Morley degree in unidimensional compact complex spaces

Abstract

Let 𝒜 be the category of all reduced compact complex spaces, viewed as a multi-sorted first order structure, in the standard way. Let 𝒰 be a sub-category of 𝒜, which is closed under the taking of products and analytic subsets, and whose morphisms include the projections. Under the assumption that Th(𝒰) is unidimensional, we show that Morley rank is equal to Noetherian dimension, in any elementary extension of 𝒰. As a result, we are able to show that Morley degree is definable in Th(𝒰), when Th(𝒰) is unidimensional.