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

作者：Bell Jason P^*; Lagarias Jeffrey C

来源：Canadian Journal of Mathematics, 2015, 67(2): 286-314.

DOI：10.4153/CJM-2013-048-3

• automorphisms
• endomorphisms
• affine space
• commutative algebras
• Skolem-Mahler-Lech theorem

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 land J of A, we show that the set S of integers m such that sigma(m) (I) superset of 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 is an element of Aut(K)(X), and Y and Z any two closed subschemes of X, the set of integers m with sigma(m) (Z) subset of Y is as above. We present examples showing that this result may fail to hold if the affine scheme Xis not of finite type, or if X is of finite type but the field K has positive characteristic.

• 出版日期2015-4