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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Desarguesian Klingenberg planes


Author: P. Y. Bacon
Journal: Trans. Amer. Math. Soc. 241 (1978), 343-355
MSC: Primary 50D35; Secondary 50A10
DOI: https://doi.org/10.1090/S0002-9947-1978-0474023-3
MathSciNet review: 0474023
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Abstract: Klingenberg planes are generalizations of Hjelmslev planes. If R is a local ring, one can construct a projective Klingenberg plane $ {\textbf{V}}(R)$ and a derived affine Klingenberg plane $ {\textbf{A}}(R)$ from R. If V is a projective Klingenberg plane, if $ {R_1},\,{R_2}$ and $ {R_3}$ are local rings, if $ {s_1},\,{s_2}$ and $ {s_3}$ are the sides of a nondegenerate triangle in V, and if each of the derived affine Klingenberg planes $ \mathcal{a}\left( {V,\,{s_i}} \right)$ is isomorphic to $ {\textbf{A}}({R_i}),\,$, $ i\, = \,1,\,2,\,3$, then the rings $ {R_1},\,{R_2}$ and $ {R_3}$ are isomorphic, and V is isomorphic to $ {\textbf{V}}({R_1});$; also, if g is a line of V, then the derived affine Klingenberg plane $ \mathcal{a}({V,\,g})$ is isomorphic to $ \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 $ {\textbf{V}}(R)$ for any local ring R; and (2) there is a flag $ (B,\,b)$ of V, and a local ring S such that each derived affine Klingenberg plane $ \mathcal{a}({V,\,m})$ is isomorphic to $ {\textbf{A}}(S)$ whenever $ m\, = \,b$, or m is a line through B which is not neighbor to b.


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DOI: https://doi.org/10.1090/S0002-9947-1978-0474023-3
Keywords: Klingenberg plane, Hjelmslev plane, local ring, desarguesian
Article copyright: © Copyright 1978 American Mathematical Society