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On groups definable in geometric fields with generic derivations

Anand Pillay, Françoise Point, Silvain Rideau-Kikuchi

Abstract

We study groups definable in existentially closed geometric fields with commuting derivations. Our main result is that such a group can be definably embedded in a group interpretable in the underlying geometric field. Compared to earlier work of the first two authors toguether with K. Peterzil, the novelty is that we also deal with infinite dimensional groups.

On groups definable in geometric fields with generic derivations

Abstract

We study groups definable in existentially closed geometric fields with commuting derivations. Our main result is that such a group can be definably embedded in a group interpretable in the underlying geometric field. Compared to earlier work of the first two authors toguether with K. Peterzil, the novelty is that we also deal with infinite dimensional groups.

Paper Structure

This paper contains 3 sections, 12 theorems, 1 equation.

Table of Contents

  1. Geometric fields with generic derivations
  2. Generic points
  3. Groups

Key Result

Theorem 1

Let $K \models T_\Delta$ and let $\Gamma$ be a group which is $L_\Delta$-definable in $K$. Then there is a group $G$ which is $L$-interpretable in (the reduct to $L$ of) $K$ and an $L_\Delta$-definable embedding $\Gamma\to G$.

Theorems & Definitions (26)

  • Theorem : \ref{['emb Gp']}
  • Definition 1.1
  • Lemma 1.2
  • proof
  • Corollary 1.3
  • proof
  • Corollary 1.4
  • proof
  • Lemma 1.5
  • proof
  • ...and 16 more