Desarguesian Klingenberg planes

Abstract

Klingenberg planes are generalizations of Hjelmslev planes. If R is a local ring, one can construct a projective Klingenberg plane V ( R ) {\textbf {V}}(R) and a derived affine Klingenberg plane A ( R ) {\textbf {A}}(R) from R. If V is a projective Klingenberg plane, if R 1 , R 2 {R_1},\,{R_2} and R 3 {R_3} are local rings, if s 1 , s 2 {s_1},\,{s_2} and s 3 {s_3} are the sides of a nondegenerate triangle in V, and if each of the derived affine Klingenberg planes a ( V , s i ) \mathcal {a}\left ( {V,\,{s_i}} \right ) is isomorphic to A ( R i ) , {\textbf {A}}({R_i}),\, , i = 1 , 2 , 3 i\, = \,1,\,2,\,3 , then the rings R 1 , R 2 {R_1},\,{R_2} and R 3 {R_3} are isomorphic, and V is isomorphic to V ( R 1 ) ; {\textbf {V}}({R_1}); ; also, if g is a line of V, then the derived affine Klingenberg plane a ( V , g ) \mathcal {a}({V,\,g}) is isomorphic to A ( R 1 ) \textbf {A}({R_1}) . Examples are given of projective Klingenberg planes V, each of which has the following two properties: (1) V is not isomorphic to V ( R ) {\textbf {V}}(R) for any local ring R; and (2) there is a flag ( B , b ) (B,\,b) of V, and a local ring S such that each derived affine Klingenberg plane a ( V , m ) \mathcal {a}({V,\,m}) is isomorphic to A ( S ) {\textbf {A}}(S) whenever m = b m\, = \,b , or m is a line through B which is not neighbor to b.