2004/06/23 by Tapani Hyttinen, Hyttinen, Tapani, Olivier Lessmann +3
Mathematics · #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Geometric and Algebraic Topology #math.LO
paper · pdf · doi:10.48550/arxiv.math/0406481
published as J. Math. Log. 5 No. 1 (2005) 1--47
arxiv created 2004/06/23 · arxiv updated 2009/12/01
This paper is concerned with extensions of geometric stability theory to some nonelementary classes. We prove the following theorem: Theorem: Let C be a large homogeneous model of a stable diagram D. Let p, q in SD(A), where p is quasiminimal and q unbounded. Let P=p(C) and Q=q(C). Suppose that there exists an integer n<omega such that dim(a1...an/A cup C)=n, for any independent a1,..., an in P and finite subset C subseteq Q, but dim(a1...an an+1/A cup C) <= n, for some independent a1,...,an,an+1 in P and some finite subset C subseteq Q. Then C interprets a group G which acts on the geometry P' obtained from P. Furthermore, either C interprets a non-classical group, or n=1,2,3 and * If n=1 then G is abelian and acts regularly on P'. * If n=2 the action of G on P' is isomorphic to the affine action of K times K^* on the algebraically closed field K. * If n = 3 the action of G on P' is isomorphic to the action of PGL2(K) on the projective line P1(K) of the algebraically closed field K .