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Generators for extensions of valuation rings

Josnei Novacoski

Abstract

For a finite valued field extension $(L/K,v)$ we describe the problem of find sets of generators for the corresponding extension $\mathcal O_L/\mathcal O_K$ of valuation rings. The main tool to obtain such sets are complete sets of (key) polynomials. We show that when the initial index coincide with the ramification index, sequences of key polynomials naturally give rise to sets of generators. We use this to prove Knaf's conjecture for pure extensions.

Generators for extensions of valuation rings

Abstract

For a finite valued field extension we describe the problem of find sets of generators for the corresponding extension of valuation rings. The main tool to obtain such sets are complete sets of (key) polynomials. We show that when the initial index coincide with the ramification index, sequences of key polynomials naturally give rise to sets of generators. We use this to prove Knaf's conjecture for pure extensions.
Paper Structure (6 sections, 16 theorems, 89 equations)

This paper contains 6 sections, 16 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. The generation of an extension of valuation rings
  4. About Theorem \ref{['mainthem']}
  5. Key polynomials and the defect formula
  6. About Knaf's conjecture

Key Result

Theorem 1.1

Let $(L/K,v)$ be a simple algebraic extension of valued fields. Assume that $\epsilon(L/K,v)= e(L/K,v)$ and take any complete set $\{Q_i(\eta)\}_{i\in I}$ for $(L/K,v)$. For each $\lambda\in\mathbb{N}_0^I$ there exists $a_\lambda\in K$ such that $\mathcal{O}_L$ is generated by as an $\mathcal{O}_K$-module.

Theorems & Definitions (27)

  • Theorem 1.1
  • Definition 1.2
  • Proposition 1.3
  • Theorem 3.1
  • Proposition 3.2
  • Lemma 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • ...and 17 more