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A Skolem–Mahler–Lech Theorem for Iterated Automorphisms of K–algebras

Published online by Cambridge University Press:  20 November 2018

Jason P. Bell  and
Jeffrey C. Lagarias
Jason P. Bell
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1. e-mail: jpbell@uwaterloo.ca
Jeffrey C. Lagarias
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043, USA. email: lagarias@umich.edu
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Abstract

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In this paper we prove a commutative algebraic extension of a generalized Skolem–Mahler– Lech theorem. Let $A$ be a finitely generated commutative $K$–algebra over a field of characteristic 0, and let $\sigma$ be a $K$–algebra automorphism of $A$. Given ideals $I$ and $J$ of $A$, we show that the set $S$ of integers $m$ such that ${{\sigma }^{m}}(I)\,\supseteq \,J$ is a finite union of complete doubly infinite arithmetic progressions in $m$, up to the addition of a finite set. Alternatively, this result states that for an affine scheme $X$ of finite type over $K$, an automorphism $\sigma \,\in \,\text{Au}{{\text{t}}_{k}}(X)$, and $Y$ and $Z$ any two closed subschemes of $X$, the set of integers $m$ with ${{\sigma }^{m}}(Z)\,\subseteq \,Y$ is as above. We present examples showing that this result may fail to hold if the affine scheme $X$ is not of finite type, or if $X$ is of finite type but the field $K$ has positive characteristic.

Keywords

11D4514R1011Y5511D88automorphismsendomorphismsaffine spacecommutative algebrasSkolem–Mahler–Lech theorem
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 67 , Issue 2 , April 2015 , pp. 286 - 314
Copyright
Copyright © Canadian Mathematical Society 2015

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