intersection theorem

{{about|projective geometry|a result on tensor products of modules|Homological conjectures in commutative algebra}}

In projective geometry, an intersection theorem or incidence theorem is a statement concerning an incidence structure – consisting of points, lines, and possibly higher-dimensional objects and their incidences – together with a pair of objects {{math|A}} and {{math|B}} (for instance, a point and a line). The "theorem" states that, whenever a set of objects satisfies the incidences (i.e. can be identified with the objects of the incidence structure in such a way that incidence is preserved), then the objects {{math|A}} and {{math|B}} must also be incident. An intersection theorem is not necessarily true in all projective geometries; it is a property that some geometries satisfy but others don't.

For example, Desargues' theorem can be stated using the following incidence structure:

Points: $\{A,B,C,a,b,c,P,Q,R,O\}$
Lines: $\{AB,AC,BC,ab,ac,bc,Aa,Bb,Cc,PQ\}$
Incidences (in addition to obvious ones such as $(A,AB)$ ): $\{(O,Aa),(O,Bb),(O,Cc),(P,BC),(P,bc),(Q,AC),(Q,ac),(R,AB),(R,ab)\}$

The implication is then $(R,PQ)$ —that point {{math|R}} is incident with line {{math|{{overbar|PQ}}}}.

Famous examples

Desargues' theorem holds in a projective plane {{math|P}} if and only if {{math|P}} is the projective plane over some division ring (skewfield) {{math|D}} — $P=\mathbb{P}_{2}D$ . The projective plane is then called desarguesian.

A theorem of Amitsur and Bergman states that, in the context of desarguesian projective planes, for every intersection theorem there is a rational identity such that the plane {{math|P}} satisfies the intersection theorem if and only if the division ring {{math|D}} satisfies the rational identity.

Pappus's hexagon theorem holds in a desarguesian projective plane $\mathbb{P}_{2}D$ if and only if {{math|D}} is a field; it corresponds to the identity $\forall a,b\in D, \quad a\cdot b=b\cdot a$ .
Fano's axiom (which states a certain intersection does not happen) holds in $\mathbb{P}_{2}D$ if and only if {{math|D}} has characteristic $\neq 2$ ; it corresponds to the identity {{math|1=a + a = 0}}.

References

{{cite book|doi=10.1016/s0079-8169(08)x6032-5|title=Polynomial Identities in Ring Theory|volume=84|series=Pure and Applied Mathematics|year=1980|isbn=9780125998505|editor-last=Rowen|editor-first=Louis Halle|publisher=Academic Press}}
{{cite journal|doi=10.1016/0021-8693(66)90004-4|title=Rational Identities and Applications to Algebra and Geometry|journal=Journal of Algebra|volume=3|issue=3|pages=304–359|year=1966|last1=Amitsur|first1=S. A.|doi-access=free}}

Category:Incidence geometry

Category:Theorems in projective geometry