Cyclotomic Completions of Polynomial Rings

Abstract

For a subset $S\subset N=\{1,2,...\}$ 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 ${\Phi }_n(q)$ with $n\in S$. 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 $Z[q{]}^N\simeq \underleftarrow{\lim }{}_{n}Z[q]/((1-q)(1-q^2)\cdots (1-q^n))$ enjoys this property.