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

Bell, Jason P. ; Lagarias, Jeffrey C.

Canadian journal of mathematics, Tome 67 (2015) no. 2, pp. 286-314

Résumé

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.

Détail

DOI : 10.4153/CJM-2013-048-3

Mots-clés : 11D45, 14R10, 11Y55, 11D88, automorphisms, endomorphisms, affine space, commutative algebras, Skolem–Mahler–Lech theorem