DOCSLIB.ORG

The Tropical Grassmannian

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Tropical Grassmannian Adv. Geom. 4 (2004), 389–411 Advances in Geometry ( de Gruyter 2004 The tropical Grassmannian David Speyer and Bernd Sturmfels* (Communicated by G. M. Ziegler) Abstract. In tropical algebraic geometry, the solution sets of polynomial equations are piecewise-linear. We introduce the tropical variety of a polynomial ideal, and we identify it with a polyhedral subcomplex of the Gro¨bner fan. The tropical Grassmannian arises in this manner from the ideal of quadratic Plu¨cker relations. It parametrizes all tropical linear spaces. Lines in tropical projective space are trees, and their tropical Grassmannian G2; n equals the space of phylogenetic trees studied by Billera, Holmes and Vogtmann. Higher Grassmannians o¤er a natural generalization of the space of trees. Their faces correspond to monomial-free initial ideals of the Plu¨cker ideal. The tropical Grassmannian G3; 6 is a simplicial complex glued from 1035 tetrahedra. 1 Introduction The tropical semiring ðR U fyg; min; þÞ is the set of real numbers augmented by in- finity with the tropical addition, which is taking the minimum of two numbers, and the tropical multiplication which is the ordinary addition [10]. These operations sat- isfy the familiar axioms of arithmetic, e.g. distributivity, with y and 0 being the a1 an Ptwo neutral elements. Tropical monomials x1 ...xn represent ordinary linear forms i aixi, and tropical polynomials X a1 a2 an H Nn A R Fðx1; x2; ...; xnÞ¼ Cax1 x2 ...xn ; with A ; Ca ; ð1Þ a A A n represent piecewise-linear convex functions FP: R ! R. To compute FðxÞ, we take n A the minimum of the a‰ne-linear forms Ca þ i¼1 aixi for a A. We define the trop- ical hypersurface TðFÞ as the set of all points x in Rn for which this minimum is at- tained at least twice, as a runs over A. Equivalently, TðFÞ is the set of all points x A Rn at which F is not di¤erentiable. Thus a tropical hypersurface is an ðn À 1Þ- dimensional polyhedral complex in Rn. The rationale behind this definition will become clear in Section 2, which gives *Partially supported by the National Science Foundation (DMS-0200729). 390 David Speyer and Bernd Sturmfels a self-contained development of the basic theory of tropical varieties. For further background and pictures see [14, §9]. Every tropical variety is a finite intersection of tropical hypersurfaces (Corollary 2.3). But not every intersection of tropical hyper- surfaces is a tropical variety (Proposition 6.3). Tropical varieties are also known as logarithmic limit sets [2], Bieri–Groves sets [4], or non-archimedean amoebas [7]. Trop- ical curves are the key ingredient in Mikhalkin’s formula [9] for planar Gromov– Witten invariants. The object of study in this paper is the tropical Grassmannian Gd; n which is a poly- n hedral fan in RðÞd defined by the ideal of quadratic Plu¨cker relations. All of our main results regarding Gd; n are stated in Section 3. The proofs appear in the subsequent sections. In Section 4 we prove Theorem 3.4 which identifies G2; n with the space of 20 phylogenetic trees in [5]. A detailed study of the fan G3; 6 H R is presented in Sec- tion 5. In Section 6 we introduce tropical linear spaces and we prove that they are parametrized by the tropical Grassmannian (Theorem 3.6). In Section 7 we show that the tropical Grassmannian G3; 7 depends on the characteristic of the ground field. 2 The tropical variety of a polynomial ideal Let K be an algebraically closed field with a valuation into the reals, denoted deg : K à ! R. We assume that 1 lies in the image of deg and we fix t A K à with degðtÞ¼1. The corresponding local ring and its maximal ideal are RK ¼fc A K : degðcÞ d 0g and MK ¼fc A K : degðcÞ > 0g: The residue field k ¼ RK =MK is algebraically closed. Given any ideal I H K½x¼K½x1; x2; ...; xn; we consider its a‰ne variety in the n-dimensional algebraic torus over K, VðIÞ¼fu A ðK ÃÞn : f ðuÞ¼0 for all f A Ig: Here K à ¼ Knf0g. In all our examples, K is the algebraic closure of the rational function field CðtÞ and ‘‘deg’’ is the standard valuation at the origin. Then k ¼ C, and if c A C½t then degðcÞ is the order of vanishing of c at 0. Every polynomial in K½x maps to a tropical polynomial as follows. If X a1 an A à A f ðx1; ...; xnÞ¼ cax1 ...xn with ca K for a A: ð2Þ a A A and Ca ¼ degðcaÞ, then tropð f Þ denotes the tropical polynomial F in (1). The following definitions are a variation on Gro¨bner basis theory [13]. Fix A Rn a1 an w . The w-weight of a term ca Á x1 ...xn in (2) is degðcaÞþa1w1 þÁÁÁþanwn. ~ The initial form inwð f Þ of a polynomial f is defined as follows. Set f ðx1; ...; xnÞ¼ w1 wn Àn ~ f ðt x1; ...; t xnÞ. Let n be the smallest weight of any term of f , so that t f is a The tropical Grassmannian 391 Àn ~ non-zero element in RK ½x. Define inwð f Þ as the image of t f in k½x. We set inwð0Þ¼0. For K ¼ CðtÞ and k ¼ C this means that the initial form inwð f Þ is a polynomial in C½x. Given any ideal I H K½x its initial ideal is defined to be inwðIÞ¼hinwð f Þ : f A Ii H k½x: Theorem 2.1. For an ideal I H K½x the following subsets of Rn coincide: (a) The closure of the set fðdegðu1Þ; ...; degðunÞÞ : ðu1; ...; unÞ A VðIÞg; (b) The intersection of the tropical hypersurfaces Tðtropð f ÞÞ where f A I; n (c) The set of all vectors w A R such that inwðIÞ contains no monomial. The set defined by the three conditions in Theorem 2.1 is denoted TðIÞ and is called the tropical variety of the ideal I. Variants of this theorem already appeared in [14, Theorem 9.17] and in [7, Theorem 6.1], without and with proof respectively. Here we present a short proof which is self-contained. Proof. First we show that (b) contains (a). As (b) is clearly closed, it is enough to consider any point w ¼ðdegðu1Þ; ...; degðunÞÞ in the set (a) and show it lies in (b). For any f A I we have f ðu1; ...; unÞ¼0 and this implies that the minimum in the definition of F ¼ tropð f Þ is attained at least twice at w. This condition is equivalent to inwð f Þ not being a monomial. This shows that (a) is contained in (b), and (b) is contained in (c). It remains to prove that (c) is contained in (a). Consider any vector à n w in (c) such that w ¼ðdegðv1Þ; ...; degðvnÞÞ for some v A ðK Þ . Since the image of the valuation is dense in R and the set defined in (a) is closed, it su‰ces to prove that w ¼ðdegðu1Þ; ...; degðunÞÞ for some u A VðIÞ. By making the change of coordinates À1 xi ¼ xi Á vi , we may assume that w ¼ð0; 0; ...; 0Þ. Since inwðIÞ contains no monomial and since k is algebraically closed, by the Null- à n stellensatz there exists a point u A VðinwðIÞÞ H ðk Þ . Let m denote the maximal ideal in k½x corresponding to u. Let S be the set of polynomials f in RK ½x whose reduc- tion modulo MK is not in m. Then S is a multiplicative set in RK ½x disjoint from I. Consider the induced map À1 À1 j : RK ! S RK ½x=S ðI V RK ½xÞ: We claim that j is injective. Suppose not, and pick c A RK nf0g with jðcÞ¼0, so we can find f A S such that cf A I. Since cÀ1 exists in K, this implies f A I which is a contradiction. The injectivity of j implies that there is some minimal prime P of the ring on the n right hand side such that P RK K is a proper ideal in K½x=I. There exists a maximal n ideal of K½x=I containing P RK K, and since K is algebraically closed, this maximal à n ideal has the form hx1 À u1; ...; xn À uni for some u A VðIÞ H ðK Þ . We claim that ui A RK and ui G ui mod MK . This will imply degðu1Þ¼ÁÁÁ¼degðunÞ¼0 and hence complete the proof. Consider any xi À ui A I. By clearing denominators, we get aix À bi A I V RK ½x 392 David Speyer and Bernd Sturmfels with bi=ai ¼ ui, and not both ai and bi lie in MK .Ifai A MK , then aix À bi G Àbi mod MK . Hence inwðIÞ contains the reduction of bi modulo MK , which is a à unit of K and hence equals the unit ideal. This is a contradiction. If ai B MK and Àbi=ai Z ui mod MK then the reduction of aix À bi modulo MK does not lie in m. À1 This means that aix À bi A S and is a unit of S RK ½x,soP is the unit ideal. But then P is not prime, also a contradiction. This completes the proof. r The key point in the previous proof can be summarized as follows: Corollary 2.2.
Load more

Recommended publications
  • Field Theory Pete L. Clark
    Field Theory Pete L. Clark Thanks to Asvin Gothandaraman and David Krumm for pointing out errors in these notes. Contents About These Notes 7 Some Conventions 9 Chapter 1. Introduction to Fields 11 Chapter 2. Some Examples of Fields 13 1. Examples From Undergraduate Mathematics 13 2. Fields of Fractions 14 3. Fields of Functions 17 4. Completion 18 Chapter 3. Field Extensions 23 1. Introduction 23 2. Some Impossible Constructions 26 3. Subfields of Algebraic Numbers 27 4. Distinguished Classes 29 Chapter 4. Normal Extensions 31 1. Algebraically closed fields 31 2. Existence of algebraic closures 32 3. The Magic Mapping Theorem 35 4. Conjugates 36 5. Splitting Fields 37 6. Normal Extensions 37 7. The Extension Theorem 40 8. Isaacs' Theorem 40 Chapter 5. Separable Algebraic Extensions 41 1. Separable Polynomials 41 2. Separable Algebraic Field Extensions 44 3. Purely Inseparable Extensions 46 4. Structural Results on Algebraic Extensions 47 Chapter 6. Norms, Traces and Discriminants 51 1. Dedekind's Lemma on Linear Independence of Characters 51 2. The Characteristic Polynomial, the Trace and the Norm 51 3. The Trace Form and the Discriminant 54 Chapter 7. The Primitive Element Theorem 57 1. The Alon-Tarsi Lemma 57 2. The Primitive Element Theorem and its Corollary 57 3 4 CONTENTS Chapter 8. Galois Extensions 61 1. Introduction 61 2. Finite Galois Extensions 63 3. An Abstract Galois Correspondence 65 4. The Finite Galois Correspondence 68 5. The Normal Basis Theorem 70 6. Hilbert's Theorem 90 72 7. Infinite Algebraic Galois Theory 74 8. A Characterization of Normal Extensions 75 Chapter 9.
    [Show full text]
  • Geometry on Grassmannians and Applications to Splitting Bundles and Smoothing Cycles
    PUBLICATIONS MATHÉMATIQUES DE L’I.H.É.S. STEVEN L. KLEIMAN Geometry on grassmannians and applications to splitting bundles and smoothing cycles Publications mathématiques de l’I.H.É.S., tome 36 (1969), p. 281-297. <http://www.numdam.org/item?id=PMIHES_1969__36__281_0> © Publications mathématiques de l’I.H.É.S., 1969, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http://www. ihes.fr/IHES/Publications/Publications.html), implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fi- chier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ GEOMETRY ON GRASSMANNIANS AND APPLICATIONS TO SPLITTING BUNDLES AND SMOOTHING CYCLES by STEVEN L. KLEIMAN (1) INTRODUCTION Let k be an algebraically closed field, V a smooth ^-dimensional subscheme of projective space over A:, and consider the following problems: Problem 1 (splitting bundles). — Given a (vector) bundle G on V, find a monoidal transformation f \ V'->V with smooth center CT such thatyG contains a line bundle P. Problem 2 (smoothing cycles). — Given a cycle Z on V, deform Z by rational equivalence into the difference Z^ — Zg of two effective cycles whose prime components are all smooth. Strengthened form. — Given any finite number of irreducible subschemes V, of V, choose a (resp. Z^, Zg) such that for all i, the intersection V^n a (resp.
    [Show full text]
  • Chapter 2 Affine Algebraic Geometry
    Chapter 2 Affine Algebraic Geometry 2.1 The Algebraic-Geometric Dictionary The correspondence between algebra and geometry is closest in affine algebraic geom- etry, where the basic objects are solutions to systems of polynomial equations. For many applications, it suffices to work over the real R, or the complex numbers C. Since important applications such as coding theory or symbolic computation require finite fields, Fq , or the rational numbers, Q, we shall develop algebraic geometry over an arbitrary field, F, and keep in mind the important cases of R and C. For algebraically closed fields, there is an exact and easily motivated correspondence be- tween algebraic and geometric concepts. When the field is not algebraically closed, this correspondence weakens considerably. When that occurs, we will use the case of algebraically closed fields as our guide and base our definitions on algebra. Similarly, the strongest and most elegant results in algebraic geometry hold only for algebraically closed fields. We will invoke the hypothesis that F is algebraically closed to obtain these results, and then discuss what holds for arbitrary fields, par- ticularly the real numbers. Since many important varieties have structures which are independent of the field of definition, we feel this approach is justified—and it keeps our presentation elementary and motivated. Lastly, for the most part it will suffice to let F be R or C; not only are these the most important cases, but they are also the sources of our geometric intuitions. n Let A denote affine n-space over F. This is the set of all n-tuples (t1,...,tn) of elements of F.
    [Show full text]
  • Vector Space Theory
    Vector Space Theory A course for second year students by Robert Howlett typesetting by TEX Contents Copyright University of Sydney Chapter 1: Preliminaries 1 §1a Logic and commonSchool sense of Mathematics and Statistics 1 §1b Sets and functions 3 §1c Relations 7 §1d Fields 10 Chapter 2: Matrices, row vectors and column vectors 18 §2a Matrix operations 18 §2b Simultaneous equations 24 §2c Partial pivoting 29 §2d Elementary matrices 32 §2e Determinants 35 §2f Introduction to eigenvalues 38 Chapter 3: Introduction to vector spaces 49 §3a Linearity 49 §3b Vector axioms 52 §3c Trivial consequences of the axioms 61 §3d Subspaces 63 §3e Linear combinations 71 Chapter 4: The structure of abstract vector spaces 81 §4a Preliminary lemmas 81 §4b Basis theorems 85 §4c The Replacement Lemma 86 §4d Two properties of linear transformations 91 §4e Coordinates relative to a basis 93 Chapter 5: Inner Product Spaces 99 §5a The inner product axioms 99 §5b Orthogonal projection 106 §5c Orthogonal and unitary transformations 116 §5d Quadratic forms 121 iii Chapter 6: Relationships between spaces 129 §6a Isomorphism 129 §6b Direct sums Copyright University of Sydney 134 §6c Quotient spaces 139 §6d The dual spaceSchool of Mathematics and Statistics 142 Chapter 7: Matrices and Linear Transformations 148 §7a The matrix of a linear transformation 148 §7b Multiplication of transformations and matrices 153 §7c The Main Theorem on Linear Transformations 157 §7d Rank and nullity of matrices 161 Chapter 8: Permutations and determinants 171 §8a Permutations 171 §8b Determinants
    [Show full text]
  • Issue 118 ISSN 1027-488X
    NEWSLETTER OF THE EUROPEAN MATHEMATICAL SOCIETY S E European M M Mathematical E S Society December 2020 Issue 118 ISSN 1027-488X Obituary Sir Vaughan Jones Interviews Hillel Furstenberg Gregory Margulis Discussion Women in Editorial Boards Books published by the Individual members of the EMS, member S societies or societies with a reciprocity agree- E European ment (such as the American, Australian and M M Mathematical Canadian Mathematical Societies) are entitled to a discount of 20% on any book purchases, if E S Society ordered directly at the EMS Publishing House. Recent books in the EMS Monographs in Mathematics series Massimiliano Berti (SISSA, Trieste, Italy) and Philippe Bolle (Avignon Université, France) Quasi-Periodic Solutions of Nonlinear Wave Equations on the d-Dimensional Torus 978-3-03719-211-5. 2020. 374 pages. Hardcover. 16.5 x 23.5 cm. 69.00 Euro Many partial differential equations (PDEs) arising in physics, such as the nonlinear wave equation and the Schrödinger equation, can be viewed as infinite-dimensional Hamiltonian systems. In the last thirty years, several existence results of time quasi-periodic solutions have been proved adopting a “dynamical systems” point of view. Most of them deal with equations in one space dimension, whereas for multidimensional PDEs a satisfactory picture is still under construction. An updated introduction to the now rich subject of KAM theory for PDEs is provided in the first part of this research monograph. We then focus on the nonlinear wave equation, endowed with periodic boundary conditions. The main result of the monograph proves the bifurcation of small amplitude finite-dimensional invariant tori for this equation, in any space dimension.
    [Show full text]
  • A Kleiman–Bertini Theorem for Sheaf Tensor Products
    A KLEIMAN{BERTINI THEOREM FOR SHEAF TENSOR PRODUCTS EZRA MILLER AND DAVID E SPEYER Abstract. Fix a variety X with a transitive (left) action by an algebraic group G. Let E and F be coherent sheaves on X. We prove that for elements g in a dense X open subset of G, the sheaf Tor i (E; gF) vanishes for all i > 0. When E and F are structure sheaves of smooth subschemes of X in characteristic zero, this follows from the Kleiman{Bertini theorem; our result has no smoothness hypotheses on the supports of E or F, or hypotheses on the characteristic of the ground field. All schemes in this note are of finite type over an arbitrary base field k. We make no restrictions on the characteristic of k. All groups G are assumed to be smooth 0 0 over k. In particular, the local rings of G ×k k are regular, for all field extensions k of k. By a transitive action of G on X, we mean one such that G × X ! X × X is scheme-theoretically surjective (this implies that X is reduced; see the Remark). The aim of this note is to prove the following result. Theorem. Let X be a variety with a transitive left action of an algebraic group G. Let E and F be coherent sheaves on X and, for all k-rational points g 2 G, let gF denote the pushforward of F along multiplication by g. Then there is a dense Zariski open subset U of G such that, for g 2 U, the sheaf Tor i(E; gF) is 0 for all i > 0.
    [Show full text]
  • 2018-06-108.Pdf
    NEWSLETTER OF THE EUROPEAN MATHEMATICAL SOCIETY Feature S E European Tensor Product and Semi-Stability M M Mathematical Interviews E S Society Peter Sarnak Gigliola Staffilani June 2018 Obituary Robert A. Minlos Issue 108 ISSN 1027-488X Prague, venue of the EMS Council Meeting, 23–24 June 2018 New books published by the Individual members of the EMS, member S societies or societies with a reciprocity agree- E European ment (such as the American, Australian and M M Mathematical Canadian Mathematical Societies) are entitled to a discount of 20% on any book purchases, if E S Society ordered directly at the EMS Publishing House. Bogdan Nica (McGill University, Montreal, Canada) A Brief Introduction to Spectral Graph Theory (EMS Textbooks in Mathematics) ISBN 978-3-03719-188-0. 2018. 168 pages. Hardcover. 16.5 x 23.5 cm. 38.00 Euro Spectral graph theory starts by associating matrices to graphs – notably, the adjacency matrix and the Laplacian matrix. The general theme is then, firstly, to compute or estimate the eigenvalues of such matrices, and secondly, to relate the eigenvalues to structural properties of graphs. As it turns out, the spectral perspective is a powerful tool. Some of its loveliest applications concern facts that are, in principle, purely graph theoretic or combinatorial. This text is an introduction to spectral graph theory, but it could also be seen as an invitation to algebraic graph theory. The first half is devoted to graphs, finite fields, and how they come together. This part provides an appealing motivation and context of the second, spectral, half. The text is enriched by many exercises and their solutions.
    [Show full text]
  • Prizes and Awards
    SAN DIEGO • JAN 10–13, 2018 January 2018 SAN DIEGO • JAN 10–13, 2018 Prizes and Awards 4:25 p.m., Thursday, January 11, 2018 66 PAGES | SPINE: 1/8" PROGRAM OPENING REMARKS Deanna Haunsperger, Mathematical Association of America GEORGE DAVID BIRKHOFF PRIZE IN APPLIED MATHEMATICS American Mathematical Society Society for Industrial and Applied Mathematics BERTRAND RUSSELL PRIZE OF THE AMS American Mathematical Society ULF GRENANDER PRIZE IN STOCHASTIC THEORY AND MODELING American Mathematical Society CHEVALLEY PRIZE IN LIE THEORY American Mathematical Society ALBERT LEON WHITEMAN MEMORIAL PRIZE American Mathematical Society FRANK NELSON COLE PRIZE IN ALGEBRA American Mathematical Society LEVI L. CONANT PRIZE American Mathematical Society AWARD FOR DISTINGUISHED PUBLIC SERVICE American Mathematical Society LEROY P. STEELE PRIZE FOR SEMINAL CONTRIBUTION TO RESEARCH American Mathematical Society LEROY P. STEELE PRIZE FOR MATHEMATICAL EXPOSITION American Mathematical Society LEROY P. STEELE PRIZE FOR LIFETIME ACHIEVEMENT American Mathematical Society SADOSKY RESEARCH PRIZE IN ANALYSIS Association for Women in Mathematics LOUISE HAY AWARD FOR CONTRIBUTION TO MATHEMATICS EDUCATION Association for Women in Mathematics M. GWENETH HUMPHREYS AWARD FOR MENTORSHIP OF UNDERGRADUATE WOMEN IN MATHEMATICS Association for Women in Mathematics MICROSOFT RESEARCH PRIZE IN ALGEBRA AND NUMBER THEORY Association for Women in Mathematics COMMUNICATIONS AWARD Joint Policy Board for Mathematics FRANK AND BRENNIE MORGAN PRIZE FOR OUTSTANDING RESEARCH IN MATHEMATICS BY AN UNDERGRADUATE STUDENT American Mathematical Society Mathematical Association of America Society for Industrial and Applied Mathematics BECKENBACH BOOK PRIZE Mathematical Association of America CHAUVENET PRIZE Mathematical Association of America EULER BOOK PRIZE Mathematical Association of America THE DEBORAH AND FRANKLIN TEPPER HAIMO AWARDS FOR DISTINGUISHED COLLEGE OR UNIVERSITY TEACHING OF MATHEMATICS Mathematical Association of America YUEH-GIN GUNG AND DR.CHARLES Y.
    [Show full text]
  • Topics in Algebraic Geometry I. Berkovich Spaces
    Math 731: Topics in Algebraic Geometry I Berkovich Spaces Mattias Jonsson∗ Fall 2016 Course Description Berkovich spaces are analogues of complex manifolds that appear when replacing complex numbers by the elements of a general valued field, e.g. p-adic numbers or formal Laurent series. They were introduced in the late 1980s by Vladimir Berkovich as a more honestly geometric alternative to the rigid spaces earlier conceived by Tate. In recent years, Berkovich spaces have seen a large and growing range of applications to complex analysis, tropical geometry, complex and arithmetic dynamics, the local Langlands program, Arakelov geometry, . The first part of the course will be devoted to the basic theory of Berkovich spaces (affinoids, gluing, analytifications). In the second part, we will discuss various applications or specialized topics, partly depending on the interest of the audience. Contents 1 September 6 5 1.1 Introduction..............................................5 1.2 Valued fields.............................................5 1.3 What is a space?...........................................7 1.4 The Berkovich affine space (as a topological space)........................7 1.4.1 The Berkovich affine line..................................8 2 September 8 9 2.1 Banach rings [Ber90, 1.1]......................................9 2.1.1 Seminormed groupsx .....................................9 2.1.2 Seminormed rings...................................... 10 2.1.3 Spectral radius and uniformization............................. 11 2.1.4 Seminormed modules..................................... 12 2.1.5 Complete tensor products.................................. 12 2.2 The spectrum [Ber90, 1.2]..................................... 12 x 3 September 13 14 3.1 The spectrum (continued) [Ber90, 1.2]............................... 14 3.2 Complex Banach algebras [Jon16, App.x G]............................. 15 3.3 The Gelfand transform [Ber90, pp.
    [Show full text]
  • On the Algebraic Theory of Fields
    Lectures on the Algebraic Theory of Fields By K.G. Ramanathan Tata Institute of Fundamental Research, Bombay 1956 Lectures on the Algebraic Theory of Fields By K.G. Ramanathan Tata Institute of Fundamental Research, Bombay 1954 Introduction There are notes of course of lectures on Field theory aimed at pro- viding the beginner with an introduction to algebraic extensions, alge- braic function fields, formally real fields and valuated fields. These lec- tures were preceded by an elementary course on group theory, vector spaces and ideal theory of rings—especially of Noetherian rings. A knowledge of these is presupposed in these notes. In addition, we as- sume a familiarity with the elementary topology of topological groups and of the real and complex number fields. Most of the material of these notes is to be found in the notes of Artin and the books of Artin, Bourbaki, Pickert and Van-der-Waerden. My thanks are due to Mr. S. Raghavan for his help in the writing of these notes. K.G. Ramanathan Contents 1 General extension fields 1 1 Extensions......................... 1 2 Adjunctions........................ 3 3 Algebraic extensions . 5 4 Algebraic Closure . 9 5 Transcendental extensions . 12 2 Algebraic extension fields 17 1 Conjugate elements . 17 2 Normal extensions . 18 3 Isomorphisms of fields . 21 4 Separability ........................ 24 5 Perfectfields ....................... 31 6 Simple extensions . 35 7 Galois extensions . 38 8 Finitefields ........................ 46 3 Algebraic function fields 49 1 F.K. Schmidt’s theorem . 49 2 Derivations ........................ 54 3 Rational function fields . 67 4 Norm and Trace 75 1 Normandtrace ...................... 75 2 Discriminant ....................... 82 v vi Contents 5 Composite extensions 87 1 Kronecker product of Vector spaces .
    [Show full text]
  • A Primer on Zeta Functions and Decomposition Spaces
    A Primer on Zeta Functions and Decomposition Spaces Andrew Kobin November 30, 2020 Abstract Many examples of zeta functions in number theory and combinatorics are special cases of a construction in homotopy theory known as a decomposition space. This article aims to introduce number theorists to the relevant concepts in homotopy theory and lays some foundations for future applications of decomposition spaces in the theory of zeta functions. 1 Introduction This expository article is aimed at introducing number theorists to the theory of decom- position spaces in homotopy theory. Briefly, a decomposition space is a certain simplicial space that admits an abstract notion of an incidence algebra and, in particular, an abstract zeta function. To motivate the connections to number theory, we show that most classical notions of zeta functions in number theory and algebraic geometry are special cases of the construction using decomposition spaces. These examples suggest a wider application of decomposition spaces in number theory and algebraic geometry, which we intend to explore in future work. Some of the ideas for this article came from a reading course on 2-Segal spaces organized by Julie Bergner at the University of Virginia in spring 2020, which resulted in the survey [ABD+], as well as another reading course over the summer, co-organized by Bergner and the arXiv:2011.13903v1 [math.NT] 27 Nov 2020 author. The author would like to thank the participants of these courses, and in particular Julie Bergner, Matt Feller and Bogdan Krstic, for helpful discussions on these and related topics. The present work also owes much to a similar article by Joachim Kock [Koc], which describes Riemann’s zeta function from the perspective taken here.
    [Show full text]
  • Affine Grassmannians and Geometric Satake Equivalences
    Affine Grassmannians and Geometric Satake Equivalences Dissertation zur Erlangung des Doktorgrades (Dr. rer. nat.) der Mathematisch-Naturwissenschaftlichen Fakult¨at der Rheinischen Friedrich-Wilhelms-Universit¨at Bonn vorgelegt von Timo Richarz aus Bad Honnef Bonn, November 2013 Angefertigt mit der Genehmigung der Mathematisch-Naturwissenschaftlichen Fakult¨at der Rheinischen Friedrich-Wilhelms-Universit¨at Bonn 1. Gutachter: Prof. Dr. Michael Rapoport 2. Gutachter: Prof. Dr. Jochen Heinloth Tag der Promotion: 30. Januar 2014 Erscheinungsjahr: 2014 In der Dissertation eingebunden: Zusammenfassung AFFINE GRASSMANNIANS AND GEOMETRIC SATAKE EQUIVALENCES BY TIMO RICHARZ This thesis consists of two parts, cf. [11] and [12]. Each part can be read independently, but the results in both parts are closely related. In the first part I give a new proof of the geometric Satake equivalence in the unramified case. In the second part I extend the theory to the ramified case using as a black box the unramified Satake equivalence. Let me be more specific. Part I. Split connected reductive groups are classified by their root data. These data come in pairs: for every root datum there is an associated dual root datum. Hence, for every split connected reductive group G, there is an associated dual group Gˆ. Following Drinfeld’s geometric interpretation of Langlands’ philosophy, the representation theory of Gˆ is encoded in the geometry of an infinite dimensional scheme canonically associated with G as follows, cf. Ginzburg [4], Mirkovi´c-Vilonen[8]. Let G be a connected reductive group over a separably closed field F . The loop group LzG is the group functor on the category of F -algebras L G: R G(R(( z))), z !−→ + where z is an additional variable.
    [Show full text]