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

Abstract

AbstractIn 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 σ 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 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 σ∊ 2 AutK(X), and Y and Z any two closed subschemes of X, the set of integers m with 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.