Fields with Lie-commuting and iterative operators

Abstract

We introduce a general framework for studying fields equipped with operators, given as co-ordinate functions of homomorphisms into a local algebra $\mathcal{D}$, satisfying various compatibility conditions that we denote by $\Gamma$ and call such structures $\mathcal{D}^{\Gamma}$-fields. These include Lie-commutativity of derivations and $\mathfrak g$-iterativity of (truncated) Hasse-Schmidt derivations. Our main result is about the existence of principal realisations of $\mathcal{D}^{\Gamma}$-kernels. As an application, we prove companionability of the theory of $\mathcal{D}^{\Gamma}$-fields and denote the companion by $\mathcal{D}^{\Gamma}$-CF. In characteristic zero, we prove that $\mathcal{D}^{\Gamma}$-CF is a stable theory that satisfies the CBP and Zilber's dichotomy for finite-dimensional types. We also prove that there is a uniform companion for model-complete theories of large $\mathcal{D}^{\Gamma}$-fields, which leads to the notion of $\mathcal{D}^{\Gamma}$-large fields and we further use this to show that PAC substructures of $\mathcal{D}^{\Gamma}$-DCF are elementary.