Plane elliptic geometry over rings
Part I of the article starts out from a projective plane over a commutative ring with a unit element. Let 2 be a unit. A symmetric bilinear form \(f: R^ 3\times R^ 3\to R\) supplies an orthogonality relation on the set of lines. If (a,a)f is a unit then the line Ra admits a reflection \(\sigma_{Ra}\). Reflections \(\sigma_{Ra}\) and \(\sigma_{Rb}\) commute if and only if \(a\sigma_{Rb}=\lambda a\) for some \(\lambda\in R\) with \(\lambda^ 2=1\). The identity \(\sigma_{Ra}\sigma_{Rb}=\sigma_{Rc}\) holds if and only if any two of the vectors a,b,c are orthogonal. If (a,a)f is a unit whenever Ra is a line (i.e. a,b,c is a basis of \(R^ 3\) for some b,c) then the projective plane together with the orthogonality induced by f is called an elliptic ring-plane \(\Pi\) (R,f). Let S be the set of reflections at lines of an elliptic ring-plane and let lower case letters denote elements of S. Let \(a| b\) stand for ab\(\in S\); \(a\underset {=} ub\) denotes that \(c| a,b\) for just one c; and \(a\hat=b\) denotes that \(a| c;c| d;d| b\) and \(a\underset {=} ub\) and \(c\underset {=} ub\) for some c,d\(\in S\). The following properties are basic: (E1) If \(a\hat=b\) and \(c| b\) then \(a\underset {=} uc\). (E2) If \(a| b\) then \(c| a\) and \(c\underset {=} ub\) and \(c\hat=b\) for some c. (E3) a,b,c\(| d\) implies abc\(\in S\). (M) If \(a\hat=b\) then ab is not an involution. - If the ring has stable rank \(\leq 3\) then S fulfills (U) Given a,b; then \(c| a\) and \(c\underset {=} ub\) for some c. Part II is a synthetic approach to elliptic ring-planes. Let G be a group and S a non-empty subset of involutions generating G which is invariant under inner automorphisms of G. Let (G,S) satisfy (E1), (E2), (E3), (M) and (U). The main theorem asserts the existence of an elliptic ring-plane \(\Pi\) (R,f) and an injective homomorphism \(\sigma\) of G into \(\{\alpha \in Aut(R^ 3,f):\det \alpha =1\}\) such that the elements of \(S\sigma\) are reflections at lines and such that \(G\sigma\) contains \(\{\alpha \in Aut(R^ 3,f):\det \alpha =1\quad and\quad \det (1+\alpha)\quad is\quad a\quad R-unit\}.\) The proof involves known methods (pseudo-plane, pseudo- semirotations) but also new ideas are developed, e.g. in order to construct transvections.
scientific article; zbMATH DE number 3909210