Orthogonality to the value group is the same as generic stability in C-minimal expansions of ACVF
نویسنده
چکیده
Proof. Suppose the first condition holds. Let B ⊇ C and let a be any realization of p|B. For γ ∈ Γ(Ba), we can write γ as f(a) for some B-definable function. Then γ |= f∗p|B. Also, p is B-invariant and f is B-definable, so the type f∗p is B-invariant. Since it is constant, it must contain the formula x = γ0 for some γ0, and γ0 must be B-definable. Therefore the formula x = γ0 is in f∗p|B, and so γ = γ0 ∈ Γ(B). As γ was an arbitrary element of Γ(Ba), we conclude that Γ(Ba) = Γ(B). Conversely, suppose that the second condition holds. Let f be an U-definable function into Γ. Let B be a set containing C, over which f is defined. Let a realize p|B. Then f(a) ∈ Γ(Ba) = Γ(B). Since f(a) |= f∗p|B, and f(a) is B-definable, the formula x = f(a) must be in f∗p|B, so f∗p is a constant type.
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تاریخ انتشار 2014