The complexity of tilting

Abstract

Many explicit constructions in algebra are not first-order definable in classical finitary logic, but are, however, definable in infinitary logic. The aim of this paper is to show how infinitary logic can be used to lift model-theoretic results in arithmetic to a quantifier-sensitive form. We apply this idea to a deep result of Jahnke and Kartas on the model theory of perfectoid fields. Our main result is that a first-order sentence $ϕ$ holds in the tilt $k^{\flat}$ of a perfectoid field $k$ if and only if a certain sentence $ϕ^{\natural}$ holds in a non-principal ultrapower of $k$, where $ϕ^{\natural}$ does not depend on $k$ and is only slightly more complicated than $ϕ$ itself. In particular, tilting preserves the Lévy hierarchy in a precise sense. To this end, we prove that the Standard Decomposition of a valued field at a constant $t$ is infinitarily interpretable by formulas of rank at most 1.