# Cyclotomic Completions of Polynomial Rings

### Kazuo Habiro

Kyoto University, Japan

## Abstract

For a subset $S⊂N={1,2,...}$ and a commutative ring $R$ with unit, let $R[q]_{S}$ denote the completion $lim _{f(q)}R[q]/(f(q))$, 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 $Z[q]_{N}≃lim _{n}Z[q]/((1−q)(1−q_{2})⋯(1−q_{n}))$ enjoys this property.

## Cite this article

Kazuo Habiro, Cyclotomic Completions of Polynomial Rings. Publ. Res. Inst. Math. Sci. 40 (2004), no. 4, pp. 1127–1146

DOI 10.2977/PRIMS/1145475444