Modern Projective Geometry Mathematics and Its Applications Managing Editor: M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands Volume 521 Modern Projective Geometry by Claude-Alain Faure and Alfred Frolicher University of Geneva, Geneva, Switzerland SPRINGER-SCIENCE+BUSINESS MEDIA, B.V. A c.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 978-90-481-5544-6 ISBN 978-94-015-9590-2 (eBook) DOI 10.1007/978-94-015-9590-2 Printed on acid-free paper All Rights Reserved © 2000 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2000 Softcover reprint of the hardcover 1st edition 2000 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. Table of Contents Preface ix Introduction xiii Chapter 1. Fundamental Notions of Lattice Theory 1 1.1 Introduction to lattices . 1 1.2 Complete lattices . 5 1.3 Atomic and atomistic lattices 7 1.4 Meet-continuous lattices 9 1.5 Modular and semimodular lattices 12 1.6 The maximal chain property 15 1. 7 Complemented lattices 17 1.8 Exercises. 21 Chapter 2. Projective Geometries and Projective Lattices 25 2.1 Definition and examples of projective geometries 26 2.2 A second system of axioms 30 2.3 Subspaces 34 2.4 The lattice .C( G) of subspaces of G 36 2.5 Correspondence of projective geometries and projective lattices 40 2.6 Quotients by subspaces and isomorphism theorems 43 2.7 Decomposition into irreducible components 47 2.8 Exercises. 49 Chapter 3. Closure Spaces and Matroids 55 3.1 Closure operators . ... 56 3.2 Examples of matroids ... 59 3.3 Projective geometries as closure spaces 63 3.4 Complete atomistic lattices 67 3.5 Quotients by a closed subset 70 3.6 Isomorphism theorems 73 3.7 Exercises. 75 Chapter 4. Dimension Theory 81 4.1 Independent subsets and bases 83 vi Table of Contents 4.2 The rank of a subspace . 86 4.3 General properties of the rank 89 4.4 The dimension theorem of degree n 92 4.5 Dimension theorems involving the corank 97 4.6 Applications to projective geometries 98 4.7 Matroids as sets with a rank function 100 4.8 Exercises. 103 Chapter 5. Geometries of degree n 107 5.1 Definition and examples 108 5.2 Degree of submatroids and quotient geometries 110 5.3 Affine geometries . 112 5.4 Embedding of a geometry of degree 1 117 5.5 Exercises. 121 Chapter 6. Morphisms of Projective Geometries 127 6.1 Partial maps 128 6.2 Definition, properties and examples of morphisms 133 6.3 Morphisms induced by semilinear maps 137 6.4 The category of projective geometries 141 6.5 Homomorphisms . 143 6.6 Examples of homomorphisms 148 6.7 Exercises. 151 Chapter 7. Embeddings and Quotient-Maps 157 7.1 Mono-sources and initial sources 158 7.2 Embeddings. 163 7.3 Epi-sinks and final sinks 169 7.4 Quotient-maps. 172 7.5 Complementary subpaces 177 7.6 Factorization of morphisms 179 7.7 Exercises. 182 Chapter 8. Endomorphisms and the Desargues Property 187 8.1 Axis and center of an endomorphism 188 8.2 Endomorphisms with a given axis 191 8.3 Endomorphisms induced by a hyperplane-embedding 195 8.4 Arguesian geometries 197 8.5 Non-arguesian planes 204 8.6 Exercises. 209 Table of Contents vii Chapter 9. Homogeneous Coordinates 215 9.1 The homothety fields of an arguesian geometry 216 9.2 Coordinates and hyperplane-embeddings . 218 9.3 The fundamental theorem for homomorphisms 221 9.4 Uniqueness of the associated fields and vector spaces 224 9.5 Arguesian planes 226 9.6 The Pappus property 228 9.7 Exercises . 230 Chapter 10. Morphisms and Semilinear Maps 235 10.1 The fundamental theorem 236 10.2 Semilinear maps and extensions of morphisms 238 10.3 The category of arguesian geometries . 242 10.4 Points in general position . 244 10.5 Projective subgeometries of an arguesian geometry 247 10.6 Exercises 249 Chapter 11. Duality 255 11.1 Duality for vector spaces 256 11.2 The dual geometry . 258 11.3 Pairings, dualities and embedding into the bidual 261 11.4 The duality functor . 264 11.5 Pairings and sesquilinear forms 267 11.6 Exercises 269 Chapter 12. Related Categories 275 12.1 The category of closure spaces 276 12.2 Galois connections and complete lattices 278 12.3 The category of complete atomistic lattices 281 12.4 Morphisms between affine geometries . 284 12.5 Characterization of strong morphisms 287 12.6 Characterization of morphisms 291 12.7 Exercises 295 Chapter 13. Lattices of Closed Subspaces 301 13.1 Topological vector spaces 302 13.2 Mackey geometries . 305 13.3 Continuous morphisms 308 13.4 Dualized geometries 310 13.5 Continuous homomorphisms 315 Vlll Table of Contents 13.6 Exercises 318 Chapter 14. Orthogonality 323 14.1 Orthogeometries . 324 14.2 Ortholattices and orthosystems 327 14.3 Orthogonal morphisms 330 14.4 The adjunction functor 334 14.5 Hilbertian geometries 337 14.6 Exercises 340 List of Problems 345 Bibliography 347 List of Axioms 357 List of Symbols 358 Index 359 Preface Projective geometry is a very classical part of mathematics and one might think that the subject is completely explored and that there is nothing new to be added. But it seems that there exists no book on projective geometry which provides a systematic treatment of morphisms. We intend to fill this gap. It is in this sense that the present monograph can be called modern. The reason why morphisms have not been studied much earlier is probably the fact that they are in general partial maps between the point sets G and G', noted 9 : G -- ~ G', i.e. maps 9 : D -4 G' whose domain Dom 9 := D is a subset of G. We give two simple examples of partial maps which ought to be morphisms. The first example is purely geometric. Let E, F be complementary subspaces of a projective geometry G. If x E G \ E, then g(x) := (E V x) n F (where E V x is the subspace generated by E U {x}) is a unique point of F, i.e. one obtains a map 9 : G \ E -4 F. As special case, if E = {z} is a singleton and F a hyperplane with z tf. F, then g: G \ {z} -4 F is the projection with center z of G onto F. The second example comes from algebra. If V is any vector space we denote by 1>(V) the associated projective geometry which has as its points the 1-dimensional vector subspaces of V. Let now V and W be two vector spaces over fields K and L respectively, and f : V -4 W a semilinear map, i.e. an additive map satisfying j(>.. x) = a('\) . f(x) for some field homomorphism (J: K -4 L. Then f induces a partial map 1> f between the associated projective geometries :J>(V) and :J>(W) as follows. Let X E 1>(V). One chooses x E X with x#- O. If f(x) = 0, then (1)f)(X) is not defined. Otherwise (1)J)(X) is the vector subspace of W generated by f(x). The semilinearity of f implies that the choice of x is irrelevant. Hence the partial map 1> f : 1>(V) -- ~ 1>(W) is well defined. The partial maps considered in these examples will be morphisms of projective geometries. However, the main criterium whether one has a suitable definition of morphism is the validity of a converse result for the second example. This goal will be achieved: every morphism g: 1>(V) -- ~ 1>(W) whose image is not contained in a line is of the form 9 = 1> f for some semilinear map f : V -4 W. Moreover, the map f is unique up to a scalar factor. The result is called Fundamental Theorem, since in the particular case where 9 is a collineation it boils down to the classical Fundamental Theorem of Projective Geometry. By introducing coordinates, Descartes opened the door for analytic geometry which allows to solve geometric problems by algebraic methods. For a projective x Preface geometry G the link with algebra is provided by homogeneous coordinates. These consist of a vector space V together with an isomorphism P(V) ~ G. We mention two examples where a problem involving algebraic arguments is equivalent with a purely geometric problem. The first one is the question: does a given irreducible projective geometry G of dimension 2: 2 admit homogeneous coordinates? It is equivalent with the so-called embedding problem, namely to find an irreducible projective geometry containing G as hyperplane. We shall describe such a geometric construction. But it seems easier to construct homogeneous coordinates first (if there exist any) and then use them in order to obtain an embedding. As second example we consider the problem of the fundamental theorem, i.e. to find for a given morphism 9 : P(V) -- ... P(W) a semilinear map f : V -+ W such that 9 = Pf. It is well known that P(V) is included as hyperplane in the projec­ tive geometry V U P(V). We then show that a map f: V -+ W is semilinear and satisfies 9 = P f if and only if the partial map f U 9 : V U P(V) -- ... W U P(W) is a morphism satisfying Ov 1-+ Ow.