Théorie des modèles et groupes, IMJ-PRG, Paris, 8th November 2022

Title. Cohen rings and existential AKE principles.

Abstract. A Cohen ring is a complete valuation ring of an unramified valuation of mixed characteristic. Every Cohen ring $A$ is determined up to isomorphism by its residue field $k$; and if $k$ is perfect, then $A$ is canonically isomorphic to the more familiar ring of Witt vectors $W(k)$. Thus Cohen rings are analogues of Witt rings over imperfect residue fields. Just as one studies truncated Witt rings to understand Witt rings, we study Cohen rings of positive characteristic as well as of characteristic zero. In the case $k=\mathbb{F}_{p}$, and more generally for perfect $k$, the work of Ax–Kochen, Ershov, and others gives an axiomatization of the complete theory of $A$. In case $k$ is imperfect, the algebraic picture changes since the embeddings between Cohen rings no longer correspond exactly to embeddings between residue fields. Nevertheless we still obtain a description of the complete theories of such valuation rings, and are able to prove stable embeddedness of the value group and residue field. We also obtain relative completeness and relative model completeness results for Cohen rings, which imply the corresponding Ax–Kochen/Ershov type results for unramified henselian valued fields also in case the residue field is imperfect. This is joint work with Franziska Jahnke.