Intersection theorem

Intersection theorem This article is about projective geometry. For a result on tensor products of modules, see 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 A and 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 A and 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: {displaystyle {A,B,C,a,b,c,P,Q,R,O}} Lines: {displaystyle {AB,AC,BC,ab,ac,bc,Aa,Bb,Cc,PQ}} Incidences (in addition to obvious ones such as {displaystyle (A,AB)} ): {displaystyle {(O,Aa),(O,Bb),(O,Cc),(P,BC),(P,bc),(Q,AC),(Q,ac),(R,AB),(R,ab)}} The implication is then {displaystyle (R,PQ)} —that point R is incident with line PQ.

Famous examples Desargues' theorem holds in a projective plane P if and only if P is the projective plane over some division ring (skewfield} D — {displaystyle 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 P satisfies the intersection theorem if and only if the division ring D satisfies the rational identity.

Pappus's hexagon theorem holds in a desarguesian projective plane {displaystyle mathbb {P} _{2}D} if and only if D is a field; it corresponds to the identity {displaystyle forall a,bin D,quad acdot b=bcdot a} . Fano's axiom (which states a certain intersection does not happen) holds in {displaystyle mathbb {P} _{2}D} if and only if D has characteristic {displaystyle neq 2} ; it corresponds to the identity a + a = 0. References Rowen, Louis Halle, ed. (1980). Polynomial Identities in Ring Theory. Pure and Applied Mathematics. Vol. 84. Academic Press. doi:10.1016/s0079-8169(08)x6032-5. ISBN 9780125998505. Amitsur, S. A. (1966). "Rational Identities and Applications to Algebra and Geometry". Journal of Algebra. 3 (3): 304–359. doi:10.1016/0021-8693(66)90004-4. Categories: Incidence geometryTheorems in projective geometry