Math 2.0

Geometry Projective geometry and linear incidence geometry Theorem

Veblen–Young theorem

Builds on Projective spaceDivision ringVector spaceProjective geometryProjective planeMoulton planeWedderburn's little theoremFundamental theorem of projective geometryPlanar ternary ring

Statement

Let SS be a set of points with a family of subsets of SS, called lines, such that:

  1. any two distinct points lie on exactly one line;
  2. every line has at least three points;
  3. (Veblen's axiom) for distinct points a,b,c,da, b, c, d, if the lines abab and cdcd meet, then the lines acac and bdbd meet.

A subspace of SS is a set of points that contains the whole line through any two of its distinct points. Suppose that the longest chains S0⊊S1⊊⋯⊊Sn=SS_0 \subsetneq S_1 \subsetneq \cdots \subsetneq S_n = S of nonempty subspaces have finite length nn, and that n≥3n \ge 3. Then there is a division ring DD such that SS is isomorphic to the projective space PG(n,D)PG(n, D). Its points are the 11-dimensional subspaces of the vector space Dn+1D^{n+1}, and its lines are the 22-dimensional subspaces. A point lies on a line when it is contained in it. The division ring DD is unique up to isomorphism.

Remarks

The key step is that Desargues's theorem holds in every projective space of dimension at least 33. Given two triangles in perspective in a plane of SS, one lifts the figure out of the plane. There the theorem follows by intersecting planes, and a projection brings it back. Desargues's theorem then lets one define an addition and a multiplication on the points of a line minus one point, which form a division ring, and coordinates follow.

The hypothesis n≥3n \ge 3 is necessary. For n=2n = 2 the axioms describe projective planes, and there are planes such as the Moulton plane that are not of the form PG(2,D)PG(2, D) for any division ring DD. If SS is finite, DD is a finite division ring, hence a field by Wedderburn's little theorem, so S≅PG(n,q)S \cong PG(n, q) for a prime power qq. The division ring DD is commutative if and only if Pappus's theorem holds in the planes of SS.

Going deeper

For a student

The theorem is the reason why the synthetic and the analytic approach to projective geometry agree in dimension at least 33. It combines with the fundamental theorem of projective geometry, which describes the isomorphisms between such spaces through semilinear maps. It explains why the theory of non-classical geometries concentrates on planes, where coordinatization needs planar ternary rings. It is also a model for later results, such as the classification of polar spaces and buildings of higher rank by Tits.

Reading path

To understand this page from the ground up, read these 21 pages in order. The foundations come first.

Show the path as a graph
Collinear points, concept, no evidence levelCollinear pointsconceptCollineation, concept, no evidence levelCollineationconceptDivision ring, concept, no evidence levelDivision ringconceptField, concept, no evidence levelFieldconceptGroup, concept, no evidence levelGroupconceptPlanar ternary ring, concept, no evidence levelPlanar ternary ringconceptProjective geometry, concept, no evidence levelProjective geometryconceptProjective plane, concept, no evidence levelProjective planeconceptProjective space, concept, no evidence levelProjective spaceconceptProjective transformation, concept, no evidence levelProjective transforma…conceptVector space, concept, no evidence levelVector spaceconceptRing, definition, no evidence levelRingdefinitionDesargues's theorem, statement, no evidence levelDesargues's theoremstatementFundamental theorem of projective geometry, statement, no evidence levelFundamental theorem o…statementCommutative ring, definition, no evidence levelCommutative ringdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionMoulton plane, example, no evidence levelMoulton planeexampleDivision ring, definition, no evidence levelDivision ringdefinitionField, definition, no evidence levelFielddefinitionWedderburn's little theorem, statement, no evidence levelWedderburn's little t…statementVeblen–Young theorem, statement, no evidence levelVeblen–Young theoremstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Collinear points concept
  2. Collineation concept
  3. Division ring concept
  4. Field concept
  5. Group concept
  6. Planar ternary ring concept
  7. Projective geometry concept
  8. Projective plane concept
  9. Projective space concept
  10. Projective transformation concept
  11. Vector space concept
  12. Ring definition
  13. Desargues's theorem statement
  14. Fundamental theorem of projective geometry statement
  15. Commutative ring definition
  16. Unit of a ring definition
  17. Moulton plane example
  18. Division ring definition
  19. Field definition
  20. Wedderburn's little theorem statement
  21. Veblen–Young theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Projective_space
About this page
TagNPNNXWMVCC
URIm2://NPNNXWMVCC
Kindstatement, theorem
WritingW0 stub
MSC51A05, 51A30, 51A25
StatusThis copy of the site has no build data. Run m2 status NPNNXWMVCC.
AboutProjective space, Projective geometry
UsesProjective space, Division ring, Vector space, Projective geometry, Projective plane, Moulton plane, Wedderburn's little theorem, Fundamental theorem of projective geometry, Planar ternary ring

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