Kleiman's theorem

rdf:langString Kleiman's theorem

rdf:langString In algebraic geometry, Kleiman's theorem, introduced by , concerns dimension and smoothness of scheme-theoretic intersection after some perturbation of factors in the intersection. Precisely, it states: given a connected algebraic group G acting transitively on an algebraic variety X over an algebraically closed field k and morphisms of varieties, G contains a nonempty open subset such that for each g in the set, 1. * either is empty or has pure dimension , where is , 2. * (Kleiman–Bertini theorem) If are smooth varieties and if the characteristic of the base field k is zero, then is smooth. Statement 1 establishes a version of Chow's moving lemma: after some perturbation of cycles on X, their intersection has expected dimension.