# 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 − *qn*)) 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