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Various Differents for 0-Dimensional Schemes and Applications Dissertation Various Differents for 0-Dimensional Schemes and Applications Le Ngoc Long Eingereicht an der Fakult¨atf¨urInformatik und Mathematik der Universit¨atPassau als Dissertation zur Erlangung des Grades eines Doktors der Naturwissenschaften Submitted to the Department of Informatics and Mathematics of the Universit¨atPassau in Partial Fulfilment of the Requirements for the Degree of a Doctor in the Domain of Science Betreuer / Advisor: Prof. Dr. Martin Kreuzer Universit¨atPassau June 2015 Various Differents for 0-Dimensional Schemes and Applications Le Ngoc Long Erstgutachter: Prof. Dr. Martin Kreuzer Zweitgutachter: Prof. Dr. J¨urgenHerzog M¨undliche Pr¨ufer: Prof. Dr. Tobias Kaiser Prof. Dr. Brigitte Forster-Heinlein Der Fakult¨atf¨urInformatik und Mathematik der Universit¨atPassau vorgelegt im Juni 2015 To my parents, my wife and my daughter Acknowledgements This thesis would not have been written without the help and support of many people and institutions. First and foremost, I am deeply indebted to my supervisor, Prof. Dr. Martin Kreuzer, for his help, guidance and encouragement during my research. I thank him for also being free with his time to answer my many questions and for his endless support inside and outside mathematics. Many arguments in this thesis are extracted from his suggestions. I would especially like to express my gratitude to him for giving me and my wife an opportunity to study together in Passau. I would like to thank the hospitality and support of the Department of Mathematics and Informatics of the University of Passau during these years, and in particular, I am grateful to the members of my examination committee, especially Prof. Dr. Tobias Kaiser and Prof. Dr. Brigitte Forster-Heinlein, for their comments and suggestions. I would like to express my gratitude to Prof. Dr. J¨urgen Herzog for his willingness to be the second referee of the thesis. I am grateful to Prof. Dr. Lorenzo Robbiano for many enlightening discussions during his visits to Passau. I am also thankful to my undergraduate advisors, Prof. Dr. Gabor Wiese and Prof. Dr. Nguyen Chanh Tu, for helpful advice, and for their support and encouragement in all my pursuits. My sincerest thanks go to my professors in the Hanoi Mathematical Institute: Prof. Dr. Ngo Viet Trung, Prof. Dr. Le Tuan Hoa, Prof. Dr. Nguyen Tu Cuong for recommending me to study abroad and for their helpful suggestions. My thanks also go to my teachers and colleagues in the Department of Mathematics at Hue University of Education for their encouragement. I am indebted to all my friends and colleagues at the Department of Mathematics and Informatics of the University of Passau for discussions and seminars, and for sharing their expertise with me during these last four years. Among many, I would like to mention Stefan Schuster, Tran N.K. Linh, Xinquang Xiu, Phillip Javanovic, iv Bilge Sipal, Markus Kriegl, Thomas Stadler, and Christian Kell. I would also like to thank all the secretaries in the department, especially Nathalie Vollst¨adt,for their help and their willingness to answer all my questions. My special appreciation goes to 322 Project of the Ministry of Education and Train- ing of Vietnam for providing financial support for my research. My appreciation also goes to Hue University for allowing me to follow this Ph.D. program. Last but not least, I wish to express my appreciation to my wife for her endless love and patience, to my daughter for being my daughter, and to my parents for their constant support and encouragement. Passau, June 2015 Contents Acknowledgements iii 1 Introduction 1 2 Preliminaries 9 2.1 Noether Differents of Algebras . 10 2.2 Fitting Ideals of Finitely Generated Modules . 22 2.3 First Properties of 0-Dimensional Schemes . 28 2.4 Trace Maps for 0-Dimensional Schemes . 41 3 Various Differents for 0-Dimensional Schemes 49 3.1 Noether Differents for 0-Dimensional Schemes . 50 3.2 Dedekind Differents for 0-Dimensional Schemes . 62 3.2.1 Dedekind Differents for Locally Gorenstein Schemes . 63 3.2.2 Dedekind Differents for 0-Dimensional Smooth Schemes . 72 3.2.3 Computing the Dedekind Differents . 75 3.3 K¨ahlerDifferents for 0-Dimensional Schemes . 85 4 Differents and Uniformity of 0-Dimensional Schemes 101 4.1 Differents and the Cayley-Bacharach Property . 103 4.2 Differents for Schemes Having Generic Hilbert Function . 121 4.3 Cayley-Bacharach Properties and Liaison . 130 4.4 Differents and Higher Uniformity of 0-Dimensional Schemes . 143 5 Differents for Some Special Cases and Applications 157 5.1 Differents for Almost Complete Intersections . 158 vi CONTENTS 5.2 The K¨ahlerDifferents of the Coordinate Ring of 0-Dimensional Schemes 168 5.3 Differents for Fat Point Schemes . 175 Appendix 189 Bibliography 201 Chapter 1 Introduction Let Ro be a commutative ring with identity. To any commutative algebra R=Ro we can associate various ideals in R which we call the differents. The most well known different is the Dedekind different which was introduced in algebraic number theory by Richard Dedekind in 1882. The other differents which are closely related to the Dedekind different are the Noether and K¨ahlerdifferents. It is useful to compare these differents to each other and to relate them to properties of the algebra R=Ro. This direction of research was initiated by Emmy Noether [Noe], and pursued by many authors in the last half century (see for instance [AB], [Fos], [Her], [Ku1], [Ku5], [Mac], [SS], and [Wal]). It turned out that many structural properties of the algebra R=Ro can be phrased as properties of its differents. For example, one can obtain several interesting criteria for the algebra in terms of these differents such as the Ramification Criterion, the Regularity Criterion, and the Smoothness Criterion (see [Ku5]). The goal of this thesis is to study the Noether, Dedekind, and K¨ahlerdifferents of a particular class of algebras. More precisely, we investigate these differents for the homogeneous coordinate ring of a 0-dimensional scheme X in the projective n-space n PK over an arbitrary field K. This approach is inspired by the work of M. Kreuzer and his coworkers ([GKR], [Kr2], [Kr4]) which suggests that, in order to study the geometry of the scheme X, one may start by considering related algebraic objects. n Given such a 0-dimensional scheme X ⊆ PK , let IX denote the homogeneous vanishing ideal of X in P = K[X0;:::;Xn]. The homogeneous coordinate ring of X is then given by R = P=IX. We always assume that no point of the support of X lies on the + hyperplane at infinity Z (X0). Then the image x0 of X0 in R is a non-zerodivisor of R and the algebra R=K[x0] is graded-free of rank deg(X) (see [Kr3] and [KR3, Section 4.3]). In order to get more information about R=K[x0], and thus also about X, it is useful to define and to consider the algebraic structure of the Noether, Dedekind, 2 1. Introduction and K¨ahlerdifferents of R=K[x0]. In particular, it is interesting to study the relations between the algebraic structure of these differents and geometric properties of the scheme X. Furthermore, we think that the Noether, Dedekind, and K¨ahler differents n for 0-dimensional schemes X ⊆ PK provide a source of excellent examples in which computer algebra methods can be applied to examine subtle structural properties of these objects. Motivating Questions Let us briefly recall the definitions of the Noether and K¨ahler differents for a n 0-dimensional scheme X in PK . As above, let IX be the homogeneous vanishing ideal of X, and let R = P=IX be the homogeneous coordinate ring of X. In the standard graded K[x0]-algebra R ⊗K[x0] R = ⊕i≥0(⊕j+k=iRj ⊗ Rk), we have the homogeneous µ ideal J = Ker(R ⊗K[x0] R −! R), where µ is the homogeneous R-linear map of degree zero given by µ(f ⊗ g) = fg. The Noether different of the algebra R=K[x0] (or for X w.r.t. x0), denoted by #N (R=K[x0]), is the image of the annihilator of J in R ⊗K[x0] R under the map µ. It is clear that #N (R=K[x0]) is a homogeneous ideal of R. Notice that the module of K¨ahlerdifferentials of the algebra R=K[x0] is the finitely generated graded R-module Ω1 = J =J 2. Moreover, we can associate with R=K[x0] Ω1 an ascending sequence of homogeneous ideals of R which are known as the R=K[x0] Fitting invariants or Fitting ideals of Ω1 . The most important one is the initial R=K[x0] Fitting ideal of Ω1 which is the homogeneous ideal of R generated by the images R=K[x0] of the maximal minors of the Jacobian matrix of the generators of IX. We denote this invariant by #K (R=K[x0]) and call it the K¨ahlerdifferent of the algebra R=K[x0] (or for X w.r.t. x0). In order to define the Dedekind different #D(R=K[x0]) of the algebra R=K[x0] (or for X w.r.t. x0), it is necessary that the scheme X satisfies some restrictive hypotheses. For example the definition in [DK] of #D(R=K[x0]) requires that X is reduced and has K-rational support. When these differents are well-defined, we ask ourselves the following questions. Question 1.0.1. Can one compute #N (R=K[x0]), #K (R=K[x0]) and #D(R=K[x0]) by using the existing functionality in a computer algebra system such as ApCoCoA? Question 1.0.2.
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