A Skolem–Mahler–Lech Theorem for Iterated Automorphisms of<i>K</i>–algebras
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Abstract
Abstract 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 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 Aut K ( 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.
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