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 (zum Beispiel, a point and a line). Das "Satz" besagt, dass, whenever a set of objects satisfies the incidences (d.h. 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.

Zum Beispiel, Desargues' theorem can be stated using the following incidence structure: Points: {Anzeigestil {EIN,B,C,a,b,c,P,Q,R,Ö}} Lines: {Anzeigestil {AB,AC,BC,ab,ac,v. Chr,Aa,Bb,Cc,PQ}} Incidences (in addition to obvious ones such as {Anzeigestil (EIN,AB)} ): {Anzeigestil {(Ö,Aa),(Ö,Bb),(Ö,Cc),(P,BC),(P,v. Chr),(Q,AC),(Q,ac),(R,AB),(R,ab)}} The implication is then {Anzeigestil (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 {Anzeigestil 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 {Anzeigestil mathbb {P} _{2}D} if and only if D has characteristic {Anzeigestil neq 2} ; it corresponds to the identity a + a = 0. References Rowen, Louis Halle, ed. (1980). Polynomial Identities in Ring Theory. Reine und Angewandte Mathematik. Vol. 84. Akademische Presse. doi:10.1016/s0079-8169(08)x6032-5. ISBN 9780125998505. Amitsur, S. EIN. (1966). "Rational Identities and Applications to Algebra and Geometry". Zeitschrift für Algebra. 3 (3): 304–359. doi:10.1016/0021-8693(66)90004-4. Kategorien: Incidence geometryTheorems in projective geometry