Dp-finite and Noetherian NIP integral domains

TL;DR

The article advances the model-theoretic understanding of Noetherian and dp-finite integral domains in NIP frameworks by developing a robust henselianity program. It proves that dp-finite and more generally finite-dp-rank domains are henselian local, and that NIP Noetherian rings decompose into finitely many henselian local components under suitable assumptions. It further shows that the integral closure of a dp-finite Noetherian domain is a definable henselian DVR and provides a trichotomy relating residue fields and characteristic. Finally, it delivers a complete classification of dp-minimal Noetherian domains into three canonical types, thereby bridging dp-minimality with classical ring-theoretic structures. These results illuminate how definability, breadth, and henselianity interact to constrain the architecture of NIP rings and pave the way for a fuller classification of Noetherian NIP domains.

Abstract

We prove some results on NIP integral domains, especially those that are Noetherian or have finite dp-rank. If $R$ is an NIP Noetherian domain that is not a field, then $R$ is a semilocal ring of Krull dimension 1, and the fraction field of $R$ has characteristic 0. Assuming the henselianity conjecture (on NIP valued fields), $R$ is a henselian local ring. Additionally, we show that integral domains of finite dp-rank are henselian local rings. Finally, we lay some groundwork for the study of Noetherian domains of finite dp-rank, and we classify dp-minimal Noetherian domains.

Theorems & Definitions (91)

• Conjecture 1.1: Henselianity conjecture
• Conjecture 1.2: Generalized henselianity conjecture
• Theorem 1.3: = Theorem \ref{['dft']}
• Theorem 1.4: = Theorem \ref{['xyz']}
• Definition 1.5
• Theorem 1.6: = Theorem \ref{['nip-w']}
• Lemma 1.7: $\subseteq$ Corollary \ref{['hah']} $\cup$ Lemma \ref{['dprbr']}
• Remark 1.8
• Remark 1.9
• Theorem 1.10: $\subseteq$ Theorem \ref{['charzero']} $\cup$ Corollary \ref{['hah']}