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Existential theories of henselian valued fields under a formal smoothness assumption

Philip Dittmann

Abstract

We study existential theories of henselian valued fields of positive characteristic with parameters from a trivially valued subfield. Compared to previous work, we relax perfectness and separability assumptions, and instead work with the weaker algebraic hypothesis of formal smoothness over the parameter field, which we discuss in detail in our setting. Assuming a weak consequence of resolution of singularities, which was already used in previous work, we obtain an axiomatisation of the existential theory of such a valued field in terms of the existential theory of the residue field, both over the same parameter field. This result has natural applications to asymptotic theories of completions of function fields of curves. We work these out in detail for the case of function fields over fairly general pseudo-algebraically closed fields, where we obtain decidability of the sets of universal/existential sentences holding in all completions or all but finitely many completions, respectively.

Existential theories of henselian valued fields under a formal smoothness assumption

Abstract

We study existential theories of henselian valued fields of positive characteristic with parameters from a trivially valued subfield. Compared to previous work, we relax perfectness and separability assumptions, and instead work with the weaker algebraic hypothesis of formal smoothness over the parameter field, which we discuss in detail in our setting. Assuming a weak consequence of resolution of singularities, which was already used in previous work, we obtain an axiomatisation of the existential theory of such a valued field in terms of the existential theory of the residue field, both over the same parameter field. This result has natural applications to asymptotic theories of completions of function fields of curves. We work these out in detail for the case of function fields over fairly general pseudo-algebraically closed fields, where we obtain decidability of the sets of universal/existential sentences holding in all completions or all but finitely many completions, respectively.
Paper Structure (4 sections, 43 theorems, 13 equations)

This paper contains 4 sections, 43 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Properties of formal smoothness
  3. Existential theories of formally smooth valuation rings
  4. Applications to function fields

Key Result

Theorem 1.1

Let $(K,v)$ and $(L,w)$ be henselian non-trivially valued fields of positive characteristic. Then the residue fields $Kv$ and $Lw$ have the same existential theory if and only if the valued fields $(K,v)$ and $(L,w)$ have the same existential theory.

Theorems & Definitions (101)

  • Theorem 1.1: AnscombeFehm_existential-equichar
  • Theorem 1.2: DittmannFehm_completions; essentially AnscombeFehm_existential-equichar
  • Theorem 1.3: ADF_existential
  • Example 1.4
  • Theorem 1.5: Corollary \ref{['cor:existential-ake']}
  • Theorem 1.6: Remark \ref{['rem:pac-aa-completions-axiomatisable']}, Corollary \ref{['cor:pac-aa-completions-decidable']}, Proposition \ref{['prop:all-completions-decidable']}
  • Theorem 1.7: Theorem \ref{['thm:almost-all-res-pac']}, Proposition \ref{['prop:almost-all-res-fields-decidable']}
  • Definition 2.1: EGA-IV-1
  • Remark 2.2
  • Lemma 2.3
  • ...and 91 more