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Flat dimension for power series over valuation rings

Adam Jones

TL;DR

The paper studies the power series ring $R[[X]]$ over a rank-1 valuation ring with dense value group, showing a counterexample to Hilbert's syzygy theorem in this non-Noetherian setting. It develops valuation-theoretic tools to extend the base valuation to $R[[X]]$, proves that $R[[X]]$ is not coherent via a valuation-based construction, and uses localizations to produce a flat $R$-module $C$ whose flat dimension over $R[[X]]$ is at least 2. The main construction relies on pure submodules and localization arguments to relate flatness over $R$ and over $R[[X]]$, revealing a breakdown of the weak Hilbert syzygy behavior in this context. The results illustrate the limitations of classical syzygy theory for power series over non-Noetherian valuation rings and demonstrate how incoherence drives higher flat-dimension phenomena.

Abstract

We examine the power series ring $R[[X]]$ over a valuation ring $R$ of rank 1, with proper, dense value group. We give a counterexample to Hilbert's syzygy theorem for $R[[X]]$, i.e. an $R[[X]]$-module $C$ that is flat over $R$ and has flat dimension at least 2 over $R[[X]]$, contradicting a previously published result. The key ingredient in our construction is an exploration of the valuation theory of $R[[X]]$. We also use this theory to give a new proof that $R[[X]]$ is not a coherent ring, a fact which is essential in our construction of the module $C$.

Flat dimension for power series over valuation rings

TL;DR

The paper studies the power series ring over a rank-1 valuation ring with dense value group, showing a counterexample to Hilbert's syzygy theorem in this non-Noetherian setting. It develops valuation-theoretic tools to extend the base valuation to , proves that is not coherent via a valuation-based construction, and uses localizations to produce a flat -module whose flat dimension over is at least 2. The main construction relies on pure submodules and localization arguments to relate flatness over and over , revealing a breakdown of the weak Hilbert syzygy behavior in this context. The results illustrate the limitations of classical syzygy theory for power series over non-Noetherian valuation rings and demonstrate how incoherence drives higher flat-dimension phenomena.

Abstract

We examine the power series ring over a valuation ring of rank 1, with proper, dense value group. We give a counterexample to Hilbert's syzygy theorem for , i.e. an -module that is flat over and has flat dimension at least 2 over , contradicting a previously published result. The key ingredient in our construction is an exploration of the valuation theory of . We also use this theory to give a new proof that is not a coherent ring, a fact which is essential in our construction of the module .
Paper Structure (5 sections, 12 theorems, 3 equations)

This paper contains 5 sections, 12 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Valuation theory for power series rings
  3. Incoherence of $R[[X]]$
  4. Localisations of $R[[X]]$
  5. Flat dimension of $R[[X]]$-modules

Key Result

Theorem A

Let $R$ be a valuation ring of rank 1, whose value group is a proper, dense subgroup of $\mathbb{R}$. Then there exists an $R[[X]]$-module $C$ such that f.dim$_R(C)=0$ and f.dim$_{R[[X]]}(C)\geq 2$.

Theorems & Definitions (26)

  • Definition 1.1
  • Theorem A
  • Definition 1.2
  • Lemma B
  • proof
  • Lemma C
  • Definition 1.3
  • Lemma A
  • proof
  • Lemma B
  • ...and 16 more