Generic stability and stability

We prove two results about generically stable types p in arbitrary theories. The first, on existence of strong germs, generalizes results from [3] on stably dominated types. The second is an equivalence of forking and dividing, assuming generic stability of p for all m. We use the latter result to answer in full generality a question posed by Hasson and Onshuus: If p(x) ∈ S(B) is stable and does not fork over A then p A is stable. (They had solved some special cases.) 1 Stable and generically stable types Our notation is standard. We work with an arbitrary complete theory T in language L. C denotes a ‘monster model’. M denotes a small elementary submodel, and A,B, . . . denote small subsets. L(C) denotes the collection of formulas with parameters from C, likewise L(A) etc. We sometimes say A-invariant for Aut(C/A)-invariant. By a global type we mean a complete type over C. We assume familiarity with notions from model theory such as heir, coheir, definable type, forking. The book [7] is a good reference, but see also [5].