The classification of compact punctually cohesive Desarguesian projective Klingenberg planes (Q752479)

A projective Klingenberg plane (PK-plane) \({\mathcal P}\) is called Desarguesian iff for some local ring \({\mathbf R}\) it is isomorphic to the ring plane \({\mathcal P}(R)\), it is called topological iff its point and line sets are equipped with topologies such that the neighbour relations \(\sim\) on the set of points, resp. lines are closed, and joining of points and intersecting of lines are continuous outside \(\sim.\) The author shows that compact Desarguesian PK-planes \({\mathcal P}(R)\) whose canonical image \({\mathcal P}(R/J)\) is non-discrete (\(J\) denotes the maximal ideal of \({\mathbf R}\)), are ordinary infinite compact projective planes with \(\sim =id\) (id denotes the identity relation on the respective set), while those which are not equal to ordinary finite or infinite projective planes (with \(\sim =id)\) correspond to compact local rings with \({\mathbf J}\neq \{0\}\). If, in the latter case furthermore \({\mathcal P}\) is punctually cohesive (i.e. any two points are incident with at least one common line), a classification is obtained using the author's unpublished results on compact right chain rings: either \({\mathcal P}\) is a finite projective Hjelmslev plane with \(\sim \neq id\) or \({\mathcal P}\) is an infinite non-discrete totally disconnected ordinary topological projective plane with \(\sim \neq id\). The local rings occurring in the second case are described as invariant compact chain domains being openly embedded into their right skewfield of fractions. The corresponding planes are also described as inverse limits of finite projective Hjelmslev planes.