Abstract

In difference algebra, basic definable sets correspond to prime ideals that are invariant under a structural endomorphism. The main idea of an article with Peterzil (Proc.  London Math.  Soc. 85:2 (2002), 257–311) was that periodic prime ideals enjoy better geometric properties than invariant ideals, and to understand a definable set, it is helpful to enlarge it by relaxing invariance to periodicity, obtaining better geometric properties at the limit. The limit in question was an intriguing but somewhat ephemeral setting called virtual ideals. However, a serious technical error was discovered by Tom Scanlon’s UCB seminar. In this text, we correct the problem via two different routes. We replace the faulty lemma by a weaker one that still allows recovering all results of the aforementioned paper for all virtual ideals. In addition, we introduce a family of difference equations (“cumulative” equations) that we expect to be useful more generally. Previous work implies that cumulative equations suffice to coordinatize all difference equations. For cumulative equations, we show that virtual ideals reduce to globally periodic ideals, thus providing a proof of Zilber’s trichotomy for difference equations using periodic ideals alone.

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