Abstract
AbstractIn this paper we prove a commutative algebraic extension of a generalized Skolem–Mahler– Lech theorem. LetAbe a finitely generated commutativeK–algebra over a field of characteristic 0, and letσbe aK–algebra automorphism ofA. Given idealsIandJofA, we show that the setSof integersmsuch thatJis a finite union of complete doubly infinite arithmetic progressions inm, up to the addition of a finite set. Alternatively, this result states that for an affine schemeXof finite type overK, an automorphismσ∊ 2 AutK(X), andYandZany two closed subschemes ofX, the set of integersmwithYis as above. We present examples showing that this result may fail to hold if the affine schemeXis not of finite type, or ifXis of finite type but the fieldKhas positive characteristic.
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