Abstract
This paper gives an axiomatic characterization of projective planes over rings of stable rank 2. These rings, which are known from algebraic K-theory, beautifully reflect simple geometric properties. The basic relations in the plane are incidence and the neighbor relation. The axioms consist of a number of axioms expressing elementary relations between points and lines such as, e. g., the existence of a unique line joining any two non-neighboring points, and a couple of axioms ensuring the existence of transvections and dilatations.
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References
Artin, E.: Geometric Algebra. Interscience Publishers, New York, London, 1957.
Barbilian, D.: ‘Zur Axiomatik der projektiven ebenen Ringgeometrien’. I, Jahresbericht D. M. V. 50 (1940), 179–229; II, Jahresbericht D. M. V. 51 (1941), 34–76.
Benz, W.: ‘Ebene Geometrie über einem Ring.’ Math. Nachr. 59 (1974), 163–193.
Bingen, F.: ‘Géométrie projective sur un anneau semiprimaire’. Acad. Roy. Belg., Bull. Cl. Sci. (5) 52 (1966), 13–24.
Cohn, P. M.: ‘Some Remarks on the Invariant Basis Property’. Topology 5 (1966), 215–228.
Cohn, P. M.: Free Rings and their Relations, Academic Press, London, New York, 1971.
Faltings, K.: ‘Modulare Verbände mit Punktsystemen’. Geometriae Dedicata 4 (1975), 105–137.
Hjelmslev, J.: ‘Einleitung in die allgemeine Kongruenzlehre’. Danske Vid. Selsk., Mat.-Fys. Medd. 8, nr. 11 (1929); 10, nr. 1 (1929); 19, nr. 12 (1942); 22, nr. 6 and nr. 13 (1945); 25, nr. 10 (1949).
Hughes, D. R. and Piper, F. C.: Projective Planes. Springer-Verlag, New York, Heidelberg, Berlin, 1973.
Kallen, W. van der: ‘Injective stability for K2’. In: Algebraic K-theory, Evanston 1976. Lecture Notes in Math., Vol. 551, Springer-Verlag, Berlin, Heidelberg, New York, 1976, pp. 77–154.
Klingenberg, W.: ‘Projektive und affine Ebenen mit Nachbarelementen’. Math. Z. 60 (1954), 384–406.
Klingenberg, W.: ‘Euklidische Ebenen mit Nachbarelementen’. Math. Z. 61 (1954), 1–25.
Klingenberg, W.: ‘Desarguesche Ebenen mit Nachbarelementen’. Abh. Math. Sem. Univ. Hamburg 20 (1955), 97–111.
Klingenberg, W.: ‘Projektive Geometrien mit Homomorphismus’. Math. Ann 132 (1956), 180–200.
Krusemeyer, M. I.: ‘Fundamental Groups, Algebraic K-theory and a Problem of Abhyankar’. Invent. Math. 19 (1973), 15–47.
Leiszner, W.: ‘Affine Barbilian-Ebenen’. I, J. Geometry 6 (1975), 31–57; II, J. Geometry 6 (1975) 105–129.
Springer, T. A., and Veldkamp, F. D.: ‘On Hjelmslev-Moufang Planes’. Math. Z. 107 (1968), 249–263.
Vasershtein, L. N.: ‘Stable Rank of Rings and Dimensionality of Topological Spaces’. Functional Anal. Appl. 5 (1971) 102–110.
Veldkamp, F. D.: ‘Collineation Groups in Hjelmslev-Moufang Planes’. Math. Z. 108 (1968), 37–52.
Veldkamp, F. D.: ‘Unitary groups in Hjelmslev-Moufang Planes’. Math. Z. 108 (1968) 288–312.
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Veldkamp, F.D. Projective planes over rings of stable rank 2. Geom Dedicata 11, 285–308 (1981). https://doi.org/10.1007/BF00149352
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DOI: https://doi.org/10.1007/BF00149352