Modern Projective Geometry

By: Claude-Alain Faure, Alfred FrÃ¶licher

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Hardcover | 30 September 2000

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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.

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Published: 15th December 2010

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Preface	p. ix
Introduction	p. xiii
Fundamental Notions of Lattice Theory	p. 1
Introduction to lattices	p. 1
Complete lattices	p. 5
Atomic and atomistic lattices	p. 7
Meet-continuous lattices	p. 9
Modular and semimodular lattices	p. 12
The maximal chain property	p. 15
Complemented lattices	p. 17
Exercises	p. 21
Projective Geometries and Projective Lattices	p. 25
Definition and examples of projective geometries	p. 26
A second system of axioms	p. 30
Subspaces	p. 34
The lattice L (G) of subspaces of G	p. 36
Correspondence of projective geometries and projective lattices	p. 40
Quotients by subspaces and isomorphism theorems	p. 43
Decomposition into irreducible components	p. 47
Exercises	p. 49
Closure Spaces and Matroids	p. 55
Closure operators	p. 56
Examples of matroids	p. 59
Projective geometries as closure spaces	p. 63
Complete atomistic lattices	p. 67
Quotients by a closed subset	p. 70
Isomorphism theorems	p. 73
Exercises	p. 75
Dimension Theory	p. 81
Independent subsets and bases	p. 83
The rank of a subspace	p. 86
General properties of the rank	p. 89
The dimension theorem of degree n	p. 92
Dimension theorems involving the corank	p. 97
Applications to projective geometries	p. 98
Matroids as sets with a rank function	p. 100
Exercises	p. 103
Geometries of degree n	p. 107
Definition and examples	p. 108
Degree of submatroids and quotient geometries	p. 110
Affine geometries	p. 112
Embedding of a geometry of degree 1	p. 117
Exercises	p. 121
Morphisms of Projective Geometries	p. 127
Partial maps	p. 128
Definition, properties and examples of morphisms	p. 133
Morphisms induced by semilinear maps	p. 137
The category of projective geometries	p. 141
Homomorphisms	p. 143
Examples of homomorphisms	p. 148
Exercises	p. 151
Embeddings and Quotient-Maps	p. 157
Mono-sources and initial sources	p. 158
Embeddings	p. 163
Epi-sinks and final sinks	p. 169
Quotient-maps	p. 172
Complementary subpaces	p. 177
Factorization of morphisms	p. 179
Exercises	p. 182
Endomorphisms and the Desargues Property	p. 187
Axis and center of an endomorphism	p. 188
Endomorphisms with a given axis	p. 191
Endomorphisms induced by a hyperplane-embedding	p. 195
Arguesian geometries	p. 197
Non-arguesian planes	p. 204
Exercises	p. 209
Homogeneous Coordinates	p. 215
The homothety fields of an arguesian geometry	p. 216
Coordinates and hyperplane-embeddings	p. 218
The fundamental theorem for homomorphisms	p. 221
Uniqueness of the associated fields and vector spaces	p. 224
Arguesian planes	p. 226
The Pappus property	p. 228
Exercises	p. 230
Morphisms and Semilinear Maps	p. 235
The fundamental theorem	p. 236
Semilinear maps and extensions of morphisms	p. 238
The category of arguesian geometries	p. 242
Points in general position	p. 244
Projective subgeometries of an arguesian geometry	p. 247
Exercises	p. 249
Duality	p. 255
Duality for vector spaces	p. 256
The dual geometry	p. 258
Pairings, dualities and embedding into the bidual	p. 261
The duality functor	p. 264
Pairings and sesquilinear forms	p. 267
Exercises	p. 269
Related Categories	p. 275
The category of closure spaces	p. 276
Galois connections and complete lattices	p. 278
The category of complete atomistic lattices	p. 281
Morphisms between affine geometries	p. 284
Characterization of strong morphisms	p. 287
Characterization of morphisms	p. 291
Exercises	p. 295
Lattices of Closed Subspaces	p. 301
Topological vector spaces	p. 302
Mackey geometries	p. 305
Continuous morphisms	p. 308
Dualized geometries	p. 310
Continuous homomorphisms	p. 315
Exercises	p. 318
Orthogonality	p. 323
Orthogeometries	p. 324
Ortholattices and orthosystems	p. 327
Orthogonal morphisms	p. 330
The adjunction functor	p. 334
Hilbertian geometries	p. 337
Exercises	p. 340
List of Problems	p. 345
Bibliography	p. 347
List of Axioms	p. 357
List of Symbols	p. 358
Index	p. 359
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Modern Projective Geometry

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