Ultraproducts and Łoś’ Theorem

Abstract

In this chapter, W denotes an infinite set, always used as an index set, on which we fix a non-principal ultrafilter.¹ Given any collection of (first-order) structures indexed by W, we can define their ultraproduct. However, in this book, we will be mainly concerned with the construction of an ultraproduct of rings, an ultraring for short, which is then defined as a certain residue ring of their Cartesian product. From this point of view, the construction is purely algebraic, although it is originally a model-theoretic one (we only provide some supplementary background on the model-theoretic perspective). We review some basic properties (deeper theorems will be proved in the later chapters), the most important of which is Łoś Theorem, relating properties of the approximations with their ultraproduct. When applied to algebraically closed fields, we arrive at a result that is pivotal in most of our applications: the Lefschetz Principle (Theorem 2.4.3), allowing us to transfer many properties between positive and zero characteristic.