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Depth of Artin-Schreier defect towers

Enric Nart, Josnei Novacoski

TL;DR

This work investigates the depth of Artin–Schreier defect towers in valued fields through Okutsu sequences. It develops a framework showing that, under rank-one Henselian assumptions, the compositum of finitely many linearly disjoint AS defect extensions often has depth one, and it provides precise criteria and constructions for depth-two AS towers within the Hahn field. The paper presents explicit examples (both Galois and non-Galois) of depth-two towers and proves depth-two in a degree $p^3$ tower, offering supporting evidence for a broader conjecture that depth-one towers arise exactly as compositums of AS extensions. By connecting depth to Okutsu data, Krasner-type criteria, and brick decompositions in the Hahn field, the results contribute to a clearer classification of defect towers and their potential role in understanding Kähler differentials in valued extensions.

Abstract

The depth of a simple algebraic extension $(L/K,v)$ of valued fields is the minimal length of the Mac Lane-Vaquié chains of the valuations on $K[x]$ determined by the choice of different generators of the extension. In a previous paper, we characterized the defectless unibranched extensions of depth one. In this paper, we analyze this problem for towers of Artin-Schreier defect extensions. Under certain conditions on $(K,v)$, we prove that the towers obtained as the compositum of linearly disjoint defect Artin-Schreier extensions of $K$ have depth one. We conjecture that these are the only depth one Artin-Schreier defect towers and we present some examples supporting this conjecture.

Depth of Artin-Schreier defect towers

TL;DR

This work investigates the depth of Artin–Schreier defect towers in valued fields through Okutsu sequences. It develops a framework showing that, under rank-one Henselian assumptions, the compositum of finitely many linearly disjoint AS defect extensions often has depth one, and it provides precise criteria and constructions for depth-two AS towers within the Hahn field. The paper presents explicit examples (both Galois and non-Galois) of depth-two towers and proves depth-two in a degree tower, offering supporting evidence for a broader conjecture that depth-one towers arise exactly as compositums of AS extensions. By connecting depth to Okutsu data, Krasner-type criteria, and brick decompositions in the Hahn field, the results contribute to a clearer classification of defect towers and their potential role in understanding Kähler differentials in valued extensions.

Abstract

The depth of a simple algebraic extension of valued fields is the minimal length of the Mac Lane-Vaquié chains of the valuations on determined by the choice of different generators of the extension. In a previous paper, we characterized the defectless unibranched extensions of depth one. In this paper, we analyze this problem for towers of Artin-Schreier defect extensions. Under certain conditions on , we prove that the towers obtained as the compositum of linearly disjoint defect Artin-Schreier extensions of have depth one. We conjecture that these are the only depth one Artin-Schreier defect towers and we present some examples supporting this conjecture.

Paper Structure

This paper contains 17 sections, 36 theorems, 146 equations.

Table of Contents

  1. Okutsu sequences and depth
  2. Artin-Schreier defect extensions
  3. Background on AS extensions
  4. Compositum of AS extensions
  5. Main theorem
  6. AS towers of degree $p^2$
  7. Subfields of the Hahn field
  8. Truncation and brick components of elements in the Hahn field
  9. Subfields of $\mathbb H$ with no finite limits
  10. AS extensions of $K$
  11. Conjecture and examples
  12. Abhyankar's AS extension.
  13. A Galois AS tower of depth two
  14. A non-Galois AS tower of depth two
  15. Depth of the tower $N/K$ of degree $p^3$
  16. ...and 2 more sections

Key Result

Theorem 1.3

The length $r$ of any Okutsu sequence of $\theta$ is equal to $\operatorname{\hbox{\rm\small depth}}(\theta)$. Moreover, for every MLV chain of $v_\theta$: and every $0\le \ell<r$, the augmentation $\mu_\ell\, \to\, \mu_{\ell+1}$ is ordinary if and only if $D_{m_\ell}$ contains a maximal element.

Theorems & Definitions (46)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Theorem 2.4
  • Remark 2.5
  • Definition 2.6
  • Remark 2.7
  • ...and 36 more