Cyclotomic Completions of Polynomial Rings

The main object of study in this paper is the completion Z[q] = lim ←−n Z[q]/((1−q)(1−q) · · · (1−q)) of the polynomial ring Z[q], which arises from the study of a new invariant of integral homology 3spheres with values in Z[q] announced by the author, which unifies all the sl2 Witten-Reshetikhin-Turaev invariants at various roots of unity. We show that any element of Z[q] is uniquely determined by its power series expansion in q − ζ for each root ζ of unity. We also show that any element of Z[q] is uniquely determined by its values at the roots of unity. These results may be interpreted that Z[q] behaves like a ring of “holomorphic functions defined on the set of the roots of unity”. We will also study the generalizations of Z[q], which are completions of the polynomial ring R[q] over a commutative ring R with unit with respect to the linear topologies defined by the principal ideals generated by products of powers of cyclotomic polynomials.