogeneous ideals in a

    polynomial ring. We also provide bounds for the regularity of the tensor product

    and Hom of two modules.

    iii

  • In chapter seven we study some inequalities on the dimension of graded com-

    ponents of Tors, and in the last chapter we present a modification of Greens

    Hyperplane Restriction Theorem. By using this restriction theorem we obtain a

    general strategy to derive variations of the Eakin-Sathaye Theorem on reductions.

    iv

  • Acknowledgment

    First of all my heartfelt thanks go to my advisor, Craig Huneke: his suggestions,

    his encouragements and his constant presence have been invaluable throughout

    these years. Most importantly, I will always keep in mind both his way of thinking

    as a mathematician and his dedication as a mentor. He really represents the kind

    of mathematician I would like to be.

    I am grateful to the whole algebra group at the University of Genoa for their

    help. I would like particularly to thank Aldo Conca for all the inspiring conver-

    sations we had. His friendly way of sharing his thoughts gave me many ideas

    and topics around which my thesis developed. Marilina Rossi and expecially Tito

    Valla have always given me important and sensible advice and continuous sup-

    port. My thanks are also extended to Enrico Sbarra for our collaboration. In fact,

    chapters four and five arose from joint work with him.

    I would like to thank Arvin Agah, Robert Brown, Daniel Katz and Bangere

    Purnaprajna, who, together with Craig Huneke, are the members of my thesis

    committee.

    I am grateful to David Eisenbud, Estela Gavosto, Mike Stillman, Bernd Sturm-

    fels and, once again, Aldo Conca and Craig Huneke for all the support they gave

  • me and the time they spent helping me to apply for postdoctoral positions. I would

    also like to thank Dale Cutkosky and Hema Srinivasan, Juan Migliore and Clau-

    dia Polini, Bernd Ulrich, Mike Stillman and Irena Peeva, and Ngo Viet Trung for

    giving me the opportunity to present my results, and for discussing the material

    with me.

    On a personal note my deepest gratitude goes to my mom, my dad, my grand-

    parents and my aunt Bina for making me feel their love and encouragement even

    when so many miles divided us.

    I would like to thank Stefano for his friendship and support in several critical

    moments. His help, together with my officemate Janets, was much appreciated.

    Thanks also to Silvia, Carmen and Ross, Olga, Mirco, Eric, Bill, Ananth,

    Manoj, Neil, Adam, Daniele, Elven, Marco, Antonio, Matteo, my cousin Rick

    and his family, and to all my other friends in Italy and everywhere else.

    During the first three years of my graduate studies, I was supported by a fel-

    lowship from the Italian Institute for Higher Mathematics Istituto Nazionale di

    Alta Matematica F. Severi. My fourth year has been supported by the Graduate

    Dissertation Fellowship from the University of Kansas. I thank both institutions.

    vi

  • Contents

    Abstract v

    Acknowledgment vii

    Introduction ix

    1 Koszul algebras 1

    1.1 Koszul filtrations . . . . . . . . . . . . . . . . . . . . . . . . . . 8

    1.2 Quadratic Grobner bases and weight functions . . . . . . . . . . . 12

    2 The Pinched Veronese is Koszul 16

    3 Castelnuovo-Mumford regularity and Hyperplane sections 24

    3.1 Castelnuovo-Mumford regularity . . . . . . . . . . . . . . . . . . 25

    3.1.1 Partial Regularities and short exact sequences . . . . . . . 25

    3.1.2 Regularity of a filter regular hyperplane section . . . . . . 27

    3.2 Equivalent definitions of regularity using hyperplane sections . . . 29

    3.2.1 Regularity and Postulation Numbers . . . . . . . . . . . . 30

    vii

  • 3.2.2 Regularity and hyperplane sections: a general approach . . 32

    3.2.3 Bayer and Stillman criterion for detecting regularity and

    some similar further criteria . . . . . . . . . . . . . . . . 34

    3.2.4 Crystallization principle . . . . . . . . . . . . . . . . . . 38

    4 Weakly Stable ideals 40

    4.1 General properties of Weakly Stable ideals . . . . . . . . . . . . . 41

    4.2 Ideals with high Castelnuovo-Mumford regularity . . . . . . . . . 51

    5 Uniform bounds for the Castelnuovo-Mumford regularity 54

    5.1 A Bound for the Castelnuovo-Mumford regularity in term of filter

    regular sections . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

    5.2 Doubly exponential bound for the Castelnuovo-Mumford regularity 59

    6 Bounds on the regularity of tensor product and Hom of modules 62

    6.1 Castelnuovo-Mumford regularity and complexes of modules . . . 63

    6.1.1 Bounds on the regularity of the tensor product . . . . . . . 65

    6.1.2 Bounds on the regularity of HomR(M,N) . . . . . . . . . 67

    7 Initial ideals, Lex-segments ideals and inequalities on Tor 69

    7.1 Initial ideals and inequalities on Tors . . . . . . . . . . . . . . . 71

    7.2 Pardues method and Macaulay estimate on the Hilbert function

    of standard graded algebras . . . . . . . . . . . . . . . . . . . . . 74

    7.2.1 Polarizations and specializations of monomial ideals . . . 74

    7.2.2 A total order on the monomial ideals . . . . . . . . . . . . 78

    7.2.3 Results of Macaulay and Pardue . . . . . . . . . . . . . . 80

    viii

  • 8 Variations on a Theorem of Eakin and Sathaye and on Greens Hy-

    perplane Restriction Theorem 83

    8.1 Macaulay representation of integer numbers . . . . . . . . . . . . 84

    8.2 Greens Hyperplane Restriction Theorem . . . . . . . . . . . . . 85

    8.3 Variations of the Theorem of Eakin and Sathaye . . . . . . . . . . 94

    References 96

    ix

  • Introduction

    In this dissertation we mainly study the three following topics: Koszul Algebras,

    bounds for the Castelnuovo Mumford regularity, and methods involving the use of

    generic changes of coordinates and generic hyperplane restrictions. The general

    approach used in this thesis that unifies the three subjects is trying to reduce the

    problems we consider to the study of initial ideals. More generally, the whole

    thesis is tied together by the constant effort to compare homological invariants of

    a special fiber of a flat family to the ones of a generic fiber.

    The first Chapter gives an introduction to the study of Koszul algebra. All the

    results presented there are quite standard except for an extension of the notion of

    Koszul filtration which is needed for the second chapter.

    In the second Chapter we prove that a certain toric ring:

    R = K[X3,X2Y,XY 2,Y 3,X2Z,Y 2Z,XZ2,Y Z2,Z3],

    called the Pinched Veronese, is Koszul. For about ten years this ring has been

    the most famous example for which the Koszulness was unknown . The problem

    about the Koszulness of the pinched Veronese was raised by B.Sturmfels in 1993

    in a conversation with Irena Peeva. Ever since it has been circulating as a concrete

  • example to test the efficiency of the new theorems and techniques concerning

    Koszul algebras. As far as we know, the main approach employed to attack this

    problem was the use of techniques particularly suited for studying semigroup rings

    and their associated polytopes. Following a different strategy we achieved our goal

    by using a combination of flat deformations and Koszul filtrations.

    The Chapters three, four, five and six are dedicated to the study of the Castel-

    nuovo-Mumford regularity.

    In Chapter three we give a collection of techniques and formulas to compute

    the regularity by using hyperplane sections. Some results presented here include

    a formula for the regularity that involves the postulation numbers, and several