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 $K^n$, 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.