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Imaginaries in perfect bounded pseudo algebraically closed fields with finitely many independent valuations

Bryan González Leandro

Abstract

In this paper, we prove weak elimination of imaginaries for perfect bounded pseudo-algebraically closed fields equipped with finitely many independent valuations. Our approach combines an extension result for types to invariant types with an amalgamation theorem. As a special case, we obtain full elimination of imaginaries when the field is equipped with a single valuation.

Imaginaries in perfect bounded pseudo algebraically closed fields with finitely many independent valuations

Abstract

In this paper, we prove weak elimination of imaginaries for perfect bounded pseudo-algebraically closed fields equipped with finitely many independent valuations. Our approach combines an extension result for types to invariant types with an amalgamation theorem. As a special case, we obtain full elimination of imaginaries when the field is equipped with a single valuation.

Paper Structure

This paper contains 9 sections, 24 theorems, 38 equations.

Table of Contents

  1. Preliminaries
  2. The language and the theory
  3. Coding of finite sets from one valuation
  4. The criterion for weak elimination of imaginaries
  5. Orthogonality of the geometric sorts
  6. Description of the algebraic closure
  7. Extending $\mathcal{L}$-types by invariant $\mathcal{L}_i$-types
  8. The 3-amalgamation theorem over geometric parameters
  9. Bringing everything together

Key Result

Lemma 3.1

Let $\varepsilon\in\mathop{\mathrm{dcl}}\nolimits_{\mathcal{L}}^{\overline{M}^{\text{a}}}(M)$. Then, there is $\eta\in M$ such that $\varepsilon$ and $\eta$ are interdefinable in the pair $(\overline{M}^{\text{a}},M)$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (59)

  • Remark 1.4
  • Definition 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 2.2
  • Remark 2.3
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • proof
  • ...and 49 more