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A henselian preparation theorem

Laurent Moret-Bailly

Abstract

We prove an analogue of the Weierstrass preparation theorem for henselian pairs, generalizing the local case recently proved by Bouthier and {\v C}esnavi{\v c}ius. As an application, we construct a henselian analogue of the resultant of p-adic series defined by Berger.

A henselian preparation theorem

Abstract

We prove an analogue of the Weierstrass preparation theorem for henselian pairs, generalizing the local case recently proved by Bouthier and {\v C}esnavi{\v c}ius. As an application, we construct a henselian analogue of the resultant of p-adic series defined by Berger.
Paper Structure (14 sections, 4 theorems, 3 equations)

This paper contains 14 sections, 4 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Related results
  3. Outline of the paper
  4. Preliminary results
  5. Review of henselian pairs
  6. Henselization
  7. Structure of henselian series rings
  8. Evaluation
  9. Schemes over henselian pairs: a decomposition result
  10. The preparation theorem
  11. Notation and assumptions
  12. Proof of Theorem \ref{['ThIntro']}
  13. Application: a henselian resultant
  14. Properties of the resultant

Key Result

Theorem 1.1

Let $R$ be a ring, $I$ an ideal of $R$. Assume that $(R,I)$ is a henselian pair. Let $d$ be a natural integer and let $f$ be an element of $R\{{t}\}$ which in $R[[t]]$ has the form $f= \sum_{i\geq0}a_{i}t^{i}$, where $a_{d}\in R^\times$ and $a_{i}\in I$ for $i<d$. Then:

Theorems & Definitions (10)

  • Theorem 1.1
  • Definition 2.1.1
  • Lemma 2.2.1
  • proof
  • Proposition 2.3.2
  • proof
  • Proposition 3.2
  • proof
  • Definition 3.3
  • Definition 4.1