Cyclotomic Completions of Polynomial Rings

Abstract

For a subset S⊂\mathbb{N}={1, 2, ...} and a commutative ring R with unit, let R[q]^S denote the completion \underleftarrow{lim}_f(q)R[q]/(f(q)), where f(q) runs over all the products of the powers of cyclotomic polynomials Φ_n(q) with n∈S. We will show that under certain conditions the completion R[q]^S can be regarded as a “ring of analytic functions” defined on the set of roots of unity of order in S. This means that an element of R[q]^S vanishes if it vanishes on a certain type of infinite set of roots of unity, or if its power series expansion at one root of unity vanishes. In particular, the completion \mathbb{Z}[q]^\mathbb{N}{=}\underleftarrow{lim}_n\mathbb{Z}[q]/((1−q)(1−q²)…(1−qⁿ)) enjoys this property.