Modern Projective Geometry

Transcription

1 Modern Projective Geometry

2 Mathematics and Its Applications Managing Editor: M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands Volume 521

3 Modern Projective Geometry by Claude-Alain Faure and Alfred Frolicher University of Geneva, Geneva, Switzerland SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

4 A c.i.p. Catalogue record for this book is available from the Library of Congress. ISBN DOI / ISBN (ebook) 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.

5 Table of Contents Preface Introduction ix xiii Chapter 1. Fundamental Notions of Lattice Theory Introduction to lattices Complete lattices Atomic and atomistic lattices Meet-continuous lattices Modular and semimodular lattices The maximal chain property Complemented lattices Exercises Chapter 2. Projective Geometries and Projective Lattices Definition and examples of projective geometries A second system of axioms Subspaces The lattice.c( G) of subspaces of G Correspondence of projective geometries and projective lattices Quotients by subspaces and isomorphism theorems 2.7 Decomposition into irreducible components 2.8 Exercises Chapter 3. Closure Spaces and Matroids 3.1 Closure operators Examples of matroids Projective geometries as closure spaces 3.4 Complete atomistic lattices 3.5 Quotients by a closed subset 3.6 Isomorphism theorems 3.7 Exercises.... Chapter 4. Dimension Theory 4.1 Independent subsets and bases

6 vi The rank of a subspace.... General properties of the rank The dimension theorem of degree n Dimension theorems involving the corank Applications to projective geometries Matroids as sets with a rank function Exercises.. Table of Contents Chapter 5. Geometries of degree n 5.1 Definition and examples 5.2 Degree of submatroids and quotient geometries 5.3 Affine geometries Embedding of a geometry of degree Exercises Chapter 6. Morphisms of Projective Geometries 6.1 Partial maps 6.2 Definition, properties and examples of morphisms 6.3 Morphisms induced by semilinear maps 6.4 The category of projective geometries 6.5 Homomorphisms Examples of homomorphisms 6.7 Exercises.. Chapter 7. Embeddings and Quotient-Maps 7.1 Mono-sources and initial sources 7.2 Embeddings Epi-sinks and final sinks 7.4 Quotient-maps Complementary subpaces 7.6 Factorization of morphisms 7.7 Exercises.... Chapter 8. Endomorphisms and the Desargues Property 8.1 Axis and center of an endomorphism 8.2 Endomorphisms with a given axis 8.3 Endomorphisms induced by a hyperplane-embedding 8.4 Arguesian geometries 8.5 Non-arguesian planes 8.6 Exercises

7 Table of Contents Chapter 9. Homogeneous Coordinates 9.1 The homothety fields of an arguesian geometry 9.2 Coordinates and hyperplane-embeddings. 9.3 The fundamental theorem for homomorphisms 9.4 Uniqueness of the associated fields and vector spaces 9.5 Arguesian planes 9.6 The Pappus property 9.7 Exercises. Chapter 10. Morphisms and Semilinear Maps 10.1 The fundamental theorem 10.2 Semilinear maps and extensions of morphisms 10.3 The category of arguesian geometries Points in general position Projective subgeometries of an arguesian geometry 10.6 Exercises Chapter 11. Duality 11.1 Duality for vector spaces 11.2 The dual geometry Pairings, dualities and embedding into the bidual 11.4 The duality functor Pairings and sesquilinear forms 11.6 Exercises Chapter 12. Related Categories 12.1 The category of closure spaces 12.2 Galois connections and complete lattices 12.3 The category of complete atomistic lattices 12.4 Morphisms between affine geometries Characterization of strong morphisms 12.6 Characterization of morphisms 12.7 Exercises Chapter 13. Lattices of Closed Subspaces 13.1 Topological vector spaces 13.2 Mackey geometries Continuous morphisms 13.4 Dualized geometries 13.5 Continuous homomorphisms vii

8 Vlll Table of Contents 13.6 Exercises Chapter 14. Orthogonality 14.1 Orthogeometries Ortholattices and orthosystems 14.3 Orthogonal morphisms 14.4 The adjunction functor 14.5 Hilbertian geometries 14.6 Exercises List of Problems Bibliography List of Axioms List of Symbols Index

9 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

10 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 projective 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. So the problem of the fundamental theorem is equivalent with the purely geometric problem of extending a morphism. In the original proof, this extension problem was solved first and then the fundamental theorem was deduced. We now proceed the other way round since it seems that as usual the analytical problem is easier to solve than the geometric one. One often finds in the definition of a semilinear map the additional condition that the associated map a must be an isomorphism of fields. We call these maps quasilinear. The corresponding morphisms of projective geometries will be called homomorphisms. They turn out to be rather special. In fact, any homomorphism 9 decomposes as 9 = j 0 u 0 7r where 7r is the canonical projection onto a quotient by a subspace, u an isomorphism and j the inclusion of a subspace. If one restricts the investigations to quasilinear maps and homomorphisms, then one looses many interesting results, such as e.g. a projective subgeometry of the complex projective plane P(C 3 ) which is isomorphic to p(c(n)). Since in projective geometry one has to distinguish between the notions subspace, projective subgeometry and subgeometry, we avoid terminological difficulties by calling our objects in the old-fashioned way projective geometries and not projective spaces. A projective geometry is determined up to an isomorphism by the lattice of its subspaces. This yields a correspondence between projective geometries and certain lattices. The respective lattices, called projective lattices, are known since a long time. By defining appropriate morphisms between lattices, this correspondence is extended to an equivalence between the category of projective geometries and the category of projective lattices. Similarly, a projective geometry is also determined by the closure operator that associates to an arbitrary set of points the smallest subspace containing it. This yields a correspondence between projective geometries and certain closure spaces.

11 Preface xi Again, morphisms between closure spaces are introduced and it is shown that one gets an equivalence between the category of projective geometries and a category of certain closure spaces. By considering lattices which satisfy some, but not all axioms for being projective, one obtains objects which may still have some geometric aspects. Well-known among these are the continuous geometries. In this book we make an excursion in another direction. We consider closure spaces which are more general than those which correspond to projective geometries, but are still called geometries, e.g. the affine geometries, hyperbolic geometries and Mobius geometries. The purpose of this book is the presentation of modern aspects and some recent results which are mainly due to the study of morphisms. However, we do not give a complete view of projective geometry. There are topics which are not even mentioned. We also omit the historical development. It is sometimes difficult to find out who should get credit for certain results. The selected bibliography could be helpful for readers interested in such questions. Special efforts were made in order to formulate theorems in their natural setting, i.e. without irrelevant hypotheses. Nevertheless, projective geometry is developped from scratch. Hence this book should be accessible for anyone who has some knowledge of linear algebra (vector spaces over arbitrary fields or division rings) and of partially ordered sets. Zorn's Lemma is used several times. It is stated in the appropriate version, but without proof. Some rudimentary knowledge of category theory is occasionally useful, but we have tried to make it dispensable. At the end of each chapter one finds an important section with exercises. They are of various difficulty. Some of them require only the application of results given in the preceding sections, but others introduce additional notions and form in fact complements to the chapter. Furthermore, we formulate at the end of the book a list of a few problems for which we have no answers. With pleasure we now express our gratitude to several colleagues. Our thanks go in particular to Josef Schmid for his suggestion to prove the Steinitz Exchange Theorem directly for the infinite case, and to Burchard Kaup who initiated with great skill one of the authors (A.F.) into the world of MEX and helped again and again. We also thank all colleagues who have encouraged us by their interest or by useful remarks: Ernst Binz, Francis Buekenhout, Aurelio Carboni, Horst Herrlich, Oscar Pino-Ortiz, Constantin Piron, Dieter Pumpliin and Santiago Sologuren. Finally, we thank Kluwer academic publishers for their assistance in preparing the text and for leaving us time enough to finish the manuscript carefully. March 2000 Claude-Alain Faure, Lycee de Porrentruy, CH-2900 Porrentruy Alfred Frolicher, Universite de Geneve, CH-1200 Geneve

12 Introduction This introduction is a short guide through the various chapters and includes some indications how they depend upon each other. The first chapter deals with those parts of lattice theory that playa role for the characterization of a projective geometry by its lattice of subspaces. It is possible to begin by reading Chapter 2, going back to Chapter 1 whenever it is needed. Projective geometries are introduced in Chapter 2. Two equivalent axiomatic descriptions are given. The first one consists of three simple axioms for the ternary relation collinear on the set of points. The second one consists of three axioms for the operator * that associates to a couple a, b of points the singleton {a} if a = b and the line through a and b if a =I- b. It later allows to shorten certain proofs. Subspaces of a projective geometry G constitute the main topic of the chapter. They are shown to form a lattice,c( G) having several additional properties. Since G is determined (up to an isomorphism) by the associated lattice 'c(g) one gets a correspondence between projective geometries and the so-called projective lattices. This classical result is established carefully and in a way that will be adequate for generalizing it later to an equivalence of categories. If a subspace E of a projective geometry G is given, then each one of the two intervals [<;D,E] and [E,G] of 'c(g) is a projective lattice and so one can consider the corresponding projective geometry. To [<;D, E] of course corresponds E which, due to its inclusion in G, is itself a projective geometry. The projective geometry associated to [E, G] is more interesting. It is denoted by G / E and called quotient of G by the subspace E. For these quotients geometries one obtains isomorphism theorems similar to those of group theory. Our axioms do not imply that each line of a projective geometry G contains at least three points. Since this holds if and only if the corresponding lattice,c( G) is irreducible, the respective projective geometries are called irreducible. The chapter ends with the result that every projective geometry G decomposes into irreducible components. As shown later, this decomposition is actually a coproduct. The core of Chapter 3 is the description of a projective geometry by means of the closure operator C which associates to a set A of points the smallest subspace C(A) containing A. The respective axiomatic characterization of projective geometries involves six axioms. By deleting some of these axioms one gets more general mathematical objects, such as e.g. closure spaces, matroids and geometries. Many results of projective geometry actually hold in some of these more general settings.

13 xiv Introduction The correspondence between projective geometries and projective lattices is easily extended to a correspondence between geometries and geometric lattices, and this again to one between simple closure spaces and complete atomistic lattices. In the last two sections quotients by subspaces are generalized from projective geometries to closure spaces and isomorphism theorems for this general situation are given. Matroids constitute the appropriate frame for the dimension theory developped in Chapter 4. One first defines the notions of a dependent subset and of a basis of a subspace. The key result which allows to define the rank r(e) of a subspace E as the cardinal number of a basis is a transfinite version of the Steinitz Exchange Theorem: If A is independent and A ~ C(B), then the set A injects into B. The proof uses Zorn's Lemma. The equation r(e V F) + r(e /\ F) = r(e) + r(f) holds for any two subspaces E, F of a projective geometry. For subspaces of an affine geometry it holds under the condition r(e /\ F) ~ 1 (which means En F =F (/J). A geometry for which the implication r(e /\ F) ~ n => r(e V F) + r(e /\ F) = r(e) + r(f) holds for any subspaces E, F is called a geometry of degree n. The geometries of degree 0 are exactly the projective geometries. The so-called general dimension theorem gives a dozen of equivalent conditions characterizing the geometries of degree n. A reader with some knowledge of dimension in projective geometry and who is mainly interested in the study of morphisms can jump over Chapters 4 and 5. Some results and examples concerning geometries of degree n are considered in Chapter 5. The aiiine geometries are characterized axiomatically as geometries of degree 1 satisfying an additional condition. They are closely related to projective geometries. If H is a hyperplane of a projective geometry G, then the set G \ H, considered as subgeometry of G, is an affine geometry. Conversely, if A is an affine geometry, then by adding to A its points at infinity one gets a projective geometry containing A as subgeometry. Among the geometries of degree lone also finds the usual hyperbolic geometries. The Mobius geometries are examples of geometries of degree 2, but not of degree 1. The embedding of a geometry into some projective geometry can be generalized from affine geometries to arbitrary geometries of degree 1, provided that they are of rank at least 5. There are many open questions on geometries of degree n. We hope that this short chapter, which will be used only in Chapter 12, will stimulate research in this direction. Chapter 6 begins with some basic definitions and notations concerning partial maps. Then morphisms of projective geometries are defined and characterized in several equivalent ways. Some general properties are given. Among the examples, one has in particular the morphisms of the form 'J> f : 'J>(V) 'J>(W), where f is a semilinear map V --+ W. By imposing two additional axioms one obtains a class of special morphisms, called homomorphisms. Given a semilinear map f: V --+ W one has the following result which illustrates the role of the homomorphisms: If f

14 Introduction xv is quasilinear, then P f is a homomorphism. Conversely, if P f is a homomorphism and if it is non-constant, then f is quasilinear. Chapter 7 mainly deals with embeddings of a projective geometry into another one. If a subset G' of a projective geometry G is, together with the restriction of the collinearity relation of G, a projective geometry, then it is called a projective subgeometry of G. Its inclusion into G is a morphism, but in general not a homomorphism, as the example p(jr3) ~ p(((:3) shows. An embedding can be defined as a composite i 0 u of an isomorphism u with the inclusion i of a projective subgeometry. In the special case where i is the inclusion of a subspace the embedding i 0 u is called a subspace-embedding. One might expect that if G' embeds into G, then dimg':::; dimg. This however fails: p(q(n)) embeds into p(jr3). Even more surprising is an embedding of P(((:(N)) into the complex projective plane p(((:3). An embedding 9 : G 1 ---t G 2 is an initial morphism, i.e. the following universal property holds: a partial map h: Go --4 G 1 from a projective geometry Go to G 1 is a morphism if and only if the composite 9 0 h : Go G 2 is a morphism. The generalization to initial families gi: G G i will be used in Chapter 8. Quotient-maps are dual to embeddings in the categorical sense. But the results are different and less numerous. Finally, it is shown that every morphism can be decomposed into three factors in a canonical way. Among all chapters this one is closest related to category theory. However, only a few of these results will be used later. So one can go from Chapter 6 directly to Chapter 8 if one wants to come to the Fundamental Theorem as quick as possible. In Chapter 8 we consider endomorphisms r.p: G G of an irreducible projective geometry G with dim( G) :2: 2. The notions of axis and center are generalized from collineations to endomorphisms of G. The set eh of all endomorphisms of G having a given hyperplane H ~ G as axis is studied. If G itself is a hyperplane of a projective geometry G, then one constructs by means of projections a bijection G ---t eh. Thereby eh becomes a projective geometry (isomorphic to G) and one shows that this structure is initial with respect to the evaluations eva: e H G for a E G. So eh yields a geometric construction of a hyperplane-embedding of G provided that G is embeddable. This is the case if and only if G is arguesian. It is shown that if dim G :2: 3, then G is arguesian. The chapter includes two equivalent characterizations of this property (existence of enough collineations having a given axis H, and the classical Desargues property). Examples as well as geometric and algebraic aspects of non-arguesian planes conclude the chapter. A construction of homogeneous coordinates for an arguesian (projective) geometry G is described in Chapter 9. Let H ~ G be a hyperplane and 0 E G \ H. The collineations of G having axis H and a center z E H are called translations. With the composition as operation they form an abelian group TH that operates simply transitively on the set V := G \ H. Therefore there exists a unique operation + on V such that the evaluation at 0 becomes a group isomorphism eva: TH ---t V. The

15 xvi Introduction collineations of G having axis H and a center z ff. H are called homotheties. With the composition as operation the homotheties having center 0 form a group 'liif. By adding the endomorphism no E H defined for x E G \ H by nox = 0, one then constructs a field ICj{:= 'lij{ U {no} having 'lij{ as multiplicative group. This field operates on V and therefore V becomes a vector space over ICif. The projection x 1-+ (x * 0) n H induces an isomorphism u :!J>(V) -+ H, Le. one has homogeneous coordinates for H. Together with the identity map Id : V -+ G \ H one obtains an isomorphism Id U u: V U!J>(V) -+ G. One thus gets homogeneous coordinates for G since for any vector space V over a field K one has V U!J>(V) ~!J>(V x K). So arguesian implies existence of homogeneous coordinates, and the following conditions are equivalent: G is arguesian, the classical Desargues property holds for G, G admits homogeneous coordinates, G admits a strict subspace-embedding. The chapter includes a proof (which will not be used later) of the Fundamental Theorem for the special case of homomorphisms. The final section deals with the Pappus property which gives a geometric condition equivalent with the commutativity of the homothety fields ICif. The core of Chapter 10 is the proof of the Fundamental Theorem. It says that a morphism g :!J>(V) !J>(W) which is non-degenerate (Le. whose image is not contained in aline) is of the form g =!J> f for some semilinear map f : V -+ W. The class of non-degenerate morphisms is not closed under composition, hence yields no category. However, we show that a degenerate morphism between arguesian geometries is induced by a semilinear map if and only if it can be written as the composite of two (or, equivalently, finitely many) non-degenerate morphisms. This additional lemma allows a short categorical formulation of the Fundamental Theorem. As application, a result concerning the existence of a morphism having given values on a set of points in general position is established. The final section deals with projective subgeometries of an arguesian geometry. The chapter on duality covers classical aspects as well as results involving morphisms. The dual geometry G* of a projective geometry G has the hyperplanes of G as its points. One might expect that G 1-+ G* extends to a contravariant endofunctor of the category of projective geometries and morphisms. However, one can associate to a morphism g: G G2 a partial map g*: G2* ---+ G 1* only if g is a homomorphism. Since then also g* is a homomorphism, one obtains the desired endofunctor for the respective subcategory. A pairing between G1 and G 2 consists of two partial maps gl: G G 2* and g2 : G G 1* which determine each other by simple conditions. It follows that gl, g2 both are homomorphisms. By means of the Fundamental Theorem we show that pairings between arguesian geometries correspond to sesquilinear forms. The pairing is a duality if and only if the corresponding form is non-singular. In a first part of Chapter 12 we introduce morphisms of projective lattices and then show that the so obtained category PLat is equivalent to the category Proj

16 Introduction xvii of projective geometries. Furthermore, this equivalence can be easily extended into an equivalence between simple closure spaces and complete atomistic lattices. The second part of the chapter deals with morphisms of affine geometries, in particular morphisms between vector spaces. The main result is an improved version of the fundamental theorem of affine geometry, whose proof is based on the Fundamental Theorem of projective geometry. We show that these geometric morphisms can be described algebraically as weighted semi-alline maps. Projective geometries with additional structures are considered in Chapter 13. Mackey geometries are projective geometries together with a set of distinguished subspaces: the closed subspaces of the geometry. Examples of Mackey geometries are furnished by the projective geometries associated to topological vector spaces. It is shown that the category of Mackey geometries and continuous morphisms is equivalent to two different categories of lattices. Dualized geometries are Mackey geometries for which the set of closed subspaces is determined by the set of closed hyperplanes. In that case, it turns out that a subspace is closed if and only if it is closed with respect to the associated weak topology. Moreover, a homomorphism is continuous if and only if it continuous with respect to the weak topologies. The final chapter is the continuation of the preceding one. Orthogeometries are particular dualized geometries, for which the set of closed hyperplanes is given by a polarity, or equivalently by an orthogonality relation.i. The typical example of an orthogeometry is the projective geometry associated to a vector space equipped with a non-singular reflexive sesquilinear form. The main result is a new version of Wigner's theorem. Let V1 and V 2 be two pre-hilbertian spaces over the same field 1R, C or JEll. Then every non-degenerate morphism g : P(V1) P(V2) preserving orthogonality is induced by some quasilinear isometry. Instead of such orthogonal morphisms one may consider continuous homomorphisms. These are precisely the partial maps g : G G 2 that admit an adjoint go: G G 1. And one thus gets a contravariant automorphism of the category of orthogeometries.