The model completion of the theory of modules over finitely generated commutative algebras

Abstract

We find the model completion of the theory modules over 𝔸, where 𝔸 is a finitely generated commutative algebra over a field K. This is done in a context where the field K and the module are represented by sorts in the theory, so that constructible sets associated with a module can be interpreted in this language. The language is expanded by additional sorts for the Grassmanians of all powers of Kⁿ, which are necessary to achieve quantifier elimination. The result turns out to be that the model completion is the theory of a certain class of “big” injective modules. In particular, it is shown that the class of injective modules is itself elementary. We also obtain an explicit description of the types in this theory.