In mathematics, the Veblen–Young theorem, proved by , states that a
projective space of dimension at least 3 can be constructed as the projective space associated to a vector space over a
division ring
In algebra, a division ring, also called a skew field (or, occasionally, a sfield), is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicativ ...
.
Non-Desarguesian planes give examples of 2-dimensional projective spaces that do not arise from vector spaces over division rings, showing that the restriction to dimension at least 3 is necessary.
Jacques Tits
Jacques Tits () (12 August 1930 – 5 December 2021) was a Belgian-born French mathematician who worked on group theory and incidence geometry. He introduced Tits buildings, the Tits alternative, the Tits group, and the Tits metric.
Early life ...
generalized the Veblen–Young theorem to
Tits buildings, showing that those of rank at least 3 arise from
algebraic groups.
generalized the Veblen–Young theorem to
continuous geometry In mathematics, continuous geometry is an analogue of complex projective geometry introduced by , where instead of the dimension of a subspace being in a discrete set 0, 1, \dots, \textit, it can be an element of the unit interval ,1 Von Neuman ...
, showing that a
complemented modular lattice of order at least 4 is isomorphic to the
principal right ideals of a
von Neumann regular ring
In mathematics, a von Neumann regular ring is a ring ''R'' (associative, with 1, not necessarily commutative) such that for every element ''a'' in ''R'' there exists an ''x'' in ''R'' with . One may think of ''x'' as a "weak inverse" of the eleme ...
.
Statement
A projective space ''S'' can be defined abstractly as a set ''P'' (the set of points), together with a set ''L'' of subsets of ''P'' (the set of lines), satisfying these axioms :
* Each two distinct points ''p'' and ''q'' are in exactly one line.
* Veblen's axiom: If ''a'', ''b'', ''c'', ''d'' are distinct points and the lines through ''ab'' and ''cd'' meet, then so do the lines through ''ac'' and ''bd''.
* Any line has at least 3 points on it.
The Veblen–Young theorem states that if the dimension of a projective space is at least 3 (meaning that there are two non-intersecting lines) then the projective space is isomorphic with the projective space of lines in a vector space over some
division ring
In algebra, a division ring, also called a skew field (or, occasionally, a sfield), is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicativ ...
''K''.
References
*
*
*
*
*
{{DEFAULTSORT:Veblen-Young theorem
Theorems in projective geometry
Theorems in algebraic geometry