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Revisiting virtual difference ideals

Zoé Chatzidakis, Ehud Hrushovski

Abstract

The main idea of [4] was that structures built from periodic prime ideals have better properties from the usual ones built from invariant ideals; but unable to work with periodic ideals alone, we had to generalise further to a somewhat ephemeral setting called virtual ideals. This text has two purposes. It corrects an error in [4] discovered by Tom Scanlon's UCB seminar, recovering all results for all virtual ideals. In addition, based on results in [3], we describe a wide family of difference equations where virtual ideals reduce to periodic ideals.

Revisiting virtual difference ideals

Abstract

The main idea of [4] was that structures built from periodic prime ideals have better properties from the usual ones built from invariant ideals; but unable to work with periodic ideals alone, we had to generalise further to a somewhat ephemeral setting called virtual ideals. This text has two purposes. It corrects an error in [4] discovered by Tom Scanlon's UCB seminar, recovering all results for all virtual ideals. In addition, based on results in [3], we describe a wide family of difference equations where virtual ideals reduce to periodic ideals.

Paper Structure

This paper contains 4 sections, 18 theorems, 7 equations.

Table of Contents

  1. Setting, notation, basic definitions
  2. Existence theorems for periodic ideals
  3. Going through sections 2, 3 and 4 of CHP
  4. Using the Zariski geometry to get the trichotomy

Key Result

Proposition 2.1

(Addendum to Proposition 2.4 of CHP) Let $(R,R_\sigma)$ be a pair of coordinate rings.

Theorems & Definitions (19)

  • Proposition 2.1
  • Lemma 2.2
  • Remark 2.3
  • Lemma 2.4
  • Corollary 2.5
  • Corollary 2.6
  • Lemma 2.7
  • Lemma 2.8
  • Lemma 2.9
  • Proposition 2.10
  • ...and 9 more