Supergénérix

Abstract

Given a subroup G of a stable (in the model-theoric sense) group Γ, in particular when Γ is a group of finite Morley rank, the traces on G of the definable subsets of Γ have a remarkable property: if the definable closure of G is connected, they are either supergeneric, or supergenerically complemented, in the sense of the definition given at the very beginning of this paper. An example of this situation is provided by the linear groups: for some n, G is a subgroup of Γ=GLn(K), where K is a field that we may take algebraically closed; the definable sets in the sense of GLn(K) are its constructible subsets, i.e. the boolean combinations of a finite number of its Zariski closed subsets.For any group G, the supergeneric subsets of G form a filter of large sets, which, to my best knowledge, is defined here for the first time. This paper undertakes the study of supergenericity in a general context, with no hypotheses of a model-theoric nature, but with a special attention given to the very specific properties of genericity possessed by the definable subsets of a stable group. It can be read without any knowledge of Logic, provided that one is ready to skip the proofs of the theorems showing precisely that these definable sets have these properties.